brauer groups, tamagawa measures, and rational points

BRAUER GROUPS,
TAMAGAWA MEASURES,
AND RATIONAL POINTS
ON ALGEBRAIC VARIETIES
Jörg Jahnel

Jörg Jahnel
Mathematisches Institut, Bunsenstraße 3–5, D–37073 Göttingen, Germany.
E-mail : jahnel@uni-math.gwdg.de
Url : http://www.uni-math.gwdg.de/jahnel
2000 Mathematics Subject Classification. —
11G35, 14F22, 16K50, 11-04, 14G25, 11G50.
Acknowledgements. I wish to acknowledge with gratitude my debt to Y. Tschinkel. Most of the work described in this
book was initiated by his numerous mathematical questions. During the years he spent at Göttingen, he always shared
his ideas in an extraordinarily generous manner.
I further wish to express my deep gratitude to my friend and collegue Stephan Elsenhans. He influced this book in
many ways, directly and indirectly. It is no exaggeration to say that most of what I know about computer programming,
I learned from him. The experiments which are described in Part C were carried out as a joint work. I am also indebted
to Stephan for proofreading.
Special thanks go to my parents, Regina and Alfred Jahnel, for their unflaggingbelief that, despite their incomprehension
about what I do, I must be doing that very well. Finally, I would like to thank all the people who constantly supported
and encouraged me over the years. I risk doing someone a disservice by not naming all of them here, but plead
paucity of space. Let me mention particularly Ulrich Stuhler, H.-S. Holdgrün, Thomas Fröhlich, Stefanie Schmidt,
and Carsten Thiel.
The computerpart of the workdescribed in this bookwas executed on theSun Fire V20z Serversof the GaußLaboratory
for Scientific Computing at the Göttingen Mathematical Institute. The author is grateful to Y. Tschinkel for the
permission to use these machines as well as to the system administrators for their support.
J. J.
Göttingen, Lower Saxony,
September 24, 2008

BRAUER GROUPS, TAMAGAWA MEASURES,
AND RATIONAL POINTS
ON ALGEBRAIC VARIETIES
Abstract. — We study existence and asymptotics of rational points on algebraic
varieties of Fano and intermediate type. In the first part, we study the various
versions of the Brauer group. We explain why a Brauer class may serve as an
obstruction to weak approximation or even to the Hasse principle. In the second
part, we discuss to some extent the concept of a height and formulate Manin’s
conjecture on the asymptotics of rational points on Fano varieties. The final
part describes numerical experiments.
Habilitationsschrift
vorgelegt von
Jörg Jahnel
aus Eisenberg
Göttingen, Sommer 2008

CONTENTS
Table of contents . . ....................................................
iii
Notation and fundamental conventions . . .............................. vii
Notation and conventions . . ............................................ vii
References and Citation . . .............................................. ix
Introduction . . ..........................................................
xi
Part A. The Brauer group . . ............................................ 1
I. The Brauer-Severi variety associated with a central simple algebra . . 3
1. Introductory remarks . . .............................................. 3
2. Non-abelian group cohomology . . ................. . . ............... 5
3. Galois descent . . . ......................... .......................... 8
4. Central simple algebras and non-abelian H 1 . . . . . . . . . . . . . . . . . . . . . . . . 22
5. Brauer-Severi varieties and non-abelian H 1 . . ........................ 28
6. Central simple algebras and Brauer-Severi varieties . . . ............... 35
7. Functoriality . . ........................... ........................... 39
8. The functor of points . . .............................................. 53
II. On the Brauer group of a scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
1. Azumaya algebras . . ........................ ........................ 61
2. The Brauer group . . .................................................. 65
3. The cohomological Brauer group . . .................................. 66
4. The relation to the Brauer group of the function field . . . . . . . . . . . . . . 71
5. The Brauer group and the cohomological Brauer group . . . . . . . . . . . . 74
6. Orders in central simple algebras . . ................. ................. 76
7. The Theorem of Auslander and Goldman . . . . . . . . . . . . . . . . . ......... 84

iv
CONTENTS
8. Examples . . .......................................................... 86
III. An application: The Brauer-Manin obstruction . . .................. 99
1. Adelic points . . ...................................................... 99
2. The Brauer-Manin obstruction . . .................................... 103
3. Technical lemmata . . ........................ ........................ 106
4. Computing the Brauer-Manin obstruction – The general strategy . . 110
5. The examples of Mordell . . .......................................... 113
6. The “first case” of diagonal cubic surfaces . . .......................... 128
Part B. Heights . . ...................................................... 147
IV. The concept of a height . . ............................................ 149
1. The naive height on the projective space overÉ. . . . . . . . . . . . . . . . . . . . 149
2. Generalization to number fields . . . ................ ................. 151
3. Geometric interpretation . . .......................................... 155
4. Basic arithmetic intersection theory . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 160
5. An intersection product on singular arithmetic varieties . . . . . . . . . . . . 165
6. The arithmetic Hilbert-Samuel formula . . ............................ 169
7. The adelic Picard group . . ............................................ 173
V. On the distribution of small points on abelian and toric varieties . . 189
1. Introduction . . ...................................................... 189
2. The dynamics of f . . ................................................ 192
3. Perturbing almost semiample metrics . . .............................. 199
4. Equidistribution . . ......................... ......................... 204
VI. Conjectures on the asymptotics of points of bounded height . . .... 211
1. A heuristic . . ........................................................ 211
2. The conjecture of Lang . . ............................................ 214
3. The conjecture of Batyrev and Manin . . .............................. 216
4. The conjecture of Manin . . .......................................... 220
5. Peyre’s constant . . .................................................... 224
Part C. Numerical experiments . . ...................................... 241
VII. Points of bounded height on cubic and quartic threefolds . . ...... 243
1. Introduction — Manin’s Conjecture . . ................................ 243
2. Computing the Tamagawa number . . ................................ 247

CONTENTS
v
3. On the geometry of diagonal cubic threefolds . . . . . . . . . . . . . . . . . . . . . . 252
4. Accumulating subvarieties . . ........................................ 254
5. Results . . ............................................................ 259
VIII. On the smallest point on a diagonal quartic threefold . . . . . . . . . . 265
1. A computer experiment . . .......................................... 265
2. A negative result . . .................................................. 268
3. The fundamental finiteness property . . .............................. 270
IX. General cubic surfaces . . ............................................ 281
1. Rational points on cubic surfaces . . .................................. 281
2. Background . . ........................................................ 284
3. The Galois operation on the 27 lines . . .............................. 287
4. Computation of Peyre’s constant . . .................................. 290
5. Numerical Data . . .................................................... 292
6. A concrete example . . ................................................ 297
X. On the smallest point on a diagonal cubic surface . . ................ 301
1. Introduction . . ...................................................... 301
2. The factors α and β . . ....................... . ...................... 305
3. Splitting the Picard group . . .......................................... 306
4. A technical lemma . . ................................................ 310
5. A negative result . . .................................................. 311
6. The fundamental finiteness property . . .............................. 316
XI. The Diophantine Equation x 4 + 2y 4 = z 4 + 4w 4 . . . . . . . . . . . . . . . . . . 329
1. Introduction . . ...................................................... 329
2. Congruences . . ...................................................... 330
3. Naive methods . . .................................................... 332
4. An algorithm to efficiently search for solutions . . .................... 332
5. General formulation of the method . . ................................ 335
6. Improvements I – More congruences . . .............................. 336
7. Improvements II – Adaption to our hardware . . . . . . . . . . . . . . . . . . . . . . 341
8. The solution found . . ................................................ 348
XII. New sums of three cubes . . ........................................ 349
1. Introduction . . ...................................................... 349
2. Elkies’ method . . .......................... .......................... 350
3. Implementation . . .................................................... 350

vi
CONTENTS
4. Results . . ............................................................ 351
Appendix . . .............................................................. 353
1. A script in GAP . . .................................................... 353
2. The list . . ............................................................ 356
Bibliography . . .......................................................... 363
Index . . ................................. ................................. 383

NOTATION AND FUNDAMENTAL
CONVENTIONS
Notation and conventions
We follow standard notation and conventions from Algebra, Algebraic Number
Theory, and Algebraic Geometry. More precisely:
i) We writeÆ,,É,Ê, andfor the sets of natural numbers, integers, rational
numbers, real numbers, and complex numbers, respectively.
ii) For a group G and elements σ 1 , . . . , σ n ∈ G, we denote the subgroup
generated by σ 1 , . . . , σ n by 〈σ 1 , . . . , σ n 〉 ⊆ G.
If G is abelian then G n ⊆ G is the subgroup consisting of all elements of torsion
dividing n.
iii) If a group G operates on a set M then M G denotes the invariants. We write
M σ instead of M 〈σ〉 .
iv) All rings are assumed to be associative.
v) If R is a ring then R op denotes the opposite ring. I.e., the ring that coincides
with R as an abelian group but in which one has xy = z when one had yx = z
in R.
vi) For R a ring with unit, R ∗ denotes the multiplicative group of invertible
elements in R.
vii) All homomorphisms between rings with unit are supposed to respect the
unit elements.
viii) By a field, we always mean a commutative field. I.e., a commutative ring
with unit every non-zero element of which is invertible. If K is a field then K sep
and K denote a fixed separable closure and a fixed algebraic closure, respectively.
ix) A ring with unit every non-zero element of which is invertible is called a
skew field.
x) If R is a commutative ring with unit then an R-algebra is always understood
to be a ring homomorphism j : R → A the image of which is contained in the
center of A. An R-algebra j : R → A is denoted simply by A when there seems

viii
NOTATION AND FUNDAMENTAL CONVENTIONS
to be no danger of confusion. An R-algebra being a skew field is also called a
division algebra.
xi) If σ : R → R is an automorphism of R then A σ denotes the R-algebra
R −→ σ
R −→ j
A. If M is an R-module then we put M σ := M ⊗ R R σ . M σ is an
R σ -module as well as an R-module.
xii) All central simple algebras are assumed to be finite dimensional over a
base field.
xiii) For K a number field, we write O K to denote the ring of integers in K.
If ν ∈ Val(K ) is a non-archimedean valuation ν ∈ Val(K ) then the ν-adic
completion of K is denoted by K ν and its ring of integers by O Kν .
In the particular case that K =É, we denote by ν p the normalized p-adic
valuation corresponding to a prime number p.
xiv) For R a commutative ring, we denote by SpecR the affine scheme constituted
by its spectrum.
xv) The projective space of relative dimension n over a scheme X will be denoted
by P n X . We omit the subscript when there is no danger of confusion.
xvi) If X is a scheme over a scheme T and Y is a T -scheme then we also write
X Y for the fiber product X × T Y. If Y = SpecR is affine then write X R instead
of X SpecR .
xvii) For X a scheme over a scheme T , we denote by X t the fiber of X
over t ∈ T .
If C is a scheme over the integer ring O Kν of the completion of the number
field K with respect to the valuation ν then we write C ν for the special fiber.
In the particular case ν = ν p , we write C p instead of C νp .
If C is a scheme over the integer ring O of a number field K then we use same
the notation, not mentioning the base change to O Kν .
xviii) For R any commutative ring, A a commutative R-algebra, and X an
R-scheme, a morphism x : SpecA → X of R-schemes is also called an A-valued
point on X. If A is a field then we also adopt more conventional language and
speak of a point defined or rational over A. The set of all A-valued points on X
will be denoted by X (A).
xix) If C is a scheme over a valuation ring O and x ∈ C (O) then the reduction
of x is denoted by x.

NOTATION AND FUNDAMENTAL CONVENTIONS
ix
References and Citation
When we refer to a definition, proposition, theorem, etc., in the same chapter
then we simply rely on the corresponding numbering within the chapter.
Otherwise, we add the number of the chapter.
For the purpose of citation, the articles and books being used are encoded in
the manner specified by the bibliography. In addition, we mostly give the
number of the relevant section and subsection or the number of the definition,
proposition, theorem, etc. Normally, we do not mention page numbers.

INTRODUCTION
Here, in the midst of this sad and barren landscape of the Greek accomplishments
in arithmetic, suddenly springs up a man with youthful energy:
Diophantus. Where does he come from, where does he go to? Who were his
predecessors, who his successors? We do not know. It is all one big riddle.
He lived in Alexandria. If a conjecture were permitted, I would say he was
not Greek; . . . if his writings were not in Greek, no-one would ever think
that they were an outgrowth of Greek culture . . . .
HERMANN HANKEL (1874, translated by N. Schappacher)
Diophantine equations have a long history. More than two thousand years ago,
Diophantus of Alexandria considered, among many others, the equations
x 2 + y 2 = z 2 ,
(∗)
y(6 − y) = x 3 − x ,
and
y 2 = x 2 + x 4 + x 8 .
In Diophantus’ book “Arithmetica”, we find the formula
(
(p 2 − q 2 )λ, 2pqλ, (p 2 + q 2 )λ ) (†)
which generates infinitely many solutions of (∗). For the second and
third of the equations mentioned, Diophantus gives particular solutions,
namely (1/36, 1/216) and (1/2, 9/16), respectively.
In general, a polynomial equation in several indeterminates where solutions
are sought in integers or rational numbers, is called a Diophantine equation,
in honour of Diophantus. Diophantus himself was interested in solutions in
positive integers or positive rational numbers. Contrary to the point of view
usually adopted today, he did not accept negative numbers.
It is remarkable that algebro-geometric methods have often been fruitful in order
to understand a Diophantine equation. For example, there is a simple geometric
idea behind formula (†).

xii
INTRODUCTION
Indeed, since the equation is homogeneous, it suffices to look for solutions
of X 2 + Y 2 = 1 in rationals. This equation defines the unit circle. For every
t ∈Ê, there is the line “x = −ty + 1” going through the point (1, 0).
An easy calculation shows that the second point where this line meets the
unit circle is given by ( 1 − t
2
2t
)
1 + t 2, . (‡)
1 + t 2
As every point on the unit circle may be connected with (1, 0) by a line,
one sees that the parametrization (‡) yields every rational point on the circle
(except for (−1, 0) for which the form of line equation given is not adequate).
Consequently, formula (†) delivers essentially every solution of equation (∗),
a fact which was seemingly not known to the ancient mathematicians. The morphism
P 1 −→ P 2
(p : q) ↦→ ( (p 2 − q 2 ) : 2pq : (p 2 + q 2 ) )
provides a rational parametrization of the plane conic C given by the equation
x 2 + y 2 = z 2 in P 2 . Essentially the same method works for every conic in
the plane. Further, it may be extended to several classes of singular curves of
higher degree.
Every Diophantine equation defines an algebraic variety X in an affine or projective
space. There is a one-to-one correspondence between solutions of the
Diophantine equation andÉ-rational points on X. We will prefer geometric
language to number theoretic throughout this book.
The cases when there is an obvious rational parametrization are in some sense
best possible. But even when there is nothing like that, algebraic geometry often
yields a guideline of which behaviour to expect, whether there will be no, a few
or many solutions.
The Kodaira classification distinguishes between Fano varieties, varieties of intermediate
type, and varieties of general type (at least under the additional
assumption that X is non-singular). It does not use any specifically arithmetic
information, but only information about X as a complex variety. Nevertheless,
there is overwhelming evidence for a strong connection between the
classification of X according to Kodaira and its set of rational points.
To make a vague statement, on a Fano variety, there are infinitely many rational
points expected while, on a variety of general type, there are only finitely many
rational points or even none at all. More precisely, there is the conjecture
that, on a Fano variety, there are always infinitely many rational points after a
suitable finite extension of the ground field. On the other hand, for varieties of
general type, there is the conjecture of Lang. It states that there are only finitely

INTRODUCTION
xiii
many rational points outside the union of all closed subvarieties which are not
of general type.
Another method to analyze a Diophantine equation is given by congruences.
Kurt Hensel provided a more formal framework for this method by his invention
of the p-adic numbers. As one is working over local fields, this might be
called the local method.
Consider, for example, the Diophantine equation
x 3 + 7y 3 + 49z 3 + 2u 3 + 14v 3 + 98w 3 = 0 . (§)
It has no solution inÉ6 except for (0, 0, 0, 0, 0, 0). This may be seen by a repeated
application of an argument modulo 7. A more formal reason is the fact that the
projective algebraic variety defined by (§) has no point defined overÉ7.
One might ask whether or to what extent solvability overÉp for every prime
number p together with solvability in real numbers implies the existence of
a rational solution. This question has been very inspiring for research over
many decades. As early as 1785, A.-M. Legendre gave an affirmative answer for
equations of the type
q(x, y, z) = 0
where q is a ternary quadratic form. Legendre’s result was generalized
to quadratic forms in arbitrarily many variables by Hasse and Minkowski.
The term “Hasse principle” was coined to describe the phenomenon.
A totally different sort of examples where the Hasse principle is valid is provided
by the circle method originally developed by G. H. Hardy and J. E. Littlewood.
The circle method uses tools from complex analysis to study the asymptotics
of the number of points of bounded height on complete intersections in a very
high-dimensional projective space. It provides an asymptotic formula and an
error term. The main term is of the form
τB n+1−d 1− ... −d r
for a complete intersection of multidegree (d 1 , . . . , d r ) in P n . The reader might
want to consult [Va] for a description of the method and references to the
original literature.
The exponent of the main term allows a beautiful algebro-geometric interpretation.
The anticanonical sheaf on a complete intersection of multidegree
(d 1 , . . . , d r ) in P n is precisely O(n + 1 − d 1 − . . . − d r )| X . This means,
when working with an anticanonical height instead of the naive height, the
circle method proves linear growth for theÉ-rational points.
The coefficient τ of the main term is a product of p-adic densities together with
a factor corresponding to the archimedean valuation.

xiv
INTRODUCTION
Unfortunately, it is necessary to make very restrictive assumptions on the number
of variables in comparison with the degrees of the equations. These assumptions
on the dimension of the ambient projective space are needed in order to
ensure that the provable error term is smaller than the main term. On might,
nevertheless, hope that there is a similar asymptotic under much less restrictive
conditions. This is the origin of Manin’s conjecture.
However, aswasobservedbyJ. Franke, Yu. I. Manin, andY. Tschinkel[F/M/T],
the main term as described above is not compatible with the formation of
direct products. Already on a variety as simple as P 1 × P 1 , the growth of the
number of theÉ-rational points is actually asymptotically equal to τB log B.
This may be seen by a calculation which is completely elementary.
Thus, in general, the asymptotic formula has to be modified by a log-factor.
Franke, Manin, and Tschinkel suggest the factor log rkPic(X )−1 B and prove that
this factor makes the asymptotic formula compatible with direct products.
Furthermore, it turns out that the coefficient τ has to be modified
when rkPic(X ) > 1. There appears an additional factor which is today
called α(X ). This factor is defined by a beautiful yet somewhat mysterious
elementary geometric construction.
Another problem is that the Hasse principle does not hold universally. Consider
the following elementary example which was given by C.-E. Lind in 1940.
Lind [Lin] dealt with the Diophantine equation
2u 2 = v 4 − 17w 4
defining an algebraic curve of genus 2. It is obvious that this equation is nontrivially
solvable in reals and it is easy to check that it is non-trivially solvable
inÉp for every prime number p.
On the other hand, there is no solution in rationals except for (0, 0, 0).
Indeed, assume the contrary. Then, there is a solution in integers such
that gcd(u, v, w) = 1. For such a solution, one clearly has ∤u. Since 2 is a
square but not a fourth power modulo 17, we conclude that ( u
17)
= −1. On the
other hand, for every odd prime divisor p of u, one has v 4 −17w 4 ≡ 0 (mod p).
This shows ( ) (
17
p = 1. By the low of quadratic reciprocity, p
17)
= 1. Altogether,
( u
17)
= 1 which is a contradiction.
One might argue that this example is not too interesting since, on a curve of
genus g ≥ 2, there can be only finitely manyÉ-rational points. Thus, it might
happen that there are none of them without any particular reason.
However, several other counterexamples to the Hasse principle had been invented.
Some of them were Fano varieties. For example, Sir P. Swinnerton-
Dyer[S-D62] and L.-J. Mordell [Mord] (cf. Chapter III, Section 5) constructed

INTRODUCTION
xv
examples of cubic surfaces violating the Hasse principle. A few years later,
J. W. S. Cassels and M. J. T. Guy [Ca/G] as well as A. Bremner [Bre] even found
isolated examples of diagonal cubic surfaces showing that behaviour. Typically,
the proofs were a bit less elementary than Lind’s in that sense that they
required not the quadratic but the cubic or biquadratic reciprocity low.
In the late sixties of the 20th century, Yu. I. Manin [Man] made the remarkable
discovery that all the known counterexamples to the Hasse principle could
be explained in a uniform manner. There was actually a class α ∈ Br(X ) in
the Brauer group of the underlying algebraic variety responsible for the lack
ofÉ-rational points.
This may be explained as follows. The Brauer group ofÉis relatively complicated.
One has, by virtue of global class field theory,
Br(SpecÉ) = ker ( M
s :
2/−→É/) .
pprimeÉ/⊕ M
σ : K→Ê1
Here, s is just the summation. The summandÉ/corresponding to the prime
number p is nothing but Br(SpecÉp) while the last summand is Br(SpecÊ).
Let α ∈ Br(X ) be any Brauer class of a variety X overÉ. An adelic point
x = (x ν ) ν∈Val(É) ∈ X (É)
defines restrictions of α to Br(SpecÊ) and Br(SpecÉp) for each p. If the
sum of all invariants is different from zero then, according to the computation
of Br(SpecÉ), x may not be approximated byÉ-rational points.
As α ∈ Br(X ) then “obstructs” x from being approximated by rational points,
the expression Brauer-Manin obstruction became the general standard for this
famous observation of Manin.
In the counterexamples to the Hasse principle which were known to Manin in
those days, one typically had a Brauer class the restrictions of which had a totally
degenerate behaviour. For example, on Lind’s curve, there is a Brauer class α
such that its restriction is independent of the choice of the adelic point. α restricts
to zero in Br(SpecÊ) and Br(SpecÉp) for p ≠ 17 but non-trivially
to Br(SpecÉ17). This suffices to show that there is noÉ-rational point on
that curve.
In general, the Brauer-Manin obstruction defines a subset X (É) Br ⊆ X (É)
consisting of the adelic points which are not affected by the obstruction. At least
for cubic surfaces, there is a conjecture of J.-L. Colliot-Thélène stating that
X (É) Br is equal to the set of all adelic points which may actually be approximated
byÉ-rational points.
Thus,X(É) Br = ∅whileX(É) ≠ ∅meansthatX isaprovencounterexample
to the Hasse principle. If X (É) Br X (É) then we have a counterexample to

xvi
INTRODUCTION
weak approximation. If Colliot-Thélène’s conjecture were true then one could
say that all cubic surfaces which are counterexamples to the Hasse principle or
to weak approximation are of this form.
The Brauer group of an algebraic variety X over an algebraically nonclosed
field k admits, according to the Hochschild-Serre spectral sequence,
a canonical filtration in three steps. The first step is given by the image
of Br(Speck) in Br(X ). Second, Br(X )/ Br(Speck), has a subgroup canonically
isomorphic to H 1( Gal(k/k), Pic(X k
) ) . The remaining subquotient is a subgroup
of Br(X k
) Gal(k/k) . It turns out that only the group H 1( Gal(k/k), Pic(X k
) )
is relevant for the Brauer-Manin obstruction.
In the cases where the circle method is applicable, the Noether-Lefschetz Theorem
shows that Pic(X ) =with trivial Galois operation. Consequently,
H 1( Gal(k/k), Pic(X ) ) = 0 which is clearly sufficient for the absence
of the Brauer-Manin obstruction. This coincides perfectly well with the observation
that the circle method always proves equidistribution.
By consequence, in a conjectural generalization of the results proven by the
circle method, one can work with X (É) Br instead of X (É) without making
any change in the proven cases. However, in the cases where weak
approximation fails, this does not give the correct answer as was observed
by D. R. Heath-Brown [H-B92a], in 1992. On a cubic surface such that
H 1( Gal(k/k), Pic(X ) ) =/3and a non-trivial Brauer class excludes two
thirds of the adelic points, there are nevertheless as many rational points as
naively expected. Even more, E. Peyre and Y. Tschinkel [Pe/T] showed experimentally
that if H 1( Gal(k/k), Pic(X ) ) =/3and the Brauer class does not
exclude any adelic point then there are three times more rational points than expected.
Correspondingly, in E. Peyre’s [Pe95a] definition of the conjectural
constant τ, there appears an additional factor β(X ) := #H 1( Gal(k/k), Pic(X k
) ) .
This book is concerned with Diophantine equations from the theoretical and
experimental points of view. It is divided into three parts. Part A deals with the
concepts of a Brauer group and their applications. In the first chapter, we begin
with the simplest particular case and recall the Brauer group of a field. As an
abstract group, we have Br(k) = H 2( Gal(k/k), k ∗) . Taking this as a definition,
it is not hard to show that Br(X ) classifies two a priori rather different sorts
of objects. Namely, on one hand, central simple algebras over k and, on the
other, Brauer-Severi varieties over k.
WerecallindetailthemachineryofGaloisdescent. Asanapplication, weexplain
why both central simple algebras of dimension n 2 splitting over an extension
field L as well as Brauer-Severi varieties of dimension n splitting over L are
classified by H 1( Gal(L/K ), PGL n (L) ) . In Section I.7, there is given a functor

INTRODUCTION
xvii
from central simple algebras to Brauer-Severi varieties which yields the classical
correspondence on objects.
The second chapter considers A. Grothendieck’s generalization of the
Brauer group to the case of an arbitrary scheme. We recall the concept of a
sheaf of Azumaya algebras on a scheme and explain how such a sheaf of algebras
gives rise to a class in the étale cohomology Br ′ (X ) := H 2 ét (X,m). This is what
is called the cohomological Brauer group. On the other hand, a rather naive
generalization of the definition for fields yields the concept of the Brauer group.
One has Br(X ) ⊆ Br ′ (X ). In general, the two are not equal to each other.
In Section II.7, we give a proof for the Theorem of Auslander and Goldman
stating that Br(X ) = Br ′ (X ) in the case of a smooth surface. This result was
originally shown in [A/G] before the actual invention of schemes. The proof
of Auslander and Goldman was formulated in the language of Brauer groups for
commutative rings. However, all the arguments given carry over immediately
to the case of a scheme.
The chapter is closed by computations of Brauer groups in particular examples.
In the case of a variety over an algebraically non-closed field, we study the
relationship of Br(X ) with H 1( Gal(k/k), Pic(X k
) ) . We prove Manin’s formula
expressing the latter cohomology group in terms of the Galois operation on
a specific set of divisors. For smooth cubic surfaces, one may work with the
classes given by the 27 lines.
This leads to the result of Sir P. Swinnerton-Dyer [S-D93] that, for a smooth
cubic surface, H 1( Gal(k/k), Pic(X k
) ) is one of the groups 0,/2,/3,
(/2) 2 , and (/3) 2 . Swinnerton-Dyer’s proof filled the entire article [S-D93]
and was later modified by P. K. Corn in his thesis [Cor].
We discovered that Swinnerton-Dyer’s result may be obtained in a manner which
is rather brute force but very simple. The Galois group acting on the 27 lines on
a smooth cubic surface is a subgroup of W (E 6 ). There are only 350 conjugacy
classes of subgroups of W (E 6 ). We computed H 1( Gal(k/k), Pic(X k
) ) in each of
these cases using GAP. This took 28 seconds of CPU time.
As an application of Brauer groups, the third chapter is concerned with the
Brauer-Manin obstruction. We recall the notion of an adelic point and define the
local and global evaluation maps. An adelic point x = (x ν ) ν∈Val(É) is “obstructed”
from being approximated by rational points if the global evaluation map ev gives
a non-zero value ev(α, x) for a certain Brauer class α ∈ Br(X ).
We then describe a strategy on how the Brauer-Manin obstruction may be
explicitly computed in concrete examples. We carry out this strategy for two
special types of cubic surfaces.
The first type is given as follows. Let p 0 ≡ 1 (mod 3) be a prime number
and K/Ébe the unique cubic field extension contained in the cyclotomic

xviii
INTRODUCTION
extensionÉ(ζ p0 )/É. Fix the explicit generator θ ∈ K given by
θ := trÉ(ζ p0 )/K(ζ p0 − 1) = −2n + ∑
i∈(∗ p0 ) 3 ζ i p 0
for n := p 0−1
. Then, consider the cubic surface X ⊂ P 3É, given by
6
3 ( )
T 3 (a 1 T 0 + d 1 T 3 )(a 2 T 0 + d 2 T 3 ) = T0 + θ (i) T 1 + (θ (i) ) 2 T 2 .
Here, a 1 , a 2 , d 1 , d 2 ∈. The θ (i) denote the three images of θ under Gal(K/É).
Proposition III.5.3 provides criteria to verify that such a surface is smooth
and has p-adic points for every prime p. More importantly, the Brauer-Manin
obstruction can be understood completely explicitly. At least for a generic
choice of a 1 , a 2 , d 1 , and d 2 , one has that H 1( Gal(k/k), Pic(X k
) ) =/3.
Further, there is a class α ∈ Br(X ) with the following property. For an adelic
point x = (x ν ) ν , the value of ev(α, x) depends only on the component x νp0 .
Write x νp0 =: (t 0 : t 1 : t 2 : t 3 ). Then, one has ev(α, x) = 0 if and only if
a 1 t 0 + d 1 t 3
t 3
is a cube in∗ p0
. Note that p 0 ≡ 1 (mod 3) implies that only every third element
of∗ p0
is a cube.
∏
i=1
Observe that the reduction of X modulo p 0 is given by
T 3 (a 1 T 0 + d 1 T 3 )(a 2 T 0 + d 2 T 3 ) = T 3
0 .
This means, there are three planes intersecting in a triple line. NoÉp 0
-rational
point may reduce to the triple line. Thus, there are three different planes, a
Ép 0
-rational point may reduce to. The value of ev(α, x) depends only the plane
to which its component v νp0 is mapped under reduction.
For instance (cf. Example III.5.24), for p 0 = 19, consider the cubic surface X
given by
3 ( )
T 3 (T 0 + T 3 )(12T 0 + T 3 ) = T0 + θ (i) T 1 + (θ (i) ) 2 T 2 .
Then, in19, the cubic equation
∏
i=1
T (1 + T )(12 + T ) − 1 = 0
has the three solutions 12, 15, and 17. However, in19, 13/12 = 9, 16/15 = 15,
and 18/17 = 10 which are three non-cubes. This shows that X (É) = ∅. It is
easy to check that X (É) ≠ ∅. Therefore, X is an example of a cubic surface
violating the Hasse principle.
We construct a number of similar examples. For instance, Example III.5.24
describes a cubic surface X such that H 1( Gal(k/k), Pic(X k
) ) =/3but the

INTRODUCTION
xix
generating Brauer class does not exclude a single adelic point. One would expect
that X satisfies weak approximation. Recall that, in similar examples, E. Peyre
and Y. Tschinkel [Pe/T] showed experimentally that there are three times more
É-rational points than expected.
The historically first cubic surface which could be proven to be a counterexample
to the Hasse principle was provided by Sir P. Swinnerton-Dyer [S-D62].
We recover Swinnerton-Dyer’s example (cf. Example III.5.27) for p 0 = 7,
d 1 = d 2 = 1, a 1 = 1, and a 2 = 2. L. J. Mordell [Mord] generalized Swinnerton-Dyer’s
work by giving a series of examples for p 0 = 7 and a series of
examples for p 0 = 13. Yu. I. Manin mentions Mordell’s examples explicitly in
his book [Man]. He explains these counterexamples to the Hasse principle by a
Brauer class. We generalize Mordell’s examples further to the case that p 0 is an
arbitrary prime such that p 0 ≡ 1 (mod 3).
We conclude Chapter III by a section on diagonal cubic surfaces. For these, the
Brauer-Manin obstruction was investigated in the monumental work [CT/K/S]
of J.-L. Colliot-Thélène, D. Kanevsky, and J.-J. Sansuc. We present an explicit
computation of the Brauer-Manin obstruction under a congruence condition
which corresponds more or less to the “first case” of [CT/K/S]. Our argumentation
is, however, shorter and simpler than the original one. The point is
that we make use of the fact that H 1( Gal(k/k), Pic(X k
) ) may be only 0,/2,
/3, (/2) 2 , or (/3) 2 . Further, the group (/3) 2 appears only once in
a very particular case. Thus, in order to prove H 1( Gal(k/k), Pic(X k
) ) =/3,
it is almost sufficient to construct an element of order three.
The second part of this book is devoted to the various concepts of a height.
In the fourth chapter, we start with the naive height forÉ-rational points on
projective space. Then, our goal is to give an overview of the theories which
provide natural generalizations of this simple concept.
The very first generalization is the naive height for points in projective space
defined over a finite extension ofÉ. Then, following André Weil, we introduce
the concept of a height defined by an ample invertible sheaf. This is a height
function which is defined only up to a bounded summand.
To overcome this difficulty, one has to work with arithmetic varieties and
metrized invertible sheaves. Arithmetic varieties are schemes projective
over Spec. Actually, this leads to a beautiful geometric interpretation of
the naive height.
Indeed, let X be a projective variety overÉand X a projective model of X
over Spec. Fix a hermitian line bundle L on X . Then, according to the valuative
criterion of properness, everyÉ-rational point x on X extends uniquely
to a-valued point x : Spec→X. The height function with respect to L is

xx
INTRODUCTION
then given by
h L
(x) := ̂deg ( x ∗ L ) .
Here, ̂deg denotes the Arakelov degree of a hermitian line bundle over Spec.
It turns out that this coincides exactly with the naive height when one works
with X = P n, L = O(1), and the minimum metric which is defined by
∣ ‖l‖ min := min
l ∣∣∣
i=0, ... ,n
∣ .
X i
In general, h L
admits a fundamental finiteness property as soon as L is ample.
This is the starting point of arithmetic intersection theory, a fascinating theory
which we may only touch upon. We recall the main definitions and results
of the arithmetic intersection theory due to H. Gillet and C. Soulé [G/S 90,
G/S 92, S/A/B/K]. As an application, we consider the concept of a height
introduced by J.-B. Bost, H. Gillet, and C. Soulé [B/G/S]. This is a height
function for cycles of arbitrary dimension. For a 0-dimensional cycle, the
Bost-Gillet-Soule height coincides with the usual height function defined by the
hermitian line bundle.
Wecloseour overviewofarithmeticintersectiontheoryby a sectiononthe arithmetic
Hilbert-Samuel formula. We formulate the formula as Theorem IV.6.5.
Asaconsequence, weproveS. Zhangs’s[Zh95b, (5.2)]theorem ofsuccessiveminima.
Arithmetic intersection theory seems to be a very general framework. It is,
however, still not general enough. An obvious problem is that the naive height
on P n is not covered. This problem is of technical nature. To define the intersection
product of two hermitian line bundles (or, more correctly, of the
corresponding arithmetic cycles), Gillet and Soulé had to assume that all hermitian
metrics are smooth. Unfortunately, the minimum metric is continuous,
but not smooth.
A theory which is general enough to cover the Bost-Gillet-Soule height as well
as the naive height is provided by S. Zhang’s adelic intersection theory [Zh95a].
We give a definition for a version of the adelic Picard group ˜Pic(X ). Elements of
˜Pic(X ) are called integrably metrized invertible sheaves. Then, we construct
the corresponding adelic intersection product. As an application, we explain
the height function defined by an integrably metrized invertible sheaf.
In Chapter 5, we use adelic intersection theory to give a unified proof for two
equidistribution theorems for points of small height. On one hand, there is
the equidistribution theorem of L. Szpiro, E. Ullmo and S. Zhang [S/U/Z]
for small points on abelian varieties. On the other hand, the completely parallel
theorem for the naive height on projective space which was established

INTRODUCTION
xxi
by Yuri Bilu [Bilu97]. The main result of Chapter 5 is formulated as Theorem
V.1.14. A. Chambert-Loir’s equidistribution result [C-L] for quasi-split
semi-abelian varieties can easily be deduced from this theorem.
Chapter VI is devoted to some of the most popular conjectures concerning
rational points on projective algebraic varieties. We discuss Lang’s conjecture,
the conjecture of Batyrev and Manin, and, most notably, Manin’s conjecture
about the asymptotics of points of bounded height on Fano varieties (Conjecture
VI.5.37). A large part of the chapter is concerned with E. Peyre’s Tamagawa
type number τ (X ), the coefficient expected in the asymptotic formula.
We discuss in detail all factors appearing in the definition of τ (X ).
In particular, we give a number of examples for which we explicitly compute
the factor α(X ). We mainly consider smooth cubic surfaces of arithmetic Picard
rank two.
Part C collects several reports on practical experiments. Each chapter is concerned
with a particular problem for a specific variety or family of varieties.
We tried to keep the chapters as self-contained as possible. Each one starts with
an introduction recalling those parts of the general theory described in parts A
and B which are necessary to follow that chapter.
In Chapter VII, we describe our investigations regarding the families
“ax 3 = by 3 +z 3 +v 3 +w 3 ”, a, b = 1, . . . , 100, and “ax 4 = by 4 +z 4 +v 4 +w 4 ”,
a, b = 1, . . . , 100, of projective algebraic threefolds. We report numerical evidence
for the conjecture of Manin in the refined form due to E. Peyre.
Our experiments included searching for points, computing the Tamagawa number,
and detecting the accumulating subvarieties. Concerning the programmer’s
efforts, detection of accumulating subvarieties was the most difficult part of
this project. For example, one the cubic threefolds, the following non-obvious
lines have been found. These are the only non-obvious lines we know and the
only ones containing a point of height less than 5000.
TABLE 1. Sporadic lines on cubic threefolds
a b Smallest point Point s.t. x = 0
19 18 (1 : 2 : 3 : -3 : -5) (0 : 7 : 1 : -7 : -18)
21 6 (1 : 2 : 3 : -3 : -3) (0 : 9 : 1 : -10 : -15)
22 5 (1 : -1 : 3 : 3 : -3) (0 : 27 : -4 : -60 : 49)
45 18 (1 : 1 : 3 : 3 : -3) (0 : 3 : -1 : 3 : -8)
73 17 (1 : 5 : -2 : 11 : -15) (0 : 27 : -40 : 85 : -96)
We describe all the computations which were done as well some background on
the geometry of cubic and quartic threefolds.

xxii
INTRODUCTION
In the eighth chapter, we continue our study of the family of quartic threefolds
by comparing the height m(X ) of the smallest rational point with Peyre’s constant.
We prove that there is no constant C such that m(X ) < C
τ
for every
(X )
diagonal quartic threefold, not even for those of type ax 4 = y 4 + z 4 + v 4 + w 4 .
1
We also prove that, at least for diagonal quartic threefolds, the reciprocal
τ (X )
1
behaves like a height function. I.e.,
τ
admits a fundamental finiteness property.
(X )
Chapter IX is devoted to the study of general cubic surfaces. A general cubic
surface is described by twenty coefficients. With current technology, it is impossible
to study all cubic surfaces with coefficients below a reasonable bound.
For that reason, we decided to work with coefficient vectors provided by a
random number generator. Our first sample consists of 20 000 surfaces with coefficients
randomly chosen in the interval [0, 50]. The second sample consists of
20 000 surfaces with randomly chosen coefficients from the interval [−100, 100].
We verified explicitly that each of the surfaces studied is smooth. Then, we
proved that, for each surface, the full Galois group W (E 6 ) acts on the 27 lines.
This implies that the Picard rank is always equal to 1. Also, the Brauer-Manin obstruction
is not present on any of the surfaces considered. The algorithm to
check that the group acting on the 27 lines is indeed W (E 6 ) is described in detail.
Further, we searched for rational points and computed the Tamagawa number,
thereby giving numerical evidence for Manin’s conjecture.
In the next chapter, we return to the more standard case of diagonal cubic surfaces.
The experiments areanalogous to those described in Chapters VIIand VIII
for diagonal cubic and quartic threefolds. The theory is, however, more complicated.
The geometric Picard rank is equal to 7 and, in the generic case,
there is a Brauer-Manin obstruction to weak approximation excluding precisely
two thirds of the adelic points. The factors α(X ) and β(X ) appearing in the
definition of Peyre’s constant are not always the same and need to be considered.
We demonstrate experimentally the connection of Peyre’s constant with the
height m(X ) of the smallest rational point. Under the Generalized Riemann
Hypothesis, we prove that there is no constant C such that m(X ) < C
τ for (X )
every diagonal cubic surface. We also prove that, for diagonal cubic surfaces, the
1
reciprocal
τ behaves like a height function. I.e., 1
(X ) τ
admits a fundamental
(X )
finiteness property.
Chapter XI is concerned with the Diophantine equation
x 4 + 2y 4 = z 4 + 4w 4 . ()
This equation gives an example of a K3 surface X defined overÉ. It is an open
question whether there exists a K3 surface overÉwhich has a finite non-zero
number ofÉ-rational points.

INTRODUCTION
xxiii
X might be a candidate for a K3 surface with this property. (1 : 0 : 1 : 0)
and (1 : 0 : (−1) : 0) are two obvious rational points. Sir P. Swinnerton-
Dyer [Poo/T, Problem/Question 6.c)] had publicly posed the problem to find a
third rational point on X. But no rational points different from the two obvious
ones had been found in experiments carried out by several people.
We explain our approach to efficiently search forÉ-rational points on algebraic
varieties defined by a decoupled equation. It is based on hashing, a method from
Computer Science. In the particular case of a surface in P 3É, our algorithm is of
complexity essentially O(B 2 ) for a search bound of B.
In the final implementation, we could work with the search bound B = 10 8 .
We discovered the following solution of the Diophantine equation ().
1 484 801 4 + 2 · 1 203 120 4 = 9 050 910 498 475 648 046899201
1 169 407 4 + 4 · 1 157 520 4 = 9 050 910 498 475 648 046899201
Up to changes of sign, this is the only non-obvious solution of () we know and
the only non-obvious solution of height less than 10 8 [EJ2, EJ3].
The last chapter is a report on our approach to the three cubes problem. To be
more precise, the problem is whether every rational integer n ≢ 4, 5 (mod 9)
can be written as a sum of three integral cubes.
Using an implementation of Elkies’ method, we found, among many others,
the following equations.
156 = 26 577 110 807 569 3 − 18 161 093 358 005 3 − 23 381 515 025 762 3
318 = 1 970 320 861 387 3 + 1 750 553 226 136 3 − 2 352 152 467 181 3
= 30 828 727 881 037 3 + 27 378 037 791 169 3 − 36 796 384 363 814 3
366 = 241 832 223 257 3 + 167 734 571 306 3 − 266 193 616 507 3
420 = 8 859 060 149 051 3 − 2 680 209 928 162 3 − 8 776 520 527 687 3
564 = 53 872 419 107 3 − 1 300 749 634 3 − 53 872 166 335 3
758 = 662 325 744 409 3 + 109 962 567 936 3 − 663 334 553 003 3
= 83 471 297 139 078 3 + 77 308 024 343 011 3 − 101 433 242 878 565 3
789 = 18 918 117 957 926 3 + 4 836 228 687 485 3 − 19 022 888 796 058 3
894 = 19 868 127 639 556 3 + 2 322 626 411 251 3 − 19 878 702 430 997 3
948 = 103 458 528 103 519 3 + 6 604 706 697 037 3 − 103 467 499 687 004 3
For none of the numbers on the left, a decomposition into a sum of three cubes
had been known before.
Göttingen,
Summer 2008

PART A
THE BRAUER GROUP

CHAPTER I
THE BRAUER-SEVERI VARIETY
ASSOCIATED WITH
A CENTRAL SIMPLE ALGEBRA ∗
No attention should be paid to the fact that algebra and geometry are
different in appearance.
OMAR KHAYYÁM (1070, translated by A. R. Amir-Moez)
1. Introductory remarks
This chapter is devoted to the correspondence between central simple algebras
and Brauer-Severi varieties.
This is classical material. Only Section 7 is exceptional to a certain extent.
There, we will give a functor from central simple algebras to Brauer-Severi
varieties which yields the classical correspondence on objects.
Some history. —
Central simple algebras were studied intensively by many mathematicians at the
end of the 19th and in the first half of the 20th century. We refer the reader to
N. Bourbaki [Bou-A, Note historique] for a detailed account on the history of
the subject and mention only a few important milestones here. The structure
of central simple algebras (being finite dimensional over a field K) is fairly easy.
They are full matrix rings over division algebras the center of which is equal
to K. This was finally discovered by J. H. Maclagan-Wedderburn in 1907
[MWe08] after several special cases had been treated before. T. Molien [Mol]
had considered the case of-algebras already in 1893 and the case ofÊ-algebras
had been investigated by E. Cartan [Car]. J. H. Maclagan-Wedderburn himself
had proven the structure theorem for central simple algebras over finite fields
in 1905 [MWe05, Di05].
(∗) This chapter is a revised version of the article: The Brauer-Severi variety associated with a
central simple algebra, Linear Algebraic Groups and Related Structures 52(2000), 1-60.

4 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
In 1929, R. Brauer ([Bra], see also [Deu] and [A/N/T]), using E. Noether’s
ideas about crossed products of algebras, found the group structure on the
set of similarity classes of central simple algebras over a field K. He constructed,
in today’s language, an isomorphism to the Galois cohomology
group H 2( Gal(K sep /K ), (K sep ) ∗) . Further, he discussed the structure of this
group in the case of a number field.
Relative versions of central simple algebras over base rings instead of fields were
introduced by G. Azumaya [Az] and M. Auslander and O. Goldman [A/G].
The case of an arbitrary base scheme was considered by A. Grothendieck in his
famous Bourbaki talks “Le groupe de Brauer” [GrBrI, GrBrII, GrBrIII].
Brauer-Severi varieties are twisted forms of the projective space. The term
“varieté de Brauer” appeared for the first time in 1944 in the article [Ch44]
of F. Châtelet. Nevertheless, F. Severi had proven already in 1932 that a Brauer-
Severi variety over a field K admitting a K-valued point is necessarily isomorphic
to the projective space.
It should be mentioned that Brauer-Severi varieties are important in a number
of applications. The first and definitely most striking one is the proof of the
theorem of Merkurjev-Suslin [Me/S] on the cotorsion of K 2 of a field.
The Merkurjev-Suslin theorem has an interesting further application. It provides
a certain understanding of the (torsion part of the) Chow groups of certain
algebraic varieties. The reader should consult the work of J.-L. Colliot-Thélène
[CT91] for information about that.
As a second kind of application, Brauer-Severi varieties often appear in constructions
of varieties with particular properties. For instance, the famous example
due to M. Artin and D. Mumford [A/M] of a threefold which is unirational
but not rational is a variety fibered over a rational surface such that the generic
fiber is a conic without rational points. For more historical details, especially
on the work of F. Châtelet, we refer the reader to the article of J.-L. Colliot-
Thélène [CT88].
The correspondence between central simple algebras and Brauer-Severi varieties
was first observed by E. Witt [Wi35] and H. Hasse in the special case of
quaternion algebras and plane conics.
Here, we are going to present in detail the most elementary approach to this correspondence.
It is based on non-abelian group cohomology. The main observation
is that central simple algebras of dimension n 2 over a field K as well
as (n − 1)-dimensional Brauer-Severi varieties over K can both be described
by classes in one and the same cohomology set H 1( Gal(K sep /K ), PGL n (K sep ) ) .
Note that this approach was promoted by J.-P. Serre in his books [Se62, chap. X,

Sec. 2] NON-ABELIAN GROUP COHOMOLOGY 5
§§5,6] and [Se73, Remarque III.1.3.1]. It was, however, known to F. Châtelet before.
A second approach is closer to A. Grothendieck’s style. One can give a direct
description of a functor of points P XA : {K-schemes} → {sets} in terms of data
of the central simple algebra A. It is possible to prove representability by a
projective scheme using brute force. This approach is presented explicitly in
the book of I. Kersten [Ke]. For a very detailed account, the reader may consult
the Ph.D. thesis of F. Henningsen [Hen]. We are going to prove that these two
approaches are equivalent. In fact, we will compute the functor of points of the
variety given by the first approach and show that it is naturally isomorphic to
the functor usually taken as the starting point for the second approach.
There is a third approach which we only mention here. It works via algebraic
groups and can be used to produce twisted forms not only of the projective
space but of any homogeneous space G/P where G is a semisimple algebraic
group and P ⊂ G a parabolic subgroup (see [K/R]).
2. Non-abelian group cohomology
In this section, we recall elementary facts about non-abelian group cohomology.
I.e., cohomology of discrete groups with non-abelian coefficients. Non-abelian
Galois cohomology will be the central tool for this chapter.
2.1. Definition. –––– Let G be a finite group.
i) A G-set E is a set equipped with a G-operation from the left.
A morphism of G-sets, a G-morphism for short, is a map i : E → F of G-sets
such that the diagram
G × E ·
E
commutes.
id×i
G × F
· F
ii) A G-set E carrying a group structure such that g (xy) = g x g y for every g ∈ G
and x, y ∈ E is called a G-group. Here, we write g x := g ·x for x ∈ E and g ∈ G.
If the group underlying E is abelian then E is called a G-module.
2.2. Definition. –––– Let G be a finite group.
i) For a G-set E, one puts
H 0 (G, E) := E G .
I.e., the zeroth cohomology set of G with coefficients in E is equal to the subset of
G-invariants.
i

6 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
ii) Let A be a G-group.
Then, a cocycle from G to A is a map G → A, g ↦→ a g such that a gh = a g · ga
h
for each g, h ∈ G.
Two cocycles a, a ′ are said to be cohomologous if there exists some b ∈ A such
that a ′ g = b −1 · a g · gb for every g ∈ G.
This is an equivalence relation and the quotient set, the first cohomology set of G
with coefficients in A, is denoted by H 1 (G, A). This is a pointed set as the map
g ↦→ e defines a cocycle, the so-called trivial cocycle.
2.3. Remarks. –––– i) If E is a G-group then H 0 (G, E) is a group.
ii) For a cocycle a, a ′ such that a ′ g := b−1·a g · gb for each g ∈ G is a cocycle, too.
iii) H 0 (G, A) and H 1 (G, A) are covariant functors in A. If i : A → A ′ is a
morphism of G-sets (a morphism of G-groups) then the induced map(s) will be
denoted by i ∗ : H 0 (G, A) → H 0 (G, A ′ ) (and i ∗ : H 1 (G, A) → H 1 (G, A ′ )).
iv) If A is abelian then the notions defined above coincide with the usual
group cohomology. One of the possible descriptions for H ∗ (G, A) is the cohomology
of the complex
with differentials
0 −→ A d
−→ Map(G, A)
d
−→ Map(G 2 , A)
dϕ(g 1 , . . . , g n+1 ) := g 1
ϕ(g 2 , . . . , g n+1 )
+
n
∑
j=1
d
−→ . . .
(−1) j ϕ(g 1 , . . . , g j g j+1 , . . . , g n+1 )
+ (−1) n+1 ϕ(g 1 , . . . , g n ) .
2.4. Proposition. –––– Let G be a finite group.
a) Let A ⊆ B be a G-subgroup and B/A the set of left cosets. Then, there is a natural
exact sequence of pointed sets
1 → H 0 (G, A) → H 0 (G, B) → H 0 (G, B/A) δ → H 1 (G, A) → H 1 (G, B) .
b) If A ⊆ B is a normal G-subgroup then there is a natural exact sequence of
pointed sets
1 → H 0 (G, A) → H 0 (G, B) → H 0 (G, B/A) δ → H 1 (G, A) → . . .
. . . → H 1 (G, B) → H 1 (G,B/A) .

Sec. 2] NON-ABELIAN GROUP COHOMOLOGY 7
c) If A ⊆ B is a G-module lying in the center of B then there is a natural exact
sequence of pointed sets
1 → H 0 (G, A) → H 0 (G, B) → H 0 (G, B/A) δ → H 1 (G, A) → . . .
. . . → H 1 (G, B) → H 1 (G, B/A) → δ′
H 2 (G, A) .
Here,theabeliangroupH 2 (G, A)isconsidered asapointed setwiththeunitelement.
Proof. The connection homomorphism δ is defined in the following way.
Let x ∈ H 0 (G, B/A). Take a representative x ∈ B for x and put a s := x −1 · sx.
This is a cocycle and its equivalence class is denoted by δ(x). The definition
of δ(x) is independent of the x chosen.
In the situation of c), the map δ ′ is given as follows. Let x ∈ H 1 (G, B/A).
Choose a cocycle (x g ) g∈G that represents x and lift each x g to some x g ∈ B.
Put a(g 1 , g 2 ) := g 1 xg2 · x −1
g 1 g 2 · x g1 . This is a 2-cocycle with values in A and its
equivalence class is denoted by δ ′ (x). The definition of δ ′ (x) is independent of
the choices.
Exactness has to be checked at each entry, separately.
2.5. Remark. –––– A sequence (A, a)
to be exact in (B, b) if i(A) = j −1 (c).
i
→ (B, b)
□
j
→ (C, c) of pointed sets is said
2.6. Definition. –––– Let h : G ′ → G be a homomorphism of finite groups.
Then, for an arbitrary G-set E, one has a natural pull-back map
h ∗ : H 0 (G, E) → H 0 (G ′ , E).
If E is a G-group then the pull-back map is a group homomorphism.
For an arbitrary G-group A, there is the natural pull-back map
which is a morphism of pointed sets.
h ∗ : H 1 (G, A) → H 1 (G ′ , A)
2.7. Remark. –––– If h is the inclusion of a subgroup then the pull-back
res G′
G := h∗ is usually called the restriction map.
If h : G ′ ֒→ G is the canonical projection to a quotient group then inf G′
G := h ∗
is said to be the inflation map.
The composition of res G′
G
or infG′ G with some extension of the G ′ -set E (the
G ′ -group A) is usually called the restriction, respectively inflation, too.

8 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
2.8. Remark. –––– Non-abelian group cohomology may easily be extended to
the case where G is a profinite group and A a discrete G-set (respectively
G-group) on which G operates continuously. Indeed, put
H i (G, A) := lim H i (G/G ′ , A G′ )
−→G
′
fori = 0and1. Here, thedirectlimitistakenover theinflationmapsandG ′ runs
through the open normal subgroups G ′ ⊆ G with finite quotient.
3. Galois descent
3.1. Definition. –––– Let L/K be a finite Galois extension and σ ∈ Gal(L/K ).
a) Then, a map i : V 1 → V 2 of L-vector spaces is said to be σ-linear if, for every
v, w ∈ V 1 and λ ∈ L, one has
i(v + w) = i(v) + i(w)
and
i(λv) = σ(λ)i(v) .
b) Let π 1 : X 1 → SpecL and π 2 : X 2 → SpecL be L-schemes. We say that
f : X 1 → X 2 is a morphism of L-schemes twisted by σ if the diagram
X 1
f
X 2
π 2
π 1
SpecL
S(σ)
SpecL
commutes. Here, S(σ): SpecL → SpecL denotes the morphism of affine
schemes induced by σ −1 : L → L.
3.2. Theorem. –––– Let L/K be a finite Galois extension and G := Gal(L/K ).
Then,
i) there are the following equivalences of categories,
⎧
⎫
⎨L-vector spaces
⎬
{K-vector spaces} −→ with a G-operation from the left
⎩
⎭ ,
such that every σ ∈ G operates σ-linearly
⎧
⎫
⎨L-algebras
⎬
{K-algebras} −→ with a G-operation from the left
⎩
⎭ ,
such that every σ ∈ G operates σ-linearly

Sec. 3] GALOIS DESCENT 9
⎧ ⎫ ⎧ ⎫
⎨ central simple ⎬ ⎨ central simple algebras over L ⎬
algebras
⎩ ⎭ −→ with a G-operation from the left
⎩
⎭ ,
over K
such that every σ ∈ G operates σ-linearly
⎧
⎫
{ } ⎨ commutative L-algebras
⎬
commutative
−→ with a G-operation from the left
K-algebras ⎩
⎭ ,
such that every σ ∈ G operates σ-linearly
⎧ ⎫ ⎧ ⎫
⎨ commutative ⎬ ⎨ commutative L-algebras with unit ⎬
K-algebras
⎩ ⎭ −→ with a G-operation from the left
⎩
⎭ ,
with unit such that every σ ∈ G operates σ-linearly
A ↦→ A ⊗ K L,
ii) there is the following equivalence of categories,
⎧
⎫
quasi-projective L-schemes
{ } ⎪⎨ with a G-operation from the left ⎪⎬
quasi-projective
−→ by morphisms of K-schemes ,
K-schemes
such that every σ ∈ G operates
⎪⎩
⎪⎭
by a morphism twisted by σ
X ↦→ X × SpecK SpecL .
iii) Let X be a K-scheme and r be a natural number. Then there are the following
equivalences of categories,
⎧
⎫
quasi-coherent sheaves M
{ }
quasi-coherent
−→
sheaves on X
{ }
locally free sheaves
−→
of rank r on X
on X × SpecK SpecL
⎪⎨
⎪⎬
together with a system (ι σ ) σ∈G
of isomorphisms ι σ : x ∗ σ M → M ,
satisfying ι ⎪⎩
τ ◦ x ∗ τ (ι σ) = ι στ ⎪⎭
for every σ, τ ∈ G
⎧
⎫
locally free sheaves M of rank r
on X × SpecK SpecL
⎪⎨
⎪⎬
together with a system (ι σ ) σ∈G
of isomorphisms ι σ : x ∗ σ M → M ,
satisfying ι ⎪⎩
τ ◦ x ∗ τ (ι σ) = ι στ ⎪⎭
for every σ, τ ∈ G
F ↦→ M := π ∗ F.
Here the morphisms in the categories are those respecting all the extra structures.
π : X × SpecK SpecL −→ X is the canonical morphism and
x σ : X × SpecK SpecL −→ X × SpecK SpecL

10 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
denotes the morphism induced by S(σ): SpecL → SpecL.
Proof. In each case we have to prove that the functor given is fully faithful and
essentially surjective. Full faithfulness is proven in Propositions 3.7, 3.8, and
3.9, respectively. Propositions 3.3, 3.5, and 3.6 show essential surjectivity. □
3.3. Proposition (Galois descent-algebraic version). —–
Let L/K be a finite Galois extension of fields and G := Gal(L/K ).
Let W be a vector space (an algebra, a central simple algebra, a commutative
algebra, a commutative algebra with unit, ... ) over L together with an operation
T : G × W → W of G from the left. Assume that T respects all extra structures
and that for each σ ∈ G the action of σ is a σ-linear map T σ : W → W .
Then, there is a vector space V (an algebra, a central simple algebra, a commutative
algebra, a commutative algebra with unit, ... ) over K such that there is an
isomorphism
V ⊗ K L −→ b
∼=
W .
Here, V ⊗ K L is equipped with the G-operation induced by the canonical operation
on L. b respects all extra structures and the operation of G.
Proof. Define V := W G . This is clearly a K-vector space (a K-algebra, a
commutative K-algebra, a commutative K-algebra with unit). If W is a central
simple algebra over L then V is a central simple algebra over K. The latter can
not be seen directly but follows immediately from the formula W G ⊗ K L = W
which we prove below. For that, let {l 1 , . . . , l n } be a K-basis of L.
The assertion follows from the claim below.
□
3.4. Claim. –––– There exist an index set J and a subset {x j | j ∈ J } ⊂ W G such
that {l i x j | i ∈ {1, . . . , n}, j ∈ J } is a K-basis of W .
Proof. By Zorn’s Lemma, there exists a maximal subset {x j | j ∈ J } ⊂ W G
such that {l i x j | i ∈ {1, . . . , n}, j ∈ J } ⊂ W is a system of K-linearly
independent vectors. Assume, that system is not a basis of W . Then,
〈l i x j | i ∈ {1, . . . , n}, j ∈ J 〉 K
is a proper G-invariant L-sub-vector space of W and one can choose an element
x ∈ W \ 〈l i x j | i ∈ {1, . . . , n}, j ∈ J 〉 K . For every l ∈ L, the sum
∑ T σ (lx) = ∑ σ(l) · T σ (x)
σ∈G σ∈G

Sec. 3] GALOIS DESCENT 11
is G-invariant. Further, by linear independence of characters, the matrix
⎛
⎞
σ 1 (l 1 ) . . . σ 1 (l n )
⎜
⎝
.
· · · ..
⎟ · · · ⎠
σ n (l 1 ) . . . σ n (l n )
is of maximal rank. In particular, there is some l ∈ L such that the image of
x β := ∑ σ(l) · σ(x)
σ∈G
in W /〈l i x j | i ∈ {1, . . . , n}, j ∈ J 〉 K is not equal to zero. Therefore,
{l i x j | i ∈ {1, . . . , n}, j ∈ J } ∪ {l i x β | i ∈ {1, . . . , n} }
is a K-linearly independent system of vectors contradicting the maximality
of {x j | j ∈ J }.
□
3.5. Proposition (Galois descent-geometric version). —–
Let L/K be a finite Galois extension of fields and G := Gal(L/K ).
Further, let Y be a quasi-projective L-scheme together with an operation of G from
the left by twisted morphisms. I.e., such that the diagrams
Y
T σ
Y
SpecL
S(σ)
SpecL
commute where S(σ): SpecL → SpecL is the morphism induced by σ −1 : L → L.
Then, there exists a quasi-projective K-scheme X such that there is an isomorphism
of L-schemes
X × SpecK SpecL −→ f
∼=
Y .
Here, X × SpecK SpecL is equipped with the G-operation induced by the operation
on SpecL and f is compatible with the operation of G.
Proof. Affine Case. Let Y = SpecB be an affine scheme. The G-operation
on SpecB corresponds to a G-operation from the right on B such that each
σ ∈ G operates σ −1 -linearly. Define a G-operation from the left on B
by σ · b := bσ −1 . The assertion follows immediately from Proposition 3.3.
GeneralCase. ByLemma3.10, thereexistsanaffineopencovering {Y 1 , . . . , Y n }
of Y by G-invariant schemes. Galois descent yields affine K-schemes
X 1 , . . . , X n such that there are isomorphisms
X i × SpecK SpecL ∼ =
−→ Y i

12 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
and affine K-schemes X ij , 1 ≤ i < j ≤ n, such that there are isomorphisms
X ij × SpecK SpecL ∼ =
−→ Y i ∩ Y j .
It is to be shown that every X ij admits canonical, open embeddings into X i
and X j .
In each case, on the level of rings we have a homomorphism A⊗ K L −→ B ⊗ K L
and an isomorphism B ⊗ K L ∼ =
−→ (A ⊗ K L) f such that their composition is the
localization map. Clearly, f may be assumed to be G-invariant. I.e., we may
suppose f ∈ A. Consequently, B ⊗ K L ∼ = A f ⊗ K L and, by consideration of
the G-invariants on both sides, B ∼ = A f .
The cocycle relations are obvious. Hence, we can glue the affine schemes
X 1 , . . . , X n along the affine schemes X ij for 1 ≤ i < j ≤ n to obtain the
scheme X desired. Lemma 3.12.ix) below completes the proof. □
3.6. Proposition (Galois descent for quasi-coherent sheaves). —–
Let L/K be a finite Galois extension of fields and G := Gal(L/K ).
Further, let X be a K-scheme, π : X × SpecK SpecL → X the canonical morphism
and x σ : X × SpecK SpecL → X × SpecK SpecL be the morphism induced by
S(σ): SpecL → SpecL.
Let M bea quasi-coherentsheaf over X × SpecK SpecL together with a system (ι σ ) σ∈G
of isomorphisms ι σ : x ∗ σM → M which are compatible in the sense that for each
σ, τ ∈ G there is the relation ι τ ◦ x ∗ τ(ι σ ) = ι στ .
Then, there exists a quasi-coherent sheaf F over X such that there is an isomorphism
under which the canonical isomorphism
π ∗ F ∼ =
−→ M
i σ : x ∗ σπ ∗ F = (πx σ ) ∗ F = π ∗ F id
−→ π ∗ F
is identified with ι σ for each σ. I.e., the diagrams
b
x ∗ σπ ∗ F
x∗ σ(b)
x ∗ σM
are commutative.
i σ
π ∗ F
b
M
Proof. First step. Assume X ∼ = SpecA to be an affine scheme.
Then, M = ˜M for some (A ⊗ K L)-module M. We have
x ∗ σ M =
˜
M ⊗ (A⊗K L) (A ⊗ K L σ−1 ) = M ˜ ⊗ L L σ−1 .
ι σ

Sec. 3] GALOIS DESCENT 13
Hence, x ∗ σM = ˜M σ−1 where M σ−1 coincides with M as an A-module. But its
structure of an L-vector space is given by
l ·M σ −1 m := σ−1 (l) ·M m.
Consequently, the isomorphism ι σ : x ∗ σM → M is induced by an A-module
isomorphism j σ : M → M which is σ −1 -linear. The compatibility relations,
required above, are easily translated into the condition that the maps j σ form a
G-operation on M from the right.
Define a G-operation from the left on M by σ ·m := j σ −1(m). By Galois descent
for vector spaces, theK-vector spaceM G ofG-invariants satisfies M G ⊗ K L ∼ = M.
Thisisalsoanisomorphism ofA-modulesastheG-operationonM iscompatible
with the A-operation. Putting F := ˜M G , we obtain a quasi-coherent sheaf
over X such that π ∗ F ∼ = M .
The commutativity of the diagram is a consequence of Proposition 3.3.
Second step. In general, consider an affine open covering
for X α
∼ = SpecRα .
X = [ α∈I
X α
For every intersection X α1 ∩ X α2 , we consider an affine open covering
{X α1 ,α 2 ,β ∼ = SpecR α1 ,α 2 ,β | β ∈ J α1 ,α 2
}.
By the affine case, for each α ∈ I, we are given an R α -module M α such that
π ∗ ˜M α
∼ = M |Xα × SpecK Spec L. Further, for each triple (α 1 , α 2 , β) with α 1 , α 2 ∈ I
and β ∈ J α1 ,α 2
, we have R α1 ,α 2 ,β-modules M α1 ,α 2 ,β satisfying
π ∗ ˜ Mα1 ,α 2 ,β ∼ = M | Xα1 ,α 2 ,β× SpecK SpecL .
The construction of these modules is compatible with restriction to affine subschemes.
Therefore, by Proposition 3.9, we get isomorphisms
i α1 ,α 2 ,β : M α1 ⊗ Rα1 R α1 ,α 2 ,β
∼=
−→ M α2 ⊗ Rα2 R α1 ,α 2 ,β.
It is clear that, for every α 1 , α 2 , α 3 ∈ I and every β 1 ∈ J α2 ,α 3
, β 2 ∈ J α3 ,α 1
,
and β 3 ∈ J α1 ,α 2
, these isomorphisms are compatible on the triple intersection
X α1 ,α 2 ,β 3
∩ X α3 ,α 1 ,β 2
∩ X α2 ,α 3 ,β 1
. I.e., we can glue the quasi-coherent sheaves ˜M α
along the ˜M α1 ,α 2 ,β to obtain the quasi-coherent sheaf M desired. □
3.7. Proposition (Galois descent for homomorphisms). —–
Let L/K be a finite Galois extension of fields and G := Gal(L/K ). Then, it is
equivalent

14 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
i) to give a homomorphism r : V → V ′ of K-vector spaces (of algebras over K,
of central simple algebras over K, of commutative K-algebras, of commutative
K-algebras with unit, ... ),
ii) to give a homomorphism r L : V ⊗ K L → V ′ ⊗ K L of L-vector spaces (of algebras
over L, ofcentralsimplealgebrasover L, ofcommutative L-algebras, ofcommutative
L-algebras with unit, ... ) which is compatible with the G-operations. I.e., such that
for each σ ∈ G the diagram
V ⊗ K L
r L
V ′ ⊗ K L
commutes.
σ
V ⊗ K L
r L
σ
V ′ ⊗ K L
Proof. If r is given then one defines r L := r ⊗ K L. Clearly, if r is a ring
homomorphism then r L is, too.
Conversely, in order to construct r from r L , note that the commutativity of
the diagrams implies that r L is compatible with G-invariants. Further, we
know (V ⊗ K L) G = V and (V ′ ⊗ K L) G = V ′ . Thus, we obtain a K-linear
map r : V −→ V ′ . If r L is a ring homomorphism then its restriction r is, too.
It is clear that the two procedures described are inverse to each other.
□
3.8. Proposition (Galois descent for morphisms of schemes). —–
Let L/K be a finite Galois extension of fields and G := Gal(L/K ).
Then, it is equivalent
i) to give a morphism of K-schemes f : X → X ′ ,
ii) to give a morphism of L-schemes f L : X × SpecK SpecL → X ′ × SpecK SpecL
which is compatible with the G-operations. I.e., such that for each σ ∈ G the
diagram
X × SpecK SpecL
f L
X ′ × SpecK SpecL
commutes.
σ
X × SpecK SpecL
f L
σ
X ′ × SpecK SpecL
Proof. Let f be given. Then, one defines f L := f × SpecK SpecL.
Conversely, in order to construct f from f L , observe that the question is
local in X ′ and X. Thus, we may assume we are given a homomorphism

Sec. 3] GALOIS DESCENT 15
r L : A ′ ⊗ K L −→ A ⊗ K L of L-algebras with unit making the diagrams
A ′ ⊗ K L
r L
A ⊗ K L
σ
A ′ ⊗ K L
r L
σ
A ⊗K L
commute. This is exactly the situation covered by Proposition 3.7.
It is clear that the two processes described are inverse to each other.
□
3.9. Proposition (Galois descent for morphisms of quasi-coherent sheaves). —–
Let L/K be a finite Galois extension of fields and G := Gal(L/K ). Further, let X
be a K-scheme and π : X × SpecK SpecL → X be the canonical morphism.
Then, it is equivalent
i) to give a morphism r : F → G of coherent sheaves over X,
ii) to give a morphism r L : π ∗ F → π ∗ G of quasi-coherent sheaves over
X × SpecK SpecL which is compatible with the G-operations. I.e., such that for
each σ ∈ G the diagram
commutes.
π ∗ F
x ∗ σ
π ∗ F
Proof. If r is given then one defines r L := π ∗ r.
r L
r L
π ∗ G
x ∗ σ
π ∗ G
Conversely, if r L is given then the question to construct r is local in X. Thus, assume
A is a commutative ring with unit and M and N are A-modules. We are
given a homomorphism r L : M ⊗ K L → N ⊗ K L of A ⊗ K L-modules such that
the diagrams
r L
M ⊗ K L N ⊗ K L
σ
M ⊗ K L
r L
σ
N ⊗K L
commute for each σ ∈ G. We get a morphism r : M → N as M and N are the
A-modules of G-invariants on the left and right hand side, respectively.
The two procedures described above are inverse to each other.
3.10. Lemma. –––– Let L be a field and Y be a quasi-projective L-scheme equipped
with an operation of some finite group G acting by morphisms of schemes.
Then, there exists a covering of Y by G-invariant affine open subsets.
Proof. Let y ∈ Y be an arbitrary closed point. Everything which is needed is an
affine open G-invariant subset containing y. For that, we choose an embedding
□

16 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
i : Y ֒→ P N L . By Sublemma 3.11, there exists a hypersurface H y such that
H y ⊃ i(Y ) \ i(Y ) and i(σ(y)) ∉ H y for every σ ∈ G. Here, i(Y ) denotes the
closure of i(Y ) in P N L . By construction, the morphism
i| Y \i −1 (H y ) : Y \ i −1 (H y ) −→ P N L \ H y
is a closed embedding. As P N L \ H y is an affine scheme, Y \ i −1 (H y ) must be
affine, too. Hence,
O y := \
σ −1 (Y \ i −1 (H y )) ⊂ Y
σ∈G
is the intersection of finitely many affine open subsets in a quasi-projective and,
therefore, separated scheme. Thus, O y is an affine open subset. By construction,
O y is G-invariant and contains y.
□
3.11. Sublemma. –––– Let L be a field and Z P N L be a Zariski-closed subset.
Further, let p 1 , . . . , p n ∈ P N L be finitely many closed points not contained in Z.
Then, there exists a hypersurface H ⊂ P N L which contains Z but does not contain
any of the points p 1 , . . . , p n .
Proof. We will give two proofs, an elementary one and a more canonical one
which uses cohomology of coherent sheaves.
First proof. Let S := L[X 0 , . . . , X N ] be the homogeneous coordinate ring for
the projective space P N L . S is a graded L-algebra. For d ∈Æ, we will denote by
S d the L-vector space of homogeneous elements of degree d.
We proceed by induction. The case n = 0 is trivial. Thus, assume that the
assertion is true for n − 1 and consider n points p 1 , . . . , p n . By induction
hypothesis, there exists a homogeneous element s ∈ S, i.e., a hypersurface
H := V (s) of a certain degree d, such that H ⊇ Z and p 1 , . . . , p n−1 ∉ H .
We may assume p n ∈ H as, otherwise, the proof would be complete.
Z ∪ {p 1 } ∪ . . . ∪ {p n−1 } is a Zariski closed subset of P N L not containing p n.
Therefore, there exist some d ′ ∈Æand some homogeneous s ′ ∈ S d ′ such that
V (s ′ ) ⊇ Z ∪ {p 1 } ∪ . . . ∪ {p n−1 } but p n ∉ V (s ′ ). Every hypersurface
V (a · s ′d + b · s d ′ ) for a, b ∈ L non-zero contains Z but neither p 1 , . . . , p n−1 ,
nor p n .
Second proof. Tensoring the canonical exact sequence
0 −→ I {p1 , ... ,p n } −→ O X −→ O {p1 , ... ,p n } −→ 0
with the ideal sheaf I Z yields an exact sequence
0 −→ I {p1 , ... ,p n }∪Z −→ I Z −→ O {p1 , ... ,p n } −→ 0

Sec. 3] GALOIS DESCENT 17
of coherent sheaves on P N L .
For each d ∈, we tensor with the invertible sheaf O(d) and find a long
cohomology exact sequence
Γ(P N L , I Z(d)) −→ Γ(P N L , O {p 1 , ... ,p n }(d)) −→ H 1 (P N L , I {p 1 , ... ,p n }∪Z(d)) .
By Serre’s vanishing theorem [Ha77, Theorem III.5.2],
for d ≫ 0. Hence, there is a surjection
H 1 (P N L , I {p 1 , ... ,p n }∪Z(d)) = 0
Γ(P N L , I Z (d)) −→ Γ(P N L , O {p1 , ... ,p n }(d)) ∼ = κ(p 1 ) ⊕ · · · ⊕ κ(p n ) .
That means that there exists a global section s of I Z (d) which does not vanish
in any of the points p 1 , . . . , p n . The section s defines a hypersurface of degree
d in P N L containing Z but not containing any of the points p 1 , . . . , p n . □
3.12. Lemma (A. Grothendieck and J. Dieudonné). —–
LetL/K beafinitefield extensionand X beaK-schemesuchthatX × SpecK SpecL is
i) reduced,
ii) irreducible,
iii) quasi-compact,
iv) locally of finite type,
v) of finite type,
vi) locally Noetherian,
vii) Noetherian,
viii) proper,
ix) quasi-projective,
x) projective,
xi) affine,
or
xii) regular.
Then, X admits the same property.
Proof. Let π : X × SpecK SpecL → X denote the canonical morphism. For iii)
through xi), we may assume L/K to be Galois. Put G := Gal(L/K ).
i) If 0 ≠ s ∈ Γ(U,O X ) were a nilpotent section of the structure sheaf over some
open subset U ⊆ X then π ♯ (s) ∈ Γ(U × SpecK SpecL, O X ×SpecK SpecL) would be a
nilpotent non-zero local section of the structure sheaf of X × SpecK SpecL.

18 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
ii) If X = X 1 ∪ X 2 were a decomposition into two closed proper subschemes
then X × SpecK SpecL = X 1 × SpecK SpecL ∪ X 2 × SpecK SpecL would be, too.
iii) Let {U α | α ∈ A } be an arbitrary affine open covering of X. Then,
{U α × SpecK SpecL | α ∈ A }
is an affine open covering of X × SpecK SpecL. Quasi-compactness guarantees
the existence of a finite sub-covering {U α × SpecK SpecL | α ∈ A 0 }.
Then, {U α | α ∈ A 0 } is a finite affine open covering for X.
iv) We may assume X = SpecA to be affine. Then, by assumption, A ⊗ K L
is a finitely generated L-algebra. Let {b 1 , . . . , b n } be a system of generators
for A ⊗ K L. Decomposing into elementary tensors, if necessary, we may
assume without restriction that all b i = a i ⊗ l i are elementary. Consider the
homomorphism
π : K [X 1 , . . . , X n ] → A
of K-algebras given by X i ↦→ a i . π induces a surjection after tensoring with L.
It is, therefore, surjective as ⊗ K L is a faithful functor.
v) This is just the combination of iii) and iv).
vi) Let X α
∼ = SpecAα be an affine open subscheme of X. We have to
show that A α is Noetherian under the hypothesis that A α ⊗ K L is. Assume
I 1 ⊂ I 2 ⊂ I 3 ⊂ . . . is an ascending chain of ideals in A α which does
not stabilize. It induces an ascending chain I 1 ⊗ K L ⊂ I 2 ⊗ K L ⊂ I 3 ⊗ K L ⊂ . . .
of ideals in A α ⊗ K L that does not stabilize, either. This is a contradiction.
vii) This is the combination of iii) and vi).
viii) By virtue of v), X is of finite type. We apply the valuative criterion
for properness. Consider a commutative diagram
U
i
T
X
SpecK
where T = SpecR is the spectrum of a valuation ring, U = SpecF is the
spectrum of its quotient field and i is the canonical morphism. Taking the base
change to SpecL, we find
U × SpecK SpecL
X × SpecK SpecL
ι
i
T × SpecK SpecL
SpecL .
Here, the morphism ι in the diagonal is the unique one making the diagram commute.
Note that U × SpecK SpecL = Spec(F ⊗ K L) is no longer the spectrum

Sec. 3] GALOIS DESCENT 19
of a field but rather the union of finitely many spectra of fields. Similarly,
T × SpecK SpecL = Spec(R ⊗ K L) is a finite union of spectra of valuation rings.
Further, there is a natural G-operation on the whole diagram without ι. As ι is
uniquely determined by the condition that it makes the diagram commute, the
diagrams
ι
T × SpecK SpecL X × SpecK SpecL
σ
T × SpecK SpecL
ι
σ
X × SpecK SpecL
are commutative for each σ ∈ G. By Proposition 3.8, ι is the base change of a
morphism T → X.
ix) and x) Taking v) and viii) into account, everything left to be shown
is the existence an ample invertible sheaf on X. By assumption, there
is an ample invertible sheaf M on X × SpecK SpecL. For σ ∈ G let
x σ : X × SpecK SpecL → X × SpecK SpecL be the morphism of schemes induced by
S(σ): SpecL → SpecL, i.e., by σ −1 on coordinate rings. The invertible sheaves
x ∗ σ M are ample, too. Therefore, M := N υ∈G x ∗ υM is an ample invertible sheaf.
For each σ ∈ G, there are canonical identifications
ι σ : x ∗ σ M = O υ∈G
x ∗ σ x∗ υ M = O υ∈G
x ∗ υσ M
id
−→ O υ∈G
x ∗ υ M = M .
Obviously, these are compatible in the sense that, for each σ, τ ∈ G, there is
the relation ι τ ◦x ∗ τ(ι σ ) = ι στ . By Galois descent for locally free sheaves, there is
an invertible sheaf L ∈ Pic(X ) such that π ∗ L ∼ = M . Lemma 3.13 shows that
L is ample.
xi) We suppose that X × SpecK SpecL ∼ =
−→ SpecB is affine. The isomorphism
defines a G-operation from the right on the L-algebra B such that each σ ∈ G
operates σ −1 -linearly. Define a G-operation from the left on B by σ·b := b·σ −1 .
Proposition 3.3 provides us a K-algebra A such that there is an isomorphism
X × SpecK SpecL ∼ =
∼=
−→ SpecB = Spec(A ⊗ K L) −→ SpecA × SpecK SpecL ,
compatible with the G-operations. Proposition 3.5 implies X ∼ = SpecA.
xii) Let (A, m) be the local ring at a point p ∈ X. By vi), we may assume (A, m)
is Noetherian. We have to show A is regular. Let us give two proofs for that,
the standard one using Serre’s homological characterization of regularity and an
elementary one.
First proof. Assume, by contradiction, (A, m) were not regular. Then, [Mat,
Theorem 19.2] shows gl. dim A = ∞. Further, by [Mat, §19, Lemma 1],

20 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
proj. dim A
A/m = ∞ and
On the other hand, we have
sup {i | Tor A i (A/m, A/m) ≠ 0 } = ∞.
Tor A i (A/m, A/m) ⊗ K L = Tor A i (A/m, A/m) ⊗ A (A ⊗ K L)
= Tor A i (A/m, (A ⊗ K L/m ⊗ K L))
= Tor A⊗ KL
i ((A ⊗ K L/m ⊗ K L), (A ⊗ K L/m ⊗ K L))
= 0
for i > dim A ⊗ K L as A ⊗ K L is regular. This is a contradiction.
Note that we only used that π is faithfully flat.
Second proof. Put d := dim A. The ring A ⊗ K L is not local, in general.
But the quotient (A ⊗ K L)/(m⊗ K L) ∼ = A/m⊗ K L is a direct product of finitely
many fields since L/K is a finite, separable field extension. The quotients
(A ⊗ K L)/(m n ⊗ K L) are Artin rings as they are flat of relative dimension zero
over A/m n . Consequently, they are direct products of Artin local rings,
(A ⊗ K L)/(m n ⊗ K L)
∼=
−→ A (n)
1
× · · · × A (n)
l .
Under this isomorphism, (m ⊗ K L)/(m n ⊗ K L) is mapped to the product of the
maximal ideals m (n)
1
× · · · × m (n)
l as it is a nilpotent ideal and the quotient has to
be a direct product of fields.
For each i ∈ {1, . . . , l}, let f i ∈ A ⊗ K L be the element which maps to the
standard vector e i = (0, . . . , 0, 1, 0, . . . , 0). Then,
A (n)
i
∼= ((A ⊗ K L)/(m n ⊗ K L)) fi
∼ = (A ⊗K L) fi /m n (A ⊗ K L) fi .
By assumption, (A ⊗ K L) fi is a regular local ring and m(A ⊗ K L) fi is its maximal
ideal. (A ⊗ K L) fi is of dimension d as it is flat of relative dimension zero
over A. The Hilbert-Samuel function of a regular local ring is standard. We have
length ( A (n)
i
) = length ((A ⊗K L) fi /m n (A ⊗ K L) fi )
= length ((A ⊗ K L) fi /(m(A ⊗ K L) fi ) n )
( ) n + d − 1
= .
d
Consequently, for the function H A such that H A (n) := dim K (A/m n ), one gets
(
H A (n) = dim L (A ⊗K L)/(m n ⊗ K L) )
( )
= dim L A
(n)
1
× · · · × A (n)
l

Sec. 3] GALOIS DESCENT 21
=
=
l
∑
i=1
l
∑
i=1
( )
dim L A
(n)
i
[(
A
(n)
i
/m (n)
i
) ] ( )
: L · length A
(n)
i .
)
. How-
This shows that H A is a constant multiple of n ↦→ ( n+d−1
d
ever, since H A (1) = 1, there is no ambiguity about constant factors. A is regular.
□
3.13. Lemma. –––– Let L/K be an arbitrary field extension, X a K-scheme of
finite type, π : X × SpecK SpecL → X the canonical morphism and L ∈ Pic(X ) an
invertible sheaf. Suppose that the pull-back π ∗ L ∈ Pic(X × SpecK SpecL) is ample.
Then, L is ample.
Proof. We have to prove that, for every coherent sheaf F on X and every
closed point x ∈ X, the canonical map px n : Γ(X, F ⊗ L ⊗n ) → F x ⊗ Lx
⊗n is
surjective for n ≫ 0. For this, it is obviously sufficient to prove surjectivity of
But, as is easy to see,
p n x ⊗ K L : Γ(X, F ⊗ L ⊗n ) ⊗ K L → (F x ⊗ L ⊗n
x ) ⊗ K L .
Γ(X, F ⊗ L ⊗n ) ⊗ K L = Γ(X × SpecK SpecL, π ∗ F ⊗ π ∗ L ⊗n )
while (F x ⊗ L n
x ) ⊗ K L = Γ(π −1 (x), π ∗ F ⊗ π ∗ L ⊗n ). Here, π −1 (x) denotes the
fiber of π above x. For n ≫ 0, the map p n x ⊗ K L is surjective as π ∗ L is ample.
□
3.14. Lemma. –––– Let R be a ring, F ≠ 0 a free R-module of finite rank and M
an arbitraryR-module. Suppose that M⊗ R F is a locally free R-module of finite rank.
Then, M is locally free of finite rank.
Proof. The R-module M⊗ R F is locally free of finite rank. Therefore, there exists
some affine open covering {SpecR f1 , . . . , SpecR fn } of SpecR such that each
(M ⊗ R F )⊗ R R fi = (M ⊗ R R fi ) ⊗ R F
is a free R fi -module. M ⊗ R R fi = M fi is a direct summand. Therefore, M fi is a
projective R fi -module.
We have a surjection (M ⊗ R F )⊗ R R fi ։ M fi , the kernel K of which is a direct
summand of (M ⊗ R F ) ⊗ R R fi , too. In particular, K is a finitely generated
R fi -module. Hence, M fi is finitely presented and, therefore, locally free by
[Mat, Theorem 7.12 and Theorem 4.10].
□

22 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
3.15. Remark. –––– Galois descent is a central technique in André Weil’s foundation
of Algebraic Geometry. The Grothendieck school gave a far-reaching
generalization of it, the so-called faithful flat descent. It turns out that Galois
descent is sufficient for our presentation of the relationship between central simple
algebras and Brauer-Severi varieties. For that reason, we decided to present
it in detail. Faithful flat descent will play a main role in the next chapter.
4. Central simple algebras and non-abelian H 1
We are going to make use of the following well-known facts about central
simple algebras.
4.1. Lemma (J. H. Maclagan-Wedderburn, R. Brauer). —– Let K be a field.
a) Let A be a central simple algebra over K. Then, there exist a skew field D with
center K and a natural number n such that A ∼ = M n (D) is isomorphic to the full
algebra of n × n-matrices with entries in D.
b) Let L be a field extension of K and A be a central simple algebra over K. Then,
A ⊗ K L is a central simple algebra over L.
c) Assume K to be separably closed. Let D be a skew field being finite dimensional
over K whose center is equal to K. Then, D = K.
Proof. See the standard literature, for example [Lan93], [Bou-A], or [Ke].
4.2. Remarks. –––– a) Let A be a central simple algebra over a field K.
i) The proof of Lemma 4.1.a) shows that in the presentation A ∼ = M n (D) the
skew field D is unique up to isomorphism of K-algebras and the natural number
n is unique.
ii) A ⊗ K K sep is isomorphic to a full matrix algebra over K sep . In particular,
dim K A is a perfect square. The natural number ind(A) := √ dim K (D) is called
the index of A.
b) Let A 1 , A 2 be central simple algebras over a field K. Then A 1 ⊗ K A 2 can be
shown to be a central simple algebra over K. Further, if A is a central simple
algebra over a field K then A ⊗ K A op ∼ = Aut K-Vect (A). I.e., it is isomorphic to a
matrix algebra.
c) Two central simple algebras A 1
∼ = Mn1 (D 1 ), A 2
∼ = Mn2 (D 2 ) over a field K are
said to be similar if the corresponding skew fields D 1 and D 2 are isomorphic
as K-algebras. This is an equivalence relation on the set of all isomorphism
classes of central simple algebras over K. The tensor product induces a group
structure on the set of similarity classes of central simple algebras over K, this
is the so-called Brauer group Br(K ) of the field K.
□

Sec. 4] CENTRAL SIMPLE ALGEBRAS AND NON-ABELIAN h 1 23
4.3. Definition. –––– Let K be a field and A be a central simple algebra over K.
A field extension L of K admitting the property that A ⊗ K L is isomorphic to
a full matrix algebra is said to be a splitting field for A. In this case, one says that
A splits over L.
4.4. Lemma (Theorem of Skolem-Noether). —–
Let R be a commutative ring with unit. Then, GL n (R) operates on M n (R) by
conjugation,
(g, m) ↦→ gmg −1 .
If R = L is a field then this defines an isomorphism
PGL n (L) := GL n (L)/L ∗ ∼ =
−→ Aut L (M n (L)) .
Proof. One has L = Zent(M n (L)). Therefore, the mapping is well-defined and
injective.
Surjectivity. Let j : M n (L) → M n (L) be an automorphism. We consider the
algebra
M := M n (L) ⊗ L M n (L) op ( ∼ = M n 2(L)).
M n (L) gets equipped with the structure of a left M-module in two ways.
(A ⊗ B) • 1 C := A · C · B
(A ⊗ B) • 2 C := j(A) · C · B
Two M n 2(L)-modules of the same L-dimension are isomorphic, as the
n 2 -dimensional standard L-vector space equipped with the canonical operation
of M n 2(L) is the only simple left M n 2(L)-module and there are no non-trivial
extensions. Thus, there is an isomorphism h : (M n (L), • 1 ) → (M n (L), • 2 ).
Let us put I := h(E) to be the image of the identity matrix.
M ∈ M n (L) we have
h(M ) = h((E ⊗ M ) • 1 E) = (E ⊗ M ) • 2 h(E) = h(E) · M = I · M.
In particular, I ∈ GL n (L). Therefore,
I · M = h(M ) = h((M ⊗ E) • 1 E) = (M ⊗ E) • 2 h(E) = j(M ) · I
For every
for each M ∈ M n (L) and j(M ) = IMI −1 .
□
4.5. Definition. –––– Let n be a natural number.
i) If K is a field then we will denote by Az K n the set of all isomorphism classes
of central simple algebras A of dimension n 2 over K.

24 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
ii) Let L/K be a field extension. Then Azn
L/K will denote the set of all isomorphism
classes of central simple algebras A which are of dimension n 2 over K
and split over L. Obviously, Az K n := S Az L/K
L/K
n .
4.6. Theorem (cf. J.-P. Serre: Corps locaux [Se62, chap. X, §5]). —–
Let L/K be a finite Galois extension of fields, G := Gal(L/K ) its Galois group,
and n ∈Æ.
Then, there is a natural bijection of pointed sets
a = a L/K
n
: Az L/K
n
∼=
−→ H 1 (G, PGL n (L)) ,
A ↦→ a A .
Proof. Let A be a central simple algebra over K which splits over L,
The diagrams
A ⊗ K L ∼ =
−→ M n (L) .
f
A ⊗ K L f M n (L)
σ
σ
do not commute, in general.
A ⊗ K L f M n (L)
For each σ ∈ G, define a σ ∈ PGL n (L) by putting ( f ◦ σ) = a σ ◦ (σ ◦ f ).
It turns out that
f ◦ στ = ( f ◦ σ) ◦ τ
= a σ ◦ (σ ◦ f ) ◦ τ
= a σ ◦ σ ◦ ( f ◦ τ )
= a σ ◦ σ ◦ (a τ ◦ (τ ◦ f ))
= a σ ◦ σ a τ ◦ (στ ◦ f ) .
I.e., a στ = a σ · σa τ and (a σ ) σ∈G is a cocycle.
If one starts with another isomorphism f ′ : A⊗ K L −→ M n (L) then there exists
some b ∈ PGL n (L) such that f = b ◦ f ′ . The equality ( f ◦ σ) = a σ ◦ (σ ◦ f )
implies
f ′ ◦ σ = b −1 ◦ f ◦ σ = b −1 · a σ ◦ (σ ◦ (b ◦ f ′ )) = b −1 · a σ · σb ◦ (σ ◦ f ′ ) .
Thus, the isomorphism f ′ yields a cocycle cohomologous to (a σ ) σ∈G . The
mapping a is well-defined.

Sec. 4] CENTRAL SIMPLE ALGEBRAS AND NON-ABELIAN h 1 25
Injectivity. Assume that A and A ′ are chosen in such a way that the construction
above yields one and the same cohomology class a A = a A ′ ∈ H 1 (G, PGL n (L)).
After choosing suitable isomorphisms f and f ′ , one has the equalities
( f ◦ σ) = a σ ◦ (σ ◦ f ) and ( f ′ ◦ σ) = a σ ◦ (σ ◦ f ′ ) in the diagram
A ⊗ K L f f
M n (L) ′
A ′ ⊗ K L
σ
A ⊗ K L f
σ
M n (L)
f ′
σ
A ′ ⊗ K L .
Consequently, f ◦ σ ◦ f −1 ◦ σ −1 = f ′ ◦ σ ◦ f ′−1 ◦ σ −1 and, therefore,
f ◦ σ ◦ f −1 ◦ f ′ ◦ σ −1 ◦ f ′−1 = id .
The outer part of the diagram commutes. Taking the G-invariants on both sides
yields A ∼ = A ′ .
Surjectivity. Let a cocycle (a σ ) σ∈G for H 1 (G, PGL n (L)) be given. We define a
new G-operation on M n (L) as follows.
Let σ ∈ G act as
a σ ◦ σ : M n (L)
σ a
−→ M n (L) −→
σ
Mn (L) .
Note that this is a σ-linear mapping. Further, one has
(a σ ◦ σ) ◦ (a τ ◦ τ ) = a σ ◦ σ a τ ◦ στ = a στ ◦ στ .
I.e., we constructed a group operation from the left. Galois descent yields the
desired algebra.
□
4.7. Corollary. –––– Let L/K be a finite Galois extension of fields and let n be a
natural number.
a) Let L ′ be a field extension of L such that L ′ /K is Galois, too. Then, the following
diagram of morphisms of pointed sets commutes,
Az L/K
n
a L/K
n
H 1 (Gal (L/K ), PGL n (L))
nat. incl.
inf Gal (L′ /K)
Gal(L/K)
Az L′ /K
n
a L′ /K
n
H 1( Gal(L ′ /K ), PGL n (L ′ ) ) .

26 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
b)LetK ′ beanintermediatefield oftheextensionL/K. Then,thefollowingdiagram
of morphisms of pointed sets commutes,
Az L/K
n
a L/K
n
H 1( Gal(L/K ), PGL n (L) )
⊗ K K ′
res Gal(L/K′ )
Gal(L/K)
Az L/K ′
n
a L/K ′
n
H 1( Gal(L/K ′ ), PGL n (L) ) .
Proof. These are direct consequences of the construction of the bijections a ∗ n .
□
4.8. Corollary. –––– Let K be a field and n be a natural number. Then, there is a
unique natural bijection
a = a K n : Az K n −→ H 1( Gal(K sep /K ), PGL n (K sep ) )
such that a K n | Az
L/K
n
= a L/K
n for each finite Galois extension L/K in K sep .
Proof. In order to get connected to the definition of the cohomology of a
profinite group, only one technical point is to be proven. That is the formula
PGL n (K sep ) Gal(K sep /K ′) = PGL n (K ′ )
for K ⊆ K ′ ⊆ K sep any intermediate field. To see this, observe that the
exact sequence
1 −→ (K sep ) ∗ −→ GL n (K sep ) −→ PGL n (K sep ) −→ 1
induces a long exact sequence
1→(K ′ ) ∗ → GL n (K ′ ) → PGL n (K sep ) Gal(K sep /K ′) → H 1( Gal(K sep /K ′ ), (K sep ) ∗)
in cohomology. Finally, the right entry vanishes by Hilbert’s Theorem 90.
Cf. Lemma 5.11.
□

Sec. 4] CENTRAL SIMPLE ALGEBRAS AND NON-ABELIAN h 1 27
4.9. Proposition. –––– Let K be a field and m and n be natural numbers.
Then, there is a commutative diagram
Az K n
a K n
H 1( Gal(K sep /K ), PGL n (K sep ) )
A↦→M m (A)
Az K mn
(i n mn) ∗
a K mn
H 1( Gal(K sep /K ), PGL mn (K sep ) ) .
Here, (imn) n ∗ is the map induced by the block-diagonal embedding
imn n : PGL n (K sep ) −→ PGL mn (K sep )
⎛ ⎞
E 0 · · · 0
0 E · · · 0
E ↦→ ⎜
⎝ . .
..
⎟ . . ⎠ .
0 0 · · · E
Proof. Let A ∈ Az K n . By the construction above, a cycle representing
the cohomology class a K n (A) is given as follows. Choose an isomorphism
f : A ⊗ K K sep → M n (K sep ) and put a σ := ( f ◦ σ) ◦ (σ ◦ f ) −1 ∈ Aut(M n (K sep ))
for each σ ∈ Gal(K sep /K ).
On the other hand, for M m (A) ∈ Az K mn one may choose the isomorphism
M m ( f ): M m (A) ⊗ K K sep = M m (A ⊗ K K sep ) −→ M m (M n (K sep )) ∼ = M mn (K sep ).
For each σ ∈ Gal(K sep /K ), this yields the automorphism ã σ of M m (M n (K sep ))
which operates as a σ on each block. If a σ is given by conjugation with a matrix
A σ then ã σ is given by conjugation with
⎛ ⎞
A σ 0 · · · 0
0 A σ · · · 0
⎜
⎝ . .
..
⎟ . . ⎠ .
0 0 · · · A σ
This is exactly what was to be proven.
□
4.10. Remark. –––– The proposition above shows
Br(K ) ∼ = lim −→n
H 1( Gal(K sep /K ), PGL n (K sep ) ) .

28 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
Further, for each m and n, there is a commutative diagram of exact sequences
as follows,
1 (K sep ) ∗ GL n (K sep )
PGL n (K sep )
1
1 (K sep ) ∗ GL mn (K sep ) PGL mn (K sep ) 1 .
We note that (K sep ) ∗ is mapped into the centers of GL n (K sep ) and GL mn (K sep ), respectively.
Therefore, there are boundary maps to the second group cohomology
group and they are compatible with each other to give a map
lim H
−→ 1( Gal(K sep /K ), PGL n (K sep ) ) −→ H 2( Gal(K sep /K ), (K sep ) ∗) .
n
It is not complicated to show that this map is injective and surjective. Cf. Corollary
II.5.3.
j n mn
5. Brauer-Severi varieties and non-abelian H 1
5.1. Definition. –––– Let K be a field.
i) A scheme X over K is called a Brauer-Severi variety if there exists a finite,
separable field extension L/K such that X × SpecK SpecL is isomorphic to a
projective space P N L .
ii) A field extension L of K admitting the property that X × SpecK SpecL ∼ = P N L
for some n ∈Æis said to be a splitting field for X. In this case, one says that
X splits over L.
5.2. Proposition. –––– Let X be a Brauer-Severi variety over a field K. Then,
i) X is a variety. I.e., X is reduced and irreducible as a scheme.
ii) X is projective and regular.
iii) X is geometrically integral.
iv) One has Γ(X, O X ) = K.
v) K is algebraically closed in the function field X (K ).
Proof. We will denote the dimension of X by N .
i) and ii) are direct consequences of Lemma 3.12.i),ii), x) and xii).
iii) is clear from the definition.
iv) We have
Γ(X, O X ) ⊗ K K sep = Γ(X × SpecK SpecK sep , O X ×SpecK SpecK sep)
= Γ(P N K sep, O P N K sep)
= K sep .

Sec. 5] BRAUER-SEVERI VARIETIES AND NON-ABELIAN h 1 29
Consequently, Γ(X, O X ) = K.
v) Assume, g ∈ K (X ) is a rational function which is algebraic over K. We fix
an algebraic closure K of K.
The pull-back g K
is a rational function on X × SpecK SpecK ∼ = P N which is
K
algebraic over K. Thus, g K
is a constant function. Consequently, g itself has
definitely no poles. I.e., g is a regular function on X. By iv), we see g ∈ K. □
5.3. Lemma. –––– Let R be a commutative ring with unit.
a) Then, there is an operation of the group GL n (R) on P n−1
R
R-schemes as follows.
by morphisms of
A ∈ GL n (R) gives rise to the morphism given by the graded automorphism
R[X 0 , . . . , X n−1 ] −→ R[X 0 , . . . , X n−1 ]
f (X 0 , . . . , X n−1 ) ↦→ f ((X 0 , X 1 , . . . , X n−1 ) · A t )
of the homogeneous coordinate ring.
b) If R = L is a field then this induces an isomorphism
PGL n (L)
∼=
−→ Aut L-schemes (P n−1
L ) .
Proof. a) is clear. For b), see [Ha77, Example II.7.1.1]. The proof given there
works equally well without the assumption on L to be algebraically closed. □
5.4. Remark. –––– Let S be any commutative R-algebra with unit. Then, on
the set P n−1
R (S) of S-valued points, the definition above yields the naive operation
⎛ ⎞
a 11 . . . a 1n
⎜
⎝
.
. ..
⎟ . ⎠
a n1 . . . a nn
⊤
↓
(x 0 : . . . : x n−1 ) ↦→ ( (a 11 x 0 + . . . + a 1n x n−1 ) : . . . : (a n1 x 0 + . . . + a nn x n−1 ) )
of GL n (S).
5.5. Definition. –––– Let r be natural number.
i) If K is a field then we will denote by BS K r the set of all isomorphism classes of
Brauer-Severi varieties X of dimension r over K.
ii) Let L/K be a field extension. Then, BS L/K
r will denote the set of all isomorphism
classes of Brauer-Severi varieties X over K which are of dimension r and
split over L. Obviously, BS K r := S BS L/K
r .
L/K

30 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
5.6. Theorem (cf. J.-P. Serre: Corps locaux [Se62, chap. X, §6). —–
Let L/K be a finite Galois extension, G := Gal(L/K ) its Galois group, and n be
a natural number. Then, there exists a natural bijection of pointed sets
α = α L/K
n−1 : BSL/K n−1
∼=
−→ H 1 (G, PGL n (L)),
X ↦→ α X .
Proof. Let X be a Brauer-Severi variety over K that splits over L,
X × SpecK SpecL ∼ =
−→ P n−1
L .
On X × SpecK SpecL, as well as on P n−1
L , there are operations of G from
the left by morphisms of K-schemes. The action of σ ∈ G is induced by
S(σ): SpecL → SpecL in both cases. Unfortunately, the diagrams
do not commute, in general.
X × SpecK SpecL
σ
X × SpecK SpecL
f
f
f
P n−1
L
σ
P n−1
L
For each σ ∈ G, define α σ ∈ PGL n (L) by putting ( f ◦ σ) = α σ ◦ (σ ◦ f ).
It turns out that
I.e., (α σ ) σ∈G is a cocycle.
f ◦ στ = ( f ◦ σ) ◦ τ
= α σ ◦ (σ ◦ f ) ◦ τ
= α σ ◦ σ ◦ ( f ◦ τ )
= α σ ◦ σ ◦ (α τ ◦ (τ ◦ f ))
= α σ ◦ σ α τ ◦ (στ ◦ f ) .
If one starts with another isomorphism f ′ : X × SpecK SpecL −→ P n−1
L then
there exists some b ∈ PGL n (L) such that f = b ◦ f ′ . The equality
( f ◦ σ) = α σ ◦ (σ ◦ f ) implies
f ′ ◦ σ = b −1 ◦ f ◦ σ = b −1 · α σ ◦ (σ ◦ (b ◦ f ′ )) = b −1 · α σ · σb ◦ (σ ◦ f ′ ) .
Thus, the isomorphism f ′ yields a cocycle cohomologous to (α σ ) σ∈G . By consequence,
the mapping α is well-defined.
Injectivity. Assume X and X ′ are chosen such that the same cohomology class
α X = α X ′ ∈ H 1 (G, PGL n (L)) arises. After choosing suitable isomorphisms f
and f ′ , one has the formulas ( f ◦σ) = α σ ◦ (σ ◦ f ) and ( f ′ ◦σ) = α σ ◦ (σ ◦ f ′ )

Sec. 5] BRAUER-SEVERI VARIETIES AND NON-ABELIAN h 1 31
in the diagram
X × SpecK SpecL
σ
X × SpecK SpecL
f
f
P n−1
L
σ
P n−1
L
f ′
f ′
X ′ × SpecK SpecL
σ
X ′ × SpecK SpecL .
Therefore, f ◦ σ ◦ f −1 ◦ σ −1 = f ′ ◦ σ ◦ f ′−1 ◦ σ −1 and, consequently,
f ◦ σ ◦ f −1 ◦ f ′ ◦ σ −1 ◦ f ′−1 = id .
The outer part of the diagram commutes. Galois descent yields X ∼ = X ′ .
Surjectivity. Let a cocycle (α σ ) σ∈G for H 1 (G, PGL n (L)) be given. We define a
G-operation on P n−1
L by letting σ ∈ G operate as
α σ ◦ σ : P n−1
L
σ
−→ P n−1
L
α σ
−→ P
n−1
L .
This is a group operation as (α σ ) σ∈G is a cocycle. The geometric version of
Galois descent yields the desired variety.
□
5.7. Corollary. –––– Let L/K be a finite Galois extension of fields and n be a
natural number.
a) Let L ′ be a field extension of L such that L ′ /K is Galois, too. Then, the following
diagram of morphisms of pointed sets commutes.
BS L/K
n−1
α L/K
n−1
H 1( Gal(L/K ), PGL n (L) )
nat. incl.
inf Gal(L′ /K)
Gal(L/K)
BS L′ /K
n−1
α L′ /K
n−1
H 1( Gal(L ′ /K ), PGL n (L ′ ) )
b) LetK ′ beanintermediatefieldoftheextensionL/K. Then,thefollowingdiagram
of morphisms of pointed sets commutes.
BS L/K
n−1
α L/K
n−1
H 1( Gal(L/K ), PGL n (L) )
× SpecK SpecK ′
BS L/K ′
n−1
α L/K ′
res Gal(L/K′ )
Gal(L/K)
n−1
H 1( Gal(L/K ′ ), PGL n (L) )
Proof. These are direct consequences of the construction of the mappings α ∗ n−1.
□

32 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
5.8. Corollary. –––– Let K be a field and n be a natural number. Then, there is a
natural bijection
α = α K n−1 : BSK n−1 −→ H 1( Gal(K sep /K ), PGL n (K sep ) )
such that α K n−1 | BS
L/K
n−1
= α L/K
n−1 for each finite Galois extension L/K in K sep .
Proof. The proof follows the same lines as the proof of Corollary 4.8.
□
5.9. Proposition (F. Severi, cf. J.-P. Serre: [Se62, chap. X, §6, Exc. 1]). —–
Let r ∈Æ. Assume that X is a Brauer-Severi variety of dimension r over a field K.
If X (K ) ≠ ∅ then X ∼ = P r K .
Proof. Let L/K be a finite Galois extension which is a splitting field for X.
Denote its Galois group by G. Choose an isomorphism
X × SpecK SpecL ∼ =
−→ P r L .
Here, one may assume without restriction that the L-valued point
x L ∈ X × SpecK SpecL (L)
induced by the K-valued point x ∈ X (K ) is mapped to (1 : 0 : . . . : 0) ∈ P r L(L).
Therefore, the cohomology class α X ∈ H 1( G, PGL r+1 (L) ) is given by a cocycle
(α σ ) σ∈G such that every α σ admits (1 : 0 : . . . : 0) as a fixed point.
Consequently, α X belongs to the image of H 1( G, F/L ∗) under the natural
homomorphism where
⎧⎛
⎞
⎫
⎪⎨ a 11 . . . a 1,r+1
⎪⎬
⎜
F := ⎝
.
. ..
⎟
. ⎠ ∈ GL r+1 (L)
a 21 = a 31 = . . . = a r+1,1 = 0
⎪⎩
a r+1,1 . . . a r+1,r+1
∣
⎪⎭ .
But it is obvious that
⎧⎛
⎞⎫
1 a 12 . . . a 1,r+1
⎪⎨
F/L ∗ ∼ = F ′ 0 a 22 . . . a 2,r+1
⎪⎬
:= ⎜
⎝
.
. . ..
⎟ ⊆ GL r+1 (L) .
. ⎠
⎪⎩
⎪⎭
0 a r+1,2 . . . a r+1,r+1
Further, the natural homomorphism H 1 (G, F ′ ) → H 1 (G, PGL r+1 (L)) factors
via H 1 (G, GL r+1 (L)). The assertion follows from Lemma 5.11. □
5.10. Remark. –––– One can easily show that even H 1 (G, F ′ ) = 0. Indeed,
F 1 := { (a ij ) 1≤i,j≤r+1 | a ij = 0 for i ≥ 2, i ≠ j;
a 11 = a 22 = · · · = a r+1,r+1 = 1 }

Sec. 5] BRAUER-SEVERI VARIETIES AND NON-ABELIAN h 1 33
is a normal G-subgroup of F ′ . Thus, F ′ admits a filtration by G-subgroups,
each normal in the succeeding one, such that all the subquotients occurring are
isomorphic either to GL r (L) or to (L, +).
5.11. Lemma (Theorem Hilbert 90). —– Let L/K be a finite Galois extension,
G := Gal(L/K ) its Galois group, and n ∈Æ. Then, H 1 (G, GL n (L)) = 0.
Proof. Let (a σ ) σ∈G be a cocycle with values in GL n (L). We define a G-operation
from the left on L n as follows.
We let σ ∈ G act as
a σ ◦ σ : L n
σ
−→ L n
a σ
−→ L n .
Then, it is clear that a σ ◦ σ is a σ-linear map. Galois descent yields a K-vector
space V such that there is an isomorphism
making the diagrams
V ⊗ K L ∼ =
−→
b
L n
V ⊗ K L
σ
V ⊗ K L
b
b
L n
L n a σ ◦σ
commute. In particular, one has that dim K V = n. The choice of an isomorphism
V ∼ =
−→ K n
provides us with an element b ∈ GL n (L) such that b ◦ σ = a σ ◦ σ ◦ b. I.e.,
a σ = b ◦ σ ◦ b −1 ◦ σ −1 = b ◦ σ (b −1 )
for all σ ∈ G. (a σ ) σ∈G is cohomologous to the trivial cocycle.
□
5.12. Definition. –––– Let K be a field, r ∈Æ, and X be a Brauer-Severi variety
of dimension r. Then, a linear subspace of X is a closed subvariety Y ⊂ X such
that
Y × SpecK SpecK sep ⊂ X × SpecK SpecK sep ∼ = P
r
K sep
is a linear subspace of the projective space.
This property is independent of the isomorphism chosen.
5.13. Remark. –––– A K-valued point would be a zero-dimensional linear subspace.
But, except for the trivial case X ∼ = P r K
, there are no K-valued points.

34 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
Nevertheless, it may happen that there exist linear subspaces Y X of higher dimension.
They can be investigated by cohomological methods generalizing the
argument given above.
5.14. Proposition (F. Châtelet and M. Artin). —– Let K be a field, r, d ∈Æ,
X a Brauer-Severi variety of dimension r, and Y a linear subspace of dimension d.
Then, the natural boundary maps send the cohomology classes
α K r (X ) ∈ H 1( Gal(K sep /K ), PGL r+1 (K sep ) )
and
α K d (Y ) ∈ H 1( Gal(K sep /K ), PGL d+1 (K sep ) )
to the same class in the cohomological Brauer group H 2( Gal(K sep /K ), (K sep ) ∗) .
Proof. Let L be a common splitting field for X and Y. We may assume that
L/K is a finite Galois extension. Denote its Galois group by G.
We choose an isomorphism X × SpecK SpecL ∼ =
−→ P r L . We may assume without
restriction that the linear subspace
Y × SpecK SpecL ⊂ X × SpecK SpecL ∼ = P r L
is given by the homogeneous equations X d+1 = · · · = X r = 0. Then, the
cohomology class α L/K
r (X ) ∈ H 1( G, PGL r+1 (L) ) is given by a cocycle (α σ ) σ∈G
such that “X d+1 = · · · = X r = 0” is an invariant subspace for every α σ .
Consequently, α L/K
r (X ) is in the image of H 1( G, F/L ∗) under the natural
homomorphism for
{( )
E1 H
F :=
∈ GL r+1 (L)
0 E 2
∣ E }
1 ∈ GL d+1 (L), E 2 ∈ GL r−d (L),
.
H ∈ M (d+1)×(r−d) (L)
F comes equipped with the homomorphism of G-groups
p : F −→ GL d+1 (L),
( )
E1 H
↦→ E 1 .
0 E 2
Thus, we obtain a commutative diagram that connects the three central inclusions,
GL d+1 (L)
L ∗
p
L ∗
F
L ∗ GL r+1 (L) .
i

Sec. 6] CENTRAL SIMPLE ALGEBRAS AND BRAUER-SEVERI VARIETIES 35
Therefore, there is the following commutative diagram uniting the boundary
maps,
H 1 (G, PGL d+1 (L)) H 2 (G, L ∗ )
p ∗
H 1 (G, F/L ∗ )
H 2 (G, L ∗ )
i ∗
H 1 (G, PGL r+1 (L)) H 2 (G, L ∗ ).
Finally, recall that we have a cohomology class α ∈ H 1 (G, F/L ∗ ) such that
i ∗ (α) = αr
L/K (X ) and p ∗ (α) = αd L/K (Y ). □
6. Central simple algebras and Brauer-Severi varieties
6.1. Theorem. –––– Let n ∈Æ, K a field and A a central simple algebra over K
of dimension n 2 .
a) Then,thereexistsaBrauer-SeverivarietyX A ofdimension (n−1)overK satisfying
condition (∗). (∗) determines X A uniquely up to isomorphism of K-schemes.
(∗)If L is a splitting field for A which is finite and Galois over K then L is a
splitting field for X A , too. There is one and the same cohomology class
associated with A and X A .
a A = α XA ∈ H 1( Gal(L/K ), PGL n (L) )
b) The assignment X : A ↦→ X A admits the following properties.
i) X is compatible with extensions K ′ /K of the base field. I.e.,
X A⊗K K ′ ∼ = XA × SpecK SpecK ′ .
ii) L is a splitting field for A if and only if L is a splitting field for X A .
Proof. a) Uniqueness is clear from the results of the preceding sections.
Existence. Choose a finite Galois extension L of K which is a splitting field
for A. Take condition (∗) as a definition for X A . This is independent of the
choice of L by Corollaries 4.7.a) and 5.7.a).

36 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
b.i) Let K ′ /K be an arbitrary field extension. We have the obvious diagram of
field extensions
LK ′
K ′
L
K
and the canonical inclusion Gal(LK ′ /K ′ ) ⊆ Gal(L/K ). Under the constructions
given above, one assigns to the central simple algebra A ⊗ K K ′ and the
Brauer-Severivariety X × SpecK SpecK ′ the restrictions of the cohomology classes
assigned to A, respectively X. I.e.,
a (A⊗K K ′ ) = res Gal(LK ′ /K ′ )
Gal(L/K )
(a A ),
α (X ×SpecK SpecK ′) = resGal(LK ′ /K ′ )
Gal(L/K )
(α X ).
ii) The two statements are equivalent to
and
res Gal(LK ′ /K ′ )
Gal(L/K )
(a A ) ∈ H 1( Gal(LK ′ /K ′ ), PGL n (LK ′ ) ) = 0
res Gal(LK ′ /K ′ )
Gal(L/K )
(α XA ) ∈ H 1( Gal(LK ′ /K ′ ), PGL n (LK ′ ) ) = 0,
respectively. As we have a A = α XA , the assertion follows.
□
6.2. Corollary. –––– Let K be a field and A be a central simple algebra over K.
Then, K ′ is a splitting field for A if and only if X A (K ′ ) ≠ ∅.
Proof. “=⇒” is trivial and “⇐=” is an easy consequence of Proposition 5.9. □
6.3. Corollary. –––– i) Let K be a field and n ∈Æ. Then, X induces a bijection
X K n : AzK n → BS K n−1 .
ii) Let L/K be a field extension. Then, X induces a bijection
X L/K
n
: Az L/K
n
→ BS L/K
n−1 .
iii) These mappings are compatible with extensions of the base field. I.e., the diagram
Az K n
X K n
BS K n−1
⊗ K K ′
Az K ′
n
X K ′
n
BS K ′
n−1
× SpecK SpecK ′

Sec. 6] CENTRAL SIMPLE ALGEBRAS AND BRAUER-SEVERI VARIETIES 37
commutes for every field extension K ′ /K.
Proof. i) follows immediately from Theorem 6.1.a). ii) is a consequence of
Theorem 6.1.a) together with Theorem 6.1.b.ii). iii) is simply a reformulation
of Theorem 6.1.b.i).
□
6.4. Remark. –––– It may happen that two Brauer-Severi varieties X 1 , X 2 are
birationally equivalent but not isomorphic. S. A. Amitsur [Am55] proved in
this case that the corresponding central simple algebras A 1 and A 2 generate the
same cyclic subgroup of Br(K ). It is an open question whether the converse
is true. Interesting partial results have been obtained by P. Roquette [Roq64]
and S. L. Tregub [Tr].
6.5. Proposition. –––– Let K be a field, n ∈Æ, and A a central simple algebra of
dimension n 2 over K. Then, there is an isomorphism
x A : Aut K (A)
∼=
−→ Aut K-schemes (X A ).
Proof. Choose a splitting field L for A which is finite and Galois over K.
Put G := Gal(L/K ). Choose an isomorphism A ⊗ K L −→ f
M n (L). Then, for
each σ ∈ G, there is a commutative diagram
A ⊗ K L
f
M n (L)
σ
A ⊗ K L
f
a σ ◦σ
M n (L) .
The L-linear maps a σ : M n (L) → M n (L) form a cocycle for the cohomology
class a A ∈ H 1 (G, PGL n (L)).
Further, by Galois descent, to give an element of Aut K (A) is equivalent to giving
an element of PGL n (L) which is invariant under a σ ◦ σ for every σ ∈ G.
As α XA = a A , there exists an isomorphism X A × SpecK SpecL −→ f ′
P n−1
L such
that the diagrams
X A × SpecK SpecL
f ′
P n−1
L
σ
X A × SpecK SpecL
f ′
P n−1
L
commute. Therefore, by Galois descent for morphisms of schemes, it is equivalent
to give an element of Aut K-schemes (X A ) or to give an element of PGL n (L)
which is invariant under a σ ◦ σ for every σ ∈ G.
□
a σ ◦σ

38 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
6.6. Proposition (F. Châtelet and M. Artin). —– Let K be a field, n, d ∈Æ,
and A be a central simple algebra of dimension n 2 over K.
Then, the Brauer-Severi variety X A associated with A admits a linear subspace of
dimension d if and only if d ≤ n − 1 and
d ≡ −1
(mod ind(A)).
Proof. We write A = M m (D) with a skew field D and put e := ind(A).
Clearly, n = me.
“=⇒” Let H ⊂ X A be a linear subspace of dimension d. By Proposition 5.14,
we know H ∼ = X A ′ for some central simple algebra A ′ which is similar to A.
I.e.,A ′ ∼ = M k (D)for acertaink ∈Æ. Itfollowsthatdim A ′ = k 2·dim D = k 2·e 2
and dim H = dim X A ′ = k · e − 1. This implies the congruence desired.
Further, we have dim X A = n − 1. Consequently,
d = dim H ≤ dim X A = n − 1.
“⇐=”LetL beasplittingfieldforA whichisfiniteand GaloisoverK. Denoteits
Galois group by G. Further, let k be the natural number such that d = k ·e −1.
By assumption, k · e − 1 = d ≤ n − 1 = m · e − 1, hence k ≤ m.
We consider the cohomology class a D ∈ H 1 (G, PGL e (L)), associated with D.
By Proposition 4.9, one has a A = (i e me ) ∗(a D ) where i e me : PGL e(L) → PGL me (L)
is the block-diagonal embedding. Let (a σ ) σ∈G be a cocycle representing the
cohomology class a D . Then, (i e me (a σ)) σ∈G is a cocycle that represents a A .
We define a G-operation on P me−1
L
by letting σ ∈ G act as
i e me (a σ) ◦ σ : P me−1
L
σ
P me−1
L
i e me(a σ )
This is a group operation since (i e me (a σ)) σ∈G is a cocycle.
P me−1
L .
The geometric version of Galois descent yields the Brauer-Severi variety X A .
Further, the G-operation keeps invariant the linear subspace defined by the
homogeneous equations X ke = X ke+1 = · · · = X me−1 = 0. Thus, Galois
descent may be applied to this subspace, too. It yields a variety Y of dimension
ke − 1 = d.
Galois descent for morphisms of schemes, applied to the canonical embedding,
yields a morphism Y −→ X A . This is a closed immersion as that property
descends under faithful flat base change. Consequently, Y is a linear subvariety
of X A of the dimension desired.
□

Sec. 7] FUNCTORIALITY 39
7. Functoriality
7.1. Remark. –––– The preceding results suggest that X : A ↦→ X A should
somehow be a functor. This is indeed the case. But for that, the construction
of the Brauer-Severi variety associated with a central simple algebra, which was
given above, is not sufficient. The problem is that X A is determined by its
cohomology class only up to isomorphism, not up to unique isomorphism.
Thus, in order to make X a functor, it would still be necessary to make choices.
For that reason, we aim at a more natural description of X A .
7.2. Lemma (A. Grothendieck). —– Let n ∈Æand R a commutative ring
with unit. Then, there is a bijection
{ }
submodules M in R
κ R : P n−1
R (R) −→ G(R) :=
n such that R n /M
is a locally free R-module of rank (n − 1)
subject to the conditions below.
i) κ R is natural in R. I.e., for every ring homomorphism i : R → R ′ , the diagram
commutes.
P n−1
R (R)
P n−1 (i)
P n−1
R ′ (R ′ )
κ R
κ R ′
G(R)
G(R ′ )
G(i): M ↦→M ⊗ R R ′
ii) For every a ∈ PGL n (R), the canonical operations of a on P n−1
R (R) and G(R)
are compatible with κ R . That means that the diagrams
commute.
P n−1
R (R)
a
P n−1
R
(R)
κ R
κ R
G(R)
a
G(R)
Here, a ∈ PGL n (R) acts on a submodule M ⊂ R n by matrix multiplication from
the left. I.e., by M ↦→ a · M for a ∈ GL n (R) a representative of a.
Proof. A submodule M ⊆ R n such that the quotient R n /M is locally free of
rank (n − 1) defines an R-valued point in P n−1
R .
To see this, let first m ⊆ R be an arbitrary maximal ideal. Then, Rm/M n m is a
free R m -module of rank (n − 1). In particular, it is projective. Hence, M m is
a direct summand of Rm n and, therefore, projective and of finite presentation.
Consequently, by [Mat, Theorem 7.12], M m is a free R m -module of rank one.
As R n /M is R-flat, there is an exact sequence
0 −→ M ⊗ R (R/m) −→ R n /mR n −→ R n /(M + mR n ) −→ 0.

40 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
Thus, the canonical map M/mM → R n /mR n is injective. I.e., mM = mR n ∩M.
In particular, M can not be contained in mR n . Hence, the R m -module M m
which is free of rank one is not contained in mRm.
n Consequently, if
M m = 〈(r 0 , . . . , r n−1 )〉 then there is some α ∈ {0, . . . , n − 1} such that
r α ∈ R m is a unit.
This is equivalent to
R n m/(M m + e 0 · R m + · · · + e α−1 · R m + e α+1 · R m + · · · + e n−1 · R m ) = 0
where e 0 , . . . , e n−1 ∈ R n m denote the standard elements with an index shift
by (−1). The latter is an open condition on m by [Mat, Theorem 4.10].
Therefore, there exists some f ∈ R\m such that
R n f = M f + e 1 · R f + · · · + e α−1 · R f + e α+1 · R f + · · · + e n · R f .
I.e., such that the (α + 1)-th projection M f → R f is surjective.
As, on both sides, there are locally free modules of rank one, this must
be an isomorphism. Consequently, M f is free and there is a generator of
type (r 0 , . . . , r α−1 , 1, r α+1 , . . . , r n−1 ). Taking the entries as homogeneous
coordinates, we get a morphism SpecR f → P n−1
R .
It is easy to see that all these can be glued together to give a morphism
SpecR → P n−1
R of R-schemes.
Conversely, an R-valued point SpecR → P n−1
R defines a quotient module of R n
of rank (n − 1) as follows. Cover SpecR by affine, open subsets
SpecR =
n−1
[
α=0
SpecR α
such that for each α the image i(SpecR α ) is contained in the standard affine
set “X α ≠ 0”. Then, i| SpecRα can be given in the form
(r 0 : r 1 : . . . : r α−1 : 1 : r α+1 : . . . : r n−1 ) .
But R n α/(r 0 , r 1 , . . . , r α−1 , 1, r α+1 , . . . , r n−1 ) is a free R α -module of rank (n−1)
for trivial reasons.
The compatibility stated as i) follows directly from the construction given.
ii) is clear.
7.3. Definition. –––– Let R be a commutative ring with unit.
i) Let X be an R-scheme. Then, by
P X : {R-schemes} op → {sets}
□

Sec. 7] FUNCTORIALITY 41
we will denote the functor defined by
T ↦→ Mor R-schemes (T , X )
on objects and by composition on morphisms.
ii) Let F : {R-schemes} op → {sets} be any functor. If, for some R-scheme X,
∼=
there is an isomorphism ι: F −→ P X then we will say F is represented by X.
Having fixed the isomorphism ι then, by Yoneda’s Lemma, the K-scheme representing
the functor is determined up to unique isomorphism.
7.4. Remarks. –––– a) The functors P X depend on X in a natural manner.
I.e., if p : X → X ′ is a morphism of R-schemes then there is a morphism of
functors p ∗ : P X → P X ′ given by composition with p. Therefore, there is a
covariant functor
P : {R-schemes} −→ Fun ({R-schemes} op , {sets})
described by X ↦→ P X on objects.
b) The functor P X induced by an R-scheme X is exactly what in category theory
is usually called the Hom-functor. Nevertheless, we will follow the standards in
Algebraic Geometry and refer to it as the functor of points. Note that P X (T ) is
the set of all T -valued points on X.
7.5. Corollary (A. Grothendieck). —– Let n ∈Æand R be a commutative ring
with unit. Then, the functor
P n−1
R : {R-schemes} −→ {sets}
T ↦→ P n−1
R (T ) :=
is represented by the R-scheme P n−1
R .
⎧
⎫
⎨ subsheaves M in OT n such that ⎬
OT n /M is a locally free
⎩
⎭
O T -module of rank (n − 1)
Proof. It is clear that both functors satisfy the sheaf axiom for Zariski coverings.
Therefore, it is sufficient to construct the isomorphism on the full subcategory
of affine R-schemes. This is exactly what is done in Lemma 7.2. □
7.6. Corollary-Definition. –––– Let n ∈Æ, L be a field, and F be an L-vector
space of dimension n. Put F := ˜F to be the coherent sheaf associated to F
on SpecL.

42 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
Then, the functor
P(F ): {L-schemes} −→ {sets}
⎧
⎫
⎨ subsheaves M ⊂ π ∗ F such that ⎬
(π : T → SpecL) ↦→ P(F )(π) := π ∗ F/M is a locally free
⎩
⎭
O T -module of rank (n − 1)
is representable by an L-scheme which is isomorphic to P n−1
L . We will denote
that scheme by P(F ) and call it the projective space of lines in F.
7.7. Remark. –––– If L ′ is a field containing L then P(F ⊗ L L ′ ) is naturally
isomorphic to the functor P(F )| L ′ -schemes. Therefore, there is a canonical isomorphism
P(F ⊗ L L ′ ∼=
) −→ P(F ) × SpecL SpecL ′
between the representing objects. For each L ′ -scheme U, the diagram of natural
mappings
Mor L ′ -schemes(U, P(F )) {M ⊂ πU ∗ F }
∼=
Mor L ′ -schemes(U, P(F ⊗ L L ′ )) {M ⊂ πU ∗ F }
commutes.
Consequently, for each L-scheme T , there is a commutative diagram
Mor L-schemes (T , P(F ))
{M ′ ⊂ π ∗ T F }
comp. with projection
Mor L ′ -schemes(T × SpecL SpecL ′ , P(F ))
pull-back
∼=
Mor L ′ -schemes(T × SpecL SpecL ′ , P(F ⊗ L L ′ )) {M ⊂ π ∗ T × SpecL Spec L ′F } .
Note that the column on the left is, up to the canonical isomorphism above,
exactly the base extension from SpecL to SpecL ′ .
7.8. Definition. –––– Let L/K be a finite Galois extension of fields and
G := Gal(L/K ) be its Galois group.
i) By L-Vect K , we will denote the category of all finite dimensional
L-vector spaces. As morphisms, we allow all injections which are σ-linear
for a certain σ ∈ G.
ii) L-Vect K 0 is a subcategory ofL-Vect K which consists of the same class of objects.
In L-Vect K 0 , only the bijections are taken as morphisms.

Sec. 7] FUNCTORIALITY 43
7.9. Definition. –––– Let H be a group. Then, by H we will denote the category
consisting of exactly one object ∗ such that Mor(∗, ∗) = H .
7.10. Definition. –––– Let L/K be a finite Galois extension of fields and G be
its Galois group. By Sch L/K , we will denote the category of all L-schemes with
morphisms twisted by any element of G.
We note that Sch L/K has a canonical structure of a fibered category over G.
The pull-back under σ ∈ G is given by X ↦→ X × SpecL SpecL σ−1 .
7.11. Lemma. –––– Let L/K be a finite Galois extension of fields and G be its
Galois group.
i) Then, P is a covariant functor from L-Vect K to Sch L/K . Here, a σ-linear
monomorphism i : F → F ′ for σ ∈ G induces a morphism
which is twisted by σ.
i ∗ = P(i): P(F ) −→ P(F ′ )
ii) LetF beanL-vectorspaceand i : F → F bethemultiplicationwithx ∈ L, x ≠ 0.
Then, i ∗ : P(F ) → P(F ) is equal to the identity morphism.
Proof. i) Let i : F → F ′ be a σ-linear monomorphism. We have to consider
the diagram
P(F ) P(F ′ )
SpecL
S(σ)
SpecL .
By Yoneda’s Lemma, there has to be constructed a natural transformation
i + : P(F ) → P(F ′ )(S(σ) ◦ · ) .
I.e., for each L-scheme π : T → SpecL there is a mapping
i + (π): P(F )(π) → P(F ′ )(S(σ) ◦ π)
to be given such that for each morphism p : π 1 → π 2 of L-schemes the diagram
(†)
P(F )(π 2 )
i + (π 2 )
P(F ′ )(S(σ) ◦ π 2 )
commutes.
P(F )(p)
P(F )(π 1 )
i + (π 1 )
P(F ′ )(S(σ)◦p)
P(F ′ )(S(σ) ◦ π 1 )

44 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
We construct i + (π) as follows. There is the description
P(F )(π) = M ⊂ π ∗ F | π ∗ F/M is locally free of rank (dim F − 1) }
{ }
M ⊂ (S(σ) ◦ π)
=
∗ S(σ −1 ) ∗ F |
(S(σ) ◦ π) ∗ S(σ −1 ) ∗ F/M is locally free of rank (dim F − 1) .
Here, S(σ −1 ) ∗ F = ˜F σ for F σ equal to F as an abelian group and equipped
with the scalar multiplication given by l · f := σ(l) f . Hence, i gives rise
to an L-linear monomorphism i : F σ → F ′ and, therefore, to a morphism
ĩ : S(σ −1 ) ∗ F → F ′ of sheaves on SpecL.
If M is a subsheaf of (S(σ) ◦ π) ∗ S(σ −1 ) ∗ F such that (S(σ) ◦ π) ∗ S(σ −1 ) ∗ F/M
is locally free of rank (dim F − 1) then ( (S(σ) ◦ π) ∗ (ĩ) ) (M ) is a subsheaf of
(S(σ)◦π) ∗ F ′ such that the quotient is locally free of rank (dim F ′ −1). This gives
rise to a morphism of functors i + : P(F ) −→ P(F ′ )(S(σ) ◦ · ) as desired.
By consequence, we have a morphism i ∗ : P(F ) −→ P(F ′ ) of schemes making
the diagram (†) commute.
ii) is clear from the construction.
7.12. Corollary. –––– Let L/K be a finite Galois extension of fields and
G := Gal(L/K ) its Galois group.
i) Then, P can be made into a contravariant functor from L-Vect K 0 to Sch L/K .
Here, a σ-linear isomorphism i : F → F ′ for σ ∈ G induces a morphism
which is twisted by σ −1 .
i ∗ : P(F ′ ) → P(F )
ii) LetF beanL-vectorspaceand i : F → F bethemultiplicationwithx ∈ L, x ≠ 0.
Then, i ∗ : P(F ) → P(F ) is equal the identity morphism.
Proof. For an isomorphism i : F → F ′ , put i ∗ := i −1
∗ . □
7.13. Remark. –––– The morphisms i ∗ can also be constructed directly. Indeed,
the task is to give a natural transformation i + : P(F ′ ) → P(F )(S(σ −1 ) ◦ · ).
There are the descriptions
and
P(F ′ )(π) = { M ⊂ π ∗ F ′ | π ∗ F ′ /M is locally free of rank (dim F ′ − 1) }
P(F )(S(σ −1 ) ◦ π) =
{ }
M ⊂ π ∗ S(σ −1 ) ∗ F |
π ∗ S(σ −1 ) ∗ .
F/M is locally free of rank (dim F − 1)
If M is a subsheaf of π ∗ F ′ such that the quotient π ∗ F ′ /M is locally free of
rank (dim F ′ − 1) = (dim F − 1) then π ∗ (ĩ) −1 (M ) is a subsheaf of π ∗ S(σ −1 ) ∗ F
□

Sec. 7] FUNCTORIALITY 45
such that the quotient is locally free of rank (dim F − 1). This gives rise to a
morphism of functors i + : P(F ′ ) −→ P(F )(S(σ −1 ) ◦ · ) as desired.
7.14. Remark. –––– If L is a field and A is a central simple algebra of dimension
n 2 over L then all the non-zero, simple, left A-modules are isomorphic
to each other. Further, if l is a non-zero, simple, left A-module then each
automorphism of l is given by multiplication with an element of the center
of A.
Hence, every automorphism of l induces the identity map on P(l). In particular,
two arbitrary isomorphisms i 1 , i 2 : l → l ′ between non-zero, simple,
left A-modules induce one and the same isomorphism i ∗ 1 = i ∗ 2 : P(l′ ) → P(l).
Thismeans, A determinesP(l)notonlyuptoisomorphism, butuptouniqueisomorphism.
7.15. Definition. –––– Let r ∈Æand L/K be a finite Galois extension.
Denote the Galois group Gal(L/K ) by G.
i) By Mat L/K
r , we will denote the category of all split central simple algebras of
dimension r 2 over L. I.e., of all algebras isomorphic to M r (L). As morphisms,
wetakeallhomomorphismsofK-algebraswhichareσ-linear for acertain σ ∈ G
and preserve the unit element.
ii) P L/K
r will denote the subcategory of Sch L/K consisting of all L-schemes isomorphic
to the projective space P r L. As morphisms, P L/K
r allows only the isomorphisms.
7.16. Remark. –––– In Mat K r
two objects are isomorphic.
, every morphism is an isomorphism and every
7.17. Proposition. –––– Let n ∈Æand L/K be a finite Galois extension.
Denote the Galois group Gal(L/K ) by G.
: Mat L/K
n → P L/K
n−1 .
is given by A ↦→ P(l) where l ⊂ A is a non-zero, simple,
i) There is an equivalence of categories Ξ L/K
n
On objects, Ξ L/K
n
left A-module. If i : A → A ′ is a morphism in Mat L/K
n
morphism of schemes induced by the canonical homomorphism l → A ′ ⊗ A l.
then Ξ L/K
n
(i) is the
ii) If i : A → A ′ is σ-linear for σ ∈ G then Ξ L/K
n (i) is a morphism twisted by σ.
Proof. If l is a non-zero, simple, left A-module then A ′ ⊗ A l is a non-zero, simple,
left A ′ -module. Both are n-dimensional L-vector spaces. Up to isomorphism,
these modules are unique. Therefore, the morphism of schemes
i l∗ := Ξ L/K
n
(i): Ξn
L/K (A) = P(l) −→ P(A ′ ⊗ A l) = Ξn L/K (A ′ )

46 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
induced by the homomorphism
i l : l → A ′ ⊗ A l,
x ↦→ 1 ⊗ x
is well-defined.
If i : A → A ′ is a σ-linear homomorphism then l → A ′ ⊗ A l is a σ-linear
homomorphism of vector spaces. By Lemma 7.11, the morphism Ξn
L/K (i) is
twisted by σ. This proves ii).
As i is invertible, Ξ L/K
n
(i) is an isomorphism of schemes. Consequently, Ξn
L/K is
is an equivalence of
a functor between the categories described. To prove Ξn
L/K
categories, we have to show it is fully faithful and essentially surjective. As in
P L/K
n−1
every two objects are isomorphic, essential surjectivity is clear.
For full faithfulness, it will suffice to prove that
Ξ L/K
n | Aut(A) : Aut(A) −→ Aut(X A )
is an isomorphism of groups in the case A = M n (L). For that, note
and
Thus, the functor Ξ L/K
n
Aut(A) = PGL n (L) ⋊ G
Aut(Ξn
L/K (A)) = Aut K-schemes (P n−1
L ) = PGL n (L) ⋊ G .
induces on Aut(A) a group homomorphism
Ξ: PGL n (L) ⋊ G → PGL n (L) ⋊ G .
By Lemma 7.2.ii), the restriction of Ξ to PGL n (L) is the identity. Statement
ii) shows that the quotient map G → G is the identity, too. Consequently,
Ξn L/K | Aut(A) is an isomorphism. □
7.18. Corollary. –––– The categories Mat L/K
n
and P L/K
n−1
are anti-equivalent, too.
There is an equivalence of categories ˇΞ n
L/K : Mat L/K
n → (P L/K
n−1 )op given on objects by
A ↦→ P(l) where l ⊂ A is a non-zero, simple, left A-module.
If i : A → A ′ is σ-linear for σ ∈ G then ˇΞ L/K
n (i) is twisted by σ −1 .
Proof. Put simply ˇΞ n
L/K (i) := (Ξn L/K (i)) −1 . □
7.19. Proposition. –––– Let n ∈Æ, K a field and A be a central simple algebra of
dimension n 2 over K. Then, the Brauer-Severi variety X A associated with A may
be described as follows.
Let L be a splitting field for A which is finite and Galois over K. By covariant
functoriality, the canonical Gal(L/K )-operation on A ⊗ K L induces an operation

Sec. 7] FUNCTORIALITY 47
on the projective space Ξn
L/K (A ⊗ K L). Here, σ ∈ Gal(L/K ) acts by a morphism
twisted by σ. The geometric version of Galois descent yields the K-scheme X A .
Proof. Choose an isomorphism f : A ⊗ K L → M n (L). Then, there is a cocycle
(a σ ) σ∈G from G to PGL n (L) such that for each σ ∈ G the diagram
A ⊗ K L
f
M n (L)
to the whole situation, we obtain com-
commutes. Applying the functor Ξ L/K
n
mutative diagrams
Ξ L/K
n (A ⊗ K L)
Ξ L/K
n ( f )
P n−1
L
σ
a σ ◦σ
A ⊗ K L
f
M n (L)
σ
Ξ L/K
n (A ⊗ K L)
a σ ◦σ
P n−1
L
Ξ L/K
n ( f )
such that the vertical arrows are isomorphisms. Galois descent on the lower half
of the diagram yields the description of X A given in Section 6. Galois descent
on the upper half of the diagram yields the description claimed. □
7.20. Corollary. –––– Let L/K be a field extension and A be a central simple
algebra over K.
i) Then, there is a canonical isomorphism of L-schemes
ξ L/K
A : X A⊗K L −→ X A × SpecK SpecL .
are com-
ii) For field extensions L ′ /L/K, the isomorphisms ξ L′ /K
A , ξ L′ /L
A , and ξ L/K
A
patible. I.e., the diagram
X A⊗K L ′ ξ L′ /K
A
commutes.
ξ L′ /L
X
A
A × SpecK SpecL ′
X A⊗K L × SpecL SpecL ′
ξ L/K
A × SpecLSpecL ′
Proof. Let n 2 = dim A. We choose a splitting field L ′′ for A containing L ′ .
Then, X A , X A⊗K L, and X A⊗K L ′ are constructed from
Ξ L′′ /K
n (A ⊗ K L ′′ ) = Ξ L′′ /L
n (A ⊗ K L ′′ ) = Ξ L′′ /L ′
n (A ⊗ K L ′′ )
by Galois descent using compatible descent data.
□

48 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
7.21. Definition. –––– Let r ∈Æand K be a field.
i) By Az K r , we will denote the category of all central simple algebras of dimension
r 2 over K. As morphisms, we take the homomorphisms of K-algebras
preserving the unit element. Note, in Az K r every morphism is an isomorphism.
Az /K
r will denote the category of all central simple algebras of dimension r 2
over any field extension of K. As morphisms, we take the K-algebra homomorphisms
which preserve the unit element.
ii) By BS K r , we will denote the category of all Brauer-Severi varieties of dimension
r over K. The isomorphisms of K-schemes are taken as morphisms.
BS /K
r
will denote the category of all Brauer-Severi varieties of dimension r over
any field extension of K. As morphisms, we take the compositions of an
isomorphism of K-schemes with a morphism of type
id × Speca : X × SpecL SpecL ′ −→ X.
Here, a : L → L ′ is a homomorphism of fields.
7.22. Theorem. –––– Let K be a field and n ∈Æ.
i) There is an equivalence of categories
X /K
n
satisfying the following condition.
: Az/K n −→ (BS /K
n−1 )op
For each field extension L/K, on isomorphism classes, the functor X /K
n
bijection Xn L : Az L n → BS L n−1 from Corollary 6.3.i).
induces an equiva-
ii) In particular, for each field extension L/K, the functor X /K
n
lence of categories X L n : Az L n → (BS L n−1) op .
Proof. First step. Construction of the functor.
induces the
For A ∈ Ob(Az /K
n ), we put X /K
n (A) := X A . We are going to make use of the
intrinsic description of X A given in Proposition 7.19.
If i : A → A ′ is a morphism in Az n
/K then, by restriction to the centers, i induces
a homomorphism i| Z(A) : Z(A) → Z(A ′ ) of fields extending K. Therefore, there
is a unique factorization
A cZ(A′ )
A
A ⊗ Z(A) Z(A ′ )
j
A ′
of i via the canonical inclusion c Z(A′ )
A .
By Corollary 7.20, one has the canonical isomorphism
ξ Z(A′ )/Z(A)
A : X A⊗Z(A) Z(A ′ ) −→ X A × SpecZ(A) SpecZ(A ′ ).

Sec. 7] FUNCTORIALITY 49
We put X /K
n (c Z(A′ )
A ) to be the morphism of schemes induced from ξ Z(A′ )/Z(A)
A by
projection to the first factor.
In order to construct X /K
n (i) as a functor on the category Az/K n , it remains
to describe it on the full subcategories Az K 1
n for all field extensions K 1 /K.
That means, we are left with the case that Z(A) = Z(A ′ ) and i| Z(A) is the identity.
In order to ensure functoriality, the construction has to be made compatibly
with field extensions. I.e., such that the diagram
X A ′ ⊗ K1 K 2
X /K
n (i⊗ K1 K 2 )
X /K
n (c K 2
A ′ )
X A ′
X /K
n (i)
X A⊗K1 K 2
X A
X /K
n (c K 2
A )
commutes for every field K 1 containing K, every morphism i : A → A ′ in Az K 1
n ,
and every extension K 2 /K 1 of fields containing K.
For this, we choose a splitting field L for A. We may assumeL is finite and Galois
over K 1 . As i is automatically an isomorphism, L is a splitting field for A ′ , too.
Thus, weobtainaGaloisinvarianthomomorphismi⊗ K1 L : A⊗ K1 L → A ′ ⊗ K1 L.
Applying the functor Ξ L/K 1
n , one gets a Galois invariant morphism of schemes
X A ′ × SpecK1 SpecL = Ξ L/K 1
n (A ′ ⊗ K1 L)
Ξ L/K 1
Ξ L/K 1
n (i⊗ L M )
n (A ⊗ K1 L) = X A × SpecK1 SpecL .
Galois descent for morphisms of schemes yields the morphism
X /K
n (i): X A ′ → X A
desired. X /K
n (i) is an isomorphism of schemes as i is. Consequently, X /K
n is a
functor between the categories stated.
By construction, X /K
n is essentially surjective.
We also note that, for each field extension K ′ /K, X /K
n induces a functor
X K ′
n : AzK ′
n −→ (BS K ′
n−1 )op .
Second step. Full faithfulness on automorphisms.
Let us deal with the following statement which is a special case of full faithfulness
of X /K
n .
(‡) Let K ′ be a field containing K and A be a central simple algebra of dimension
n 2 over K ′ . Then, the functor X K ′
n induces an isomorphism of groups
x A := X K ′
n | AutK ′ (A) : Aut K ′(A) −→ Aut K ′ -schemes(X A ).

50 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
In the case A = M n (K ′ ), this was proven in Proposition 7.17.
Let now A be a general central simple algebra of dimension n 2 over K ′ . Choose a
splitting field L for A which is finite and Galois over K ′ .
Injectivity. Assume a, a ′ ∈ Aut K ′(A) induce one and the same morphism
a ∗ = a ′∗ ∈ Aut K ′ -schemes(X A ) .
There are unique homomorphisms a L , a ′ L : A⊗ K L → A⊗ K L of central simple
algebras over L making the diagrams
A ⊗ K L
a L A ⊗K L A ⊗ K L
a ′ L
A ⊗ K L
c L A
A
a
A ,
c L A
commute. Applying the functor X K ′
n , we obtain the commutative diagrams below,
X A⊗K L
(c L A )∗
a ∗ L
a ∗ =a ′∗
X A⊗K L
(c L A )∗
c L A
A
X A⊗K L
(c L A )∗
a ′ L ∗
a ′
a ∗ =a ′∗
A
X A⊗K L
X A X A , X A X A .
(cA L)∗ is, up to isomorphism, the canonical morphism X A × SpecK SpecL → X A
from the fiber product to its first factor. Therefore, the morphism
a ∗ ◦ (cA L)∗
= a ′∗ ◦ (cA L)∗ admits a unique factorization into a morphism of
L-schemes and (cA) L ∗ . This implies a ∗ L = a L ′ ∗ .
SinceA⊗ K L ∼ = M n (L), wemayconcludea L = a L ′ from this. Theequalitya = a′
follows immediately.
By consequence, the homomorphism x A is injective.
Surjectivity. Let p : X A → X A be an automorphism of the K ′ -scheme X A .
(cA L)∗ is, up to isomorphism, the canonical projection X A × SpecK SpecL → X A
from the fiber product to its first factor. Therefore, the composition
p ◦ (cA L)∗ factors uniquely via (cA L)∗ . I.e., there exists a unique morphism
p L : X A⊗K L → X A⊗K L of L-schemes such that there is a commutative diagram
c L A
(c L A )∗
X A⊗K L
p L
X A⊗K L
(c L A )∗
p
X A X A .
As A⊗ K L ∼ = M n (L), Proposition 7.17 shows there exists some b ∈ Aut(A⊗ K L)
such that p L = b ∗ .
(c L A )∗

Sec. 7] FUNCTORIALITY 51
For σ ∈ Gal(L/K ′ ), let id × σ : A ⊗ K L → A ⊗ K L be the corresponding
automorphism of A ⊗ K L. Clearly, (id × σ) ◦ cA L = cA L . Therefore, the diagram
X A⊗K L
b ∗ =p L
X A⊗K L
(id×σ) ∗
X A⊗K L
b ∗ =p L
X A⊗K L
(id×σ) ∗
commutes, too. Hence,
(c L A )∗
p
X A X A .
(c L A )∗
b ∗ = (id × σ) ∗ ◦ b ∗ ◦ (id × σ −1 ) ∗ = ((id × σ −1 ) ◦ b ◦ (id × σ)) ∗ .
The injectivity of (‡) shows b = (id × σ −1 ) ◦ b ◦ (id × σ). I.e., b is invariant
with respect to the operation of Gal(L/K ′ ).
By Galois descent for homomorphisms, there exists a homomorphism
a : A → A of central simple algebras such that the diagram
A ⊗ K L
b
A ⊗ K L
commutes.
c L A
A
a
By consequence, there is a commutative diagram on the level of Brauer-Severi
varieties as follows,
X A⊗K L
(c L A )∗
b ∗ =p L
a ∗
A
c L A
X A⊗K L
X A X A .
A direct comparison with the definition of p L shows p ◦ (c L A) ∗ = a ∗ ◦ (c L A) ∗ .
Indeed, both compositions are equal to (c L A )∗ ◦ p L . Since (c L A )∗ is dominant, this
implies p = a ∗ .
The homomorphism x A is surjective, too.
Third step. Isomorphisms.
Let K ′ ⊇ K be a field and A and A ′ be central simple algebras of dimension n 2
over K ′ such that A ∼ = A ′ . Then, X K ′
n induces a bijection
(c L A )∗
Iso K ′ (A, A ′ ) −→ Iso K ′ -schemes(X A ′, X A ) .
This is an immediate consequence of the results from the previous step.

52 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
Fourth step. Faithfulness.
Assume the homomorphisms i 1 , i 2 : A → A ′ induce the same morphism of
schemes i ∗ 1 = i ∗ 2 : X A ′ → X A.
We first observe that i 1 and i 2 give rise to the same homomorphism
i 1 | Z(A) = i 2 | Z(A) : Z(A) → Z(A ′ )
between centers. Indeed, X A and X A ′ are Brauer-Severi varieties over Z(A)
and Z(A ′ ), respectively. Further, by Proposition 5.2.iv), Γ(X A , O XA ) = Z(A)
and Γ(X A ′, O XA ′) = Z(A ′ ). I.e., one can recover the centers of A and A ′ from
the Brauer-Severi varieties associated with them. By the construction of X /K
n
on morphisms, the pull-back on global sections
(i ∗ 1 )♯ = (i ∗ 2 )♯ : Z(A) = Γ(X A , O XA ) −→ Γ(X A ′, O XA ′
) = Z(A ′ )
is equal to the homomorphism Z(A) → Z(A ′ ) given by i 1 , respectively i 2 , when
restricted to the centers.
Consequently, both i 1 and i 2 can be factorized via the canonical homomorphism
c Z(A′ )
A : A → A ⊗ Z(A) Z(A ′ ) .
Let j 1 , j 2 : A⊗ Z(A) Z(A ′ ) → A ′ bethehomomorphismsofcentralsimplealgebras
over Z(A ′ ) such that j 1 ◦ c Z(A′ )
A = i 1 and j 2 ◦ c Z(A′ )
A = i 2 . Both j 1 and j 2 are
isomorphisms as they are homomorphisms of central simple algebras over the
same base field.
On the other hand, the morphisms
j ∗ 1 , j ∗ 2 : X A ′ −→ X A⊗Z(A) Z(A ′ ) = X A × SpecZ(A) SpecZ(A ′ )
coincide. Indeed, their projections to SpecZ(A ′ ) do, both being the structural
morphism. Further, the projections to X A are equal to i ∗ 1 = i ∗ 2 .
This implies j 1 = j 2 .
Fifth step. Fullness.
LetA, A ′ ∈ Ob (Az /K
n )and f : X A ′ → X A beamorphism inthecategoryBS /K
n−1 .
The corresponding map of global sections is a homomorphism Z(A) → Z(A ′ )
of fields. From the definition of the category BS /K
n−1
, we see that f gives rise to
an isomorphism of Z(A ′ )-schemes
such that f = (c Z(A′ )
A ) ∗ ◦ f .
f : X A ′ −→ X A × SpecZ(A) SpecZ(A ′ ) = X A⊗Z(A) Z(A ′ )

Sec. 8] THE FUNCTOR OF POINTS 53
By virtue of Corollary 6.3, the central simple algebras A ′ and A ⊗ Z(A) Z(A ′ )
over Z(A ′ ) are isomorphic. In the third step, we showed there is an isomorphism
a : A ⊗ Z(A) Z(A ′ ) −→ A ′
such that f = a ∗ . Consequently, f = (c Z(A′ )
A ) ∗ ◦ a ∗ = (a ◦ c Z(A′ )
A ) ∗ is in the
image of X /K
n on morphisms. X /K
n is full. □
7.23. Corollary. –––– Let K be a field and n ∈Æ. Then, there is an equivalence
of categories
ˇX K n : AzK n −→ BS K n−1 .
Proof. Compose X K n with the equivalence of categories ι: BSK n−1 → (BS K n−1 )op
given by the identity on objects and by ι(g) := g −1 on morphisms. □
8. The functor of points
8.1. Remark. –––– This section deals with the contravariant functor of points
on the category of all K-schemes defined by the Brauer-Severi variety X A .
It turns out that this functor may be described completely explicitly in terms of
the central simple algebra A.
Thus, there is a different method to introduce X A . One could start with the
functor and has to prove its representability by a scheme. It seems that this
method is closer to A. Grothendieck’s style in Algebraic Geometry than the
approach presented here. Unfortunately, the proof of representability is not
trivial at all. It is presented in detail in [Hen] or [Ke].
8.2. Definition. –––– Let K be a field, n ∈Æ, and A ∈ Ob (Az K n ).
Then, by
I A : {K-schemes} op → {sets} ,
we will denote the functor given by
{ }
sheaves of right ideals J in π
∗
T ↦→
T A such that
πT ∗ A /J is a locally free O T -module of rank n 2 − n
on objects and by pull-back on morphisms.
Here, A := Ã is the sheaf of O K-algebras associated with A on SpecK.
π = π T : T → SpecK denotes the structural morphism.

54 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
8.3. Remark. –––– The various functors I A depend on A in a natural manner.
I.e., if i : A → A ′ is a morphism in Az K n then there is a morphism of functors
i ∗ : I A ′ → I A given by the inverse image
J ↦→ π ∗ T (ĩ) −1 (J ) .
In other words, there exists a contravariant functor
I : Az K n −→ Fun ({K-schemes} op , {sets})
given on objects by A ↦→ I A .
8.4. Theorem. –––– Let K be a field and n ∈Æ. Then, there is an isomorphism
between the contravariant functors
ι: I −→ P ◦ X K n
I, P ◦ X K n : AzK n −→ Fun ({K-schemes} op , {sets}) .
8.5. Remark. –––– For a fixed central simple algebra A of dimension n 2 over K,
Theorem 8.4 asserts that there is an isomorphism ι A : I A −→ P XA between
the functors
I A , P XA : {K-schemes} op → {sets} ,
described in Definitions 8.2 and 7.3.
This means, the T -valued points on X A are in natural bijection to the
sheaves J ⊂ π ∗ T A of right ideals such that π∗ T A /J is a locally free O T -
module of rank n 2 − n. Further, if i : A → A ′ is a morphism of n 2 -dimensional
central simple algebras over K then the diagram
I A ′
ι A ′
P XA ′
is commutative.
i ∗ [X K n (i)] ∗
ι A
I A PXA
8.6. Remark. –––– I A is canonically a subfunctor of the Graßmann functor
Grass n2
n 2 −n parametrizing (n 2 − n)-dimensional quotients of the n 2 -dimensional
standard vector space. Thus, one has an embedding X A ֒→ Grass n2
n 2 −n into
the Graßmann scheme. This is the key observation for a direct proof of the
representability of I A [Hen, Ke].

Sec. 8] THE FUNCTOR OF POINTS 55
8.7. Definition. –––– Let G be a group.
By Sets G , we will denote the category of all mappings M → G where M is an
arbitrary set. As morphisms from f 1 : M 1 → G to f 2 : M 2 → G, we allow all
mappings making the diagram
M 1
M 2
f 2
f 1
G
· g
commutative for a certain g ∈ G. Here, · g : G → G denotes multiplication
by g from the right.
8.8. Notation. –––– Let L/K be a finite Galois extension. Then, Sch L/K
+ will
denote the full subcategory of Sch L/K (cf. Definition 7.10) consisting of all
non-empty L-schemes.
8.9. Fact-Definition. –––– Let L/K be a finite Galois extension and G be its
Galois group.
Then, for each X ∈ Ob (Sch L/K ), the contravariant functor
G
Hom Sch
L/K ( · , X ): Sch L/K
+ −→ {sets}
factors canonically via Sets G . The resulting functor
P L/K
X
will be called the functor of points of X.
Proof. Each morphism T → X in Sch L/K
+
: Sch L/K
+ −→ Sets G
is twisted by a unique element σ ∈ G.
□
8.10. Remark. –––– For that, we need the assumption T ≠ ∅. On the other
hand, X = ∅ could have been allowed since there are no morphisms at all in
this case.
8.11. Remarks. –––– a) P L/K
X
is closely related to the ordinary functor of
points P X introduced in Definition 7.3. For each non-empty L-scheme T ,
one has P X (T ) = [P L/K
X
(T )] −1 (id).
b) There is a covariant functor
given on objects by X ↦→ P L/K
X
.
P L/K : Sch L/K → Fun ((Sch L/K
+ )op , Sets G )

56 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
c) The functor Hom L/K Sch
( · , X ) remembers all information on X as an object
+
of Sch L/K
+ .
The functor of points
P L/K
X
: Sch L/K
+ −→ Sets G
carries additional information related to the structure of Sch L/K
+ as a fibered category.
8.12. Lemma. –––– Let n ∈Æ, L/K be a finite Galois extension of fields and
G := Gal(L/K ) its Galois group.
of functors between the com-
i) Then, there is an isomorphism jn
L/K
position
P L/K
n
: Mat L/K
n
Ξ L/K
n
P L/K
n−1
embedding
: P L/K
n
−→ I L/K
n
Sch L/K
+
P L/K Fun ((Sch L/K
+ )op , Sets G )
and the functor In
L/K given by
⎛ ⎧
⎫⎞
A ↦→ ⎝T ↦→ G ⎨ sheaves of right ideals J in (S(σ) ◦ π T ) ∗ A ⎬
such that (S(σ) ◦ π
⎩
T ) ∗ A /J
⎠
σ∈G is a locally free O T -module of rank (n 2 ⎭
− n)
on objects and by pull-back on morphisms.
Here, A := Ã denotes the sheaf of O L-algebras associated with A on SpecL.
π T : T → SpecL is the structural morphism. The set on the right hand side is
equipped with the canonical map to G.
ii) Let L ′ /L be a finite field extension such that L ′ /K is Galois. Then, the isomorphism
j L/K
n is compatible with base extension from L to L ′ .
I.e., for every A ∈ Ob (Mat L/K
n
P L/K
n (A)(T )
) and every T ∈ Ob (Sch L/K
+ ), the diagram
Mor Sch
L/K (T , Ξn
L/K (A))
j L/K
n (A)(T )
I L/K
n (A)(T )
. × SpecL SpecL ′
Mor Sch
L ′ /K (T× SpecL SpecL ′ , Ξ L′ /K
n (A⊗ L L ′ ))
pull-back
j I L′ /K
n (A⊗ L L ′ )(T× SpecL SpecL ′ )
P L′ /L
n (A ⊗ L L ′ )(T × SpecL SpecL ′ )
commutes. j is an abbreviation for j L′ /K
n (A ⊗ L L ′ )(T × SpecL SpecL ′ ).

Sec. 8] THE FUNCTOR OF POINTS 57
Proof. i) For each A ∈ Ob (Mat L/K
n ), both functors satisfy the sheaf axiom for
Zariski coverings. Thus, it will suffice to verify the assertion on affine schemes.
Let R be a commutative L-algebra with unit. By construction, we have
Ξ L/K
n (A) = P(l) for l a non-zero, simple, left A-module. By Corollary-
Definition 7.6, the set of all R-valued points on P(l) is in natural bijection
to the set of all submodules M ⊂ l⊗ L R such that the quotient l⊗ L R/M is a
locally free R-module of rank (n − 1).
As an R-module, A⊗ L R is isomorphic to the direct sum of n copies of l⊗ L R.
Under this isomorphism, an R-submodule M ⊆ l ⊗ R determines a right
ideal M ⊆ A ⊗ L R according to the definition M := L n
i=1 M. This gives a
natural bijection between the set of all right ideals in A ⊗ L R and the set of
all R-submodules M ⊆ l. Obviously,
(A ⊗ L R)/M ∼ = ((l ⊗ L R)/M ) n ∼ = ((l ⊗L R)/M ) ⊗ R R n
as R-modules. Thus, if (l⊗ L R)/M is locally free of rank (n−1) then (A⊗ L R)/M
is locally free of rank (n 2 − n). Conversely, if (A ⊗ L R)/M is locally free of
rank (n 2 − n) then, by Lemma 3.14, (l⊗ L R)/M is locally free. Clearly, it is of
rank (n − 1).
To give a morphism of L-schemes f : SpecR → Ξn
L/K (A) which is twisted by
some σ ∈ G is equivalent to giving a commutative diagram
SpecR
π SpecR
SpecL
id
S(σ)
SpecR
S(σ)◦π SpecR
SpecL
f
Ξ L/K
n (A)
id SpecL .
π L/K Ξ n (A)
Therefore, f becomes a morphism of L-schemes in the ordinary sense if one
just changes the structural morphism of SpecR into S(σ) ◦ π.
Consequently, the functor of points P L/K is isomorphic to the functor given
Ξ L/K
R (A)
in the assertion.
ii) Again, we need the concrete description of Ξ n
./K . We have Ξ L/K
n (A) = P(l)
for l a non-zero, simple, left A-module. Further, Ξ L′ /K
n (A⊗ L L ′ ) = P(l⊗ L L ′ ).
The claim now easily follows from Remark 7.7.
□
8.13. Remark. –––– Lemma 8.12 states, in particular, that for A ∈ Ob (Mat L/K
the ordinary functor of points of the L-scheme Ξn
L/K (A) is isomorphic to
{ }
sheaves of right ideals J in π
∗
T ↦→
T A such that
πT ∗ A /J is a locally free O T -module of rank (n 2 .
− n)
n )
I.e., for every L-algebra R, the set of R-valued points on Ξ L/K
n (A) is naturally
isomorphic to the set of all right ideals I in A⊗ L R such that A⊗ L R/I is a locally
free R-module of rank (n 2 − n).

58 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
8.14. Proof of Theorem 8.4. –––– For every A ∈ Ob (Az K n ), we have to construct
an isomorphism ι A : I A → P XA of functors. ι A has to be natural in A.
We will proceed in three steps.
First step. The case A ∼ = M n (K ).
We have A⊗ K R ∼ = M n (R). For X A , there is the intrinsic description given in
Proposition 7.19. X A = P(l) where l denotes a non-zero, simple, left A-module.
The assertion is true by Lemma 8.12. We note explicitly that P(l) = Ξ K/K ′
n (A)
for every subfield K ′ ⊂ K such that K/K ′ is finite Galois. In particular, the
isomorphism ι A : I A → P XA is compatible with automorphisms of A which are
only K ′ -linear.
Second step. Reduction to T -valued points for affine T .
We return to the general case that A is an arbitrary central simple algebra
over K of dimension n 2 . The functors I A and P XA satisfy the sheaf axioms for
Zariski coverings. Hence, it suffices to consider the affine case.
There is to be constructed a natural isomorphism ι A : I A → P XA of functors on
the full subcategory of affine schemes. I.e., for each commutative K-algebra R
with unit, an isomorphism
{ }
right ideals J in A ⊗K R such that A ⊗
I A (R) :=
K R/J
is a locally free R-module of rank n 2 − n
↓ι A (R)
P XA (R) := Mor K-schemes (SpecR, X A ) .
For a homomorphism r : R → R ′ of K-algebras, the corresponding diagram
I A (R)
ι A (R)
P XA (R)
is supposed to commute.
Third step. Galois descent.
Let L be a splitting field for A.
Put G := Gal(L/K ).
I A (r)
P XA (r)
I A (R ′ )
ι A (R ′ )
P XA (R ′ )
Assume that L/K is finite and Galois.
By Theorem 3.2.ii), there is a bijection
{ }
p : SpecR → XA |
P XA (R) =
p morphism of K-schemes
⎧
⎫
⎨ p : SpecR ⊗ K L → X A⊗K L |
⎬
∼= p morphism of L-schemes,
⎩
⎭ , (§)
p compatible with the G-operations on both sides
natural in the K-algebra R.

Sec. 8] THE FUNCTOR OF POINTS 59
As A⊗ K L is isomorphic to the full matrix algebra, we have
X A⊗K L = Ξn
L/K (A ⊗ K L) = P(l)
where l is a non-zero, simple, right A⊗ K L-module. Hence, there is a second
natural bijection
⎧
⎫
⎨ p : SpecR ⊗ K L → X A⊗K L |
⎬
p morphism of L-schemes,
⎩
⎭
p compatible with the G-operations on both sides
⎧
⎫
J ⊂ (A⊗ K L) ⊗ L (R⊗ K L) |
⎪⎨ J right ideal,
⎪⎬
∼= ((A⊗ K L) ⊗ L (R⊗ K L))/J locally free (R⊗ K L)-module, . ()
rk R⊗K L ((A⊗ K L) ⊗ L (R⊗ K L))/J = (n
⎪⎩
2 − n) ,
J invariant with respect to the G-operation
⎪⎭
Indeed, this is exactly the result of the first step when we note that the isomorphism
of functors I A⊗K L → P XA⊗K is compatible with K-linear automorphisms
L
of A⊗ K L.
Finally, there is a natural bijection
⎧
⎫
J ⊂ (A⊗ K L) ⊗ L (R⊗ K L) |
⎪⎨ J right ideal,
⎪⎬
((A⊗ K L) ⊗ L (R⊗ K L))/J locally free (R⊗ K L)-module,
rk R⊗K L ((A⊗ K L) ⊗ L (R⊗ K L))/J = n
⎪⎩
2 − n ,
J invariant with respect to the G-operation
⎪⎭
⎧
⎫
J ⊂ A⊗ K R |
⎪⎨
⎪⎬
J right ideal,
∼=
= I
(A⊗ ⎪⎩ K R)/J locally free R-module, A (R) (‖)
⎪⎭
rk R (A⊗ K R)/J = (n 2 − n)
by Lemma 8.16. I.e., by Galois descent for right ideals.
Indeed, everything is clear except for the statements on the ranks of the quotients.
For that, if (A⊗ K R)/J is a locally free R-module of rank (n 2 − n) then
((A⊗ K L) ⊗ L (R⊗ K L))/J ∼ = ((A⊗ K R)/J ) ⊗ R (R⊗ K L)
is a locally free (R⊗ K L)-module of the same rank. On the other hand, if
((A⊗ K R)/J ) ⊗ R (R⊗ K L)
is a locally free (R ⊗ K L)-module of rank (n 2 − n) then it is a locally free
R-module, too. By Lemma 3.14, (A ⊗ K R)/J is locally free as an R-module.
Clearly, it is of rank (n 2 − n).

60 THE BRAUER-SEVERI VARIETY OF A CENTRAL SIMPLE ALGEBRA [Chap. I
We finally note that the construction of ι A given is functorial in A. For that,
one first has to choose a splitting field L A ⊃ K for each A ∈ Ob (Az K n ) in such a
way that L A depends only on the isomorphism class of A in Az K n . If i : A → A ′
is a morphism in Az K n then A and A′ are automatically isomorphic. We execute
the constructions of ι A and ι A ′ via the splitting field chosen.
The natural bijections (§) and (‖) are applications of descent and, therefore,
compatible with the morphisms induced by i. For (), apply Lemma 8.12.i).
The proof is complete.
8.15. Remark. –––– ι A is independent of the choice of the splitting field made
in the proof. It is not difficult to deduce this from Lemma 8.12.ii).
8.16. Lemma (Galois descent for right ideals). —–
Let L/K be a finite Galois extension. Denote by G := Gal(L/K ) its Galois group.
Further, let A be a K-algebra and I ⊂ A ⊗ K L a right ideal invariant under the
canonical operation of G on A⊗ K L.
Then, there is a unique right ideal I ⊂ A such that I = I ⊗ K L.
Proof. I inherits from A⊗ K L a structure of an L-algebra. Further, I is naturally
equipped with an operation of G by homomorphisms of K-algebras. σ ∈ G acts
by a σ-linear map.
By the algebraic version of Galois descent, there exists a K-algebra I such
that I = I ⊗ K L.
Furthermore, the canonical homomorphism
I = I ⊗ K L ֒→ A⊗ K L
of L-algebras is compatible with the G-operations on both sides. Therefore,
by Galois descent for homomorphisms, we get a homomorphism I → A
of K-algebras. As the functor ⊗ K L is exact and faithful, that homomorphism
is necessarily injective.
Consider I ⊂ A as a subring. For every a ∈ A, the multiplication ·a : I → I
from the right is compatible with the operation of G. Hence, it descends to a homomorphism
I → I. That homomorphism is compatible with multiplication
by a from the right on the whole of A. Consequently, I ⊂ A is a right ideal.
Uniqueness is clear.
□
□

CHAPTER II
ON THE BRAUER GROUP OF A SCHEME
1. Azumaya algebras
1.1. Definition. –––– Let X be any scheme.
Then, a sheaf of Azumaya algebras or simply an Azumaya algebra over X is a
locally free sheaf A of locally finite rank over X equipped with the structure of
a sheaf of O X -algebras such that the following condition is fulfilled.
For every closed point x ∈ X, one has that A (x) := A ⊗ X k(x) is a central
simple algebra over the residue field k(x).
1.2. Example. –––– If X = Speck is the spectrum of a field then an Azumaya
algebra over X is nothing but a central simple algebra over k.
1.3. Proposition. –––– Let X be any scheme and A be any locally free sheaf of
locally finite rank on X having the structure of a sheaf of algebras.
Then, A is an Azumaya algebra over X if and only if the canonical homomorphism
ι A : A ⊗ X A op −→ End(A ) ,
a ⊗ b
is an isomorphism of locally free sheaves.
↦→ (x ↦→ axb)
Here, a, b, x ∈ A (U ) denote sections of A over an arbitrary open subset U of X.
Proof. “=⇒” Let A be an Azumaya algebra. Then, A (x) is a central simple
algebra over k(x) for every closed point x ∈ X. By virtue of Remark I.4.2.b),
this implies that ι A is an isomorphism at every closed point. It is therefore an
isomorphism of locally free sheaves.
“⇐=” Let x ∈ X be a closed point. We need to show that A (x) is a central
simple algebra over k(x).
By assumption, we know that
A (x) ⊗ k(x) [A (x)] op ∼ = End(k(x) n ) ∼ = M n (k(x))

62 ON THE BRAUER GROUP OF A SCHEME [Chap. II
for n := rk x (A ). Further, M n (k(x)) is a central simple algebra over k(x).
If A (x) were not central, Z(A (x)) = L k(x), then
L = L ⊗ k(x) k(x) ⊆ A (x) ⊗ k(x) [A (x)] op ∼ = Mn (k(x))
would be contained in the center of M n (k(x)). This is a contradiction.
Assume, finally, A (x) were not simple. Then, it would contain a proper twosided
ideal (0) I A (x). But, under this assumption, I ⊗ k(x) [A (x)] op would
be a non-trivial, proper two-sided ideal in A (x) ⊗ k(x) [A (x)] op ∼ = M n (k(x)).
We obtained a contradiction. A is, therefore, a central simple algebra.
1.4. Definition. –––– Let X be any scheme and A be a locally free sheaf on X
having the structure of a sheaf of algebras.
a) Then, for a point x ∈ X, we will say that A is Azumaya in x if A (x) is a
central simple algebra over k(x).
b) The set of all points x ∈ X such that A is not Azumaya in x will be called
the non-Azumaya locus of A .
1.5. Corollary. –––– Let X be any scheme and A be any locally free sheaf on X
having the structure of a sheaf of algebras.
a) Then, the non-Azumaya locus T ⊂ X of A is the support of a Cartier divisor.
In particular, it is a closed set in X.
b) Assume, X is regular and connected. Then, the non-Azumaya locus of A is either
the whole of X or a subset pure of codimension one.
Proof. a) The assertion is local in X. We may, therefore, assume that A is a free
sheaf, say, of rank n.
ι A : A ⊗ X A op ∼ = O
n 2
X −→ End(A ) ∼ = OX
n2
is then a morphism of free sheaves which are both of rank n 2 .
Whether ι A ⊗ X k(x) is an isomorphism is equivalent to the non-vanishing of its
determinant in x. In the trivialization, the latter is a section s of the structure
sheaf O X . div s is, by definition, the support of a Cartier divisor.
b) This is an immediate consequence of a). □
1.6. Fact. –––– Let A and B be two Azumaya algebras on a scheme X.
Then, their tensor product A ⊗ OX B as locally free sheaves over X, equipped with
the obvious structure of an O X -algebra, is again an Azumaya algebra.
Proof. On sections over an open subset U ⊆ X, the algebra structure is given by
(a ⊗ b)(a ′ ⊗ b ′ ) := aa ′ ⊗ bb ′ .
□

Sec. 1] AZUMAYA ALGEBRAS 63
It is clear that A ⊗ OX B is a locally free O X -module.
The property of being Azumaya may be tested in closed points. One has
(A ⊗ OX B)(x) = A ⊗ OX B⊗ OX k(x) = (A ⊗ OX k(x)) ⊗ k(x) (B⊗ OX k(x)) .
On the right hand side, the tensor product of two central simple algebras is
again a central simple algebra.
□
1.7. Fact. –––– Let f : X → Y be a morphism of schemes and A be an Azumaya
algebra on Y.
Then, the pull-back f ∗ A of A as a locally free sheaf, equipped with the obvious
structure of an O X -algebra, is an Azumaya algebra on X.
Proof. As the assertion is local in both, Y and X, we may assume that
f : X = SpecS → Y = SpecR is a morphism of affine schemes and that
A is given by an R-algebra A which is free of finite rank as an R-module.
Then, f ∗ A is given by the S-algebra A ⊗ R S. It algebra structure is given,
analogously to the one on the tensor product above, by
(a ⊗ s)(a ′ ⊗ s ′ ) := aa ′ ⊗ ss ′ .
Again, we test the property of being Azumaya in closed points.
For that, let x ∈ X and put y := f (x). Thus, k(x) is a field extension of k(y).
What we have to show is that A⊗ R S ⊗ S k(x) is a central simple algebra over k(x).
But,
A⊗ R S ⊗ S k(x) = A⊗ R k(x) = A⊗ R k(y)⊗ k(y) k(x)
and A⊗ R k(y) is, by assumption, a central simple algebra over k(y).
The assertion follows from Lemma I.4.1.b).
□
1.8. Proposition. –––– Let f : X → Y be a morphism of schemes which is faithfully
flat. Further, let A be a quasi-coherent sheaf on Y, equipped with the structure
of a sheaf of O Y -algebras.
Then, A is an Azumaya algebra on Y if and only if f ∗ A is an Azumaya algebra
on X.
Proof. “=⇒” This is Fact 1.7.
“⇐=” The assumption that f ∗ A is an Azumaya algebra on X includes that
f ∗ A is a locally free O X -module, locally of finite rank. We aim first at showing
that this implies A is a locally free O Y -module, locally of finite rank.
For that, let y ∈ Y and choose x ∈ X such that f (x) = y. We have a flat local
homomorphism of local rings i : O Y,y → O X,x . By [Mat, Theorem 7.2], O X,x is
faithfully flat over O Y,y .

64 ON THE BRAUER GROUP OF A SCHEME [Chap. II
Further, ( f ∗ A ) x = A y ⊗ OY,y O X,x is a locally free O X,x -module of finite rank.
Lemma I.3.14 implies that A y is indeed a locally free O Y,y -module and of finite
rank. This means, A is a locally free O Y -module, locally of finite rank.
In the rest of the proof, flatness will no more be used.
To verify A is an Azumaya algebra over Y, we will use the criterion provided
by Proposition 1.3. We have f ∗ A = A ⊗ OY O X and, therefore,
f ∗ A ⊗ OX [ f ∗ A ] op = (A ⊗ OY O X ) ⊗ OX (O X ⊗ OY [A ] op )
∼= A ⊗ OY [A ] op ⊗ OY O X
= f ∗ (A ⊗ OY [A ] op ) .
Further,
End( f ∗ A ) = f ∗ (End(A ))
and the isomorphisms are compatible in such a way that
f ∗ ι A = ι f ∗ A .
Thus, we are given that f ∗ ι A = ι f ∗ A is an isomorphism of locally free sheaves.
We have to prove this implies that the original ι A was an isomorphism.
For that, note that det( f ∗ ι A ) = f # det(ι A ). Thus, we know that f # det(ι A ) is
a nowhere vanishing section of the invertible sheaf
f ∗ [Λ max (A ⊗ OY [A ] op ) ∨ ⊗ Λ max (End(A ))] .
We have to show that det(ι A ) is nowhere vanishing.
Again, for y ∈ Y, we choose x ∈ X such that f (x) = y. Then, we have a local
homomorphism of local rings i : O Y,y → O X,x . As we work in stalks, we may
choose a trivialization and write det(ι A ) ∈ O Y,y . Then,
f # det(ι A ) = i(det(ι A )) ∈ O X,x .
If det(ι A ) were vanishing at y then this would mean simply det(ι A ) ∈ m Y,y .
Then, i(det(ι A )) ∈ m X,x . I.e., f # det(ι A ) vanishes at x.
This is a contradiction.
1.9. Proposition. –––– Let f : X → Y be a morphism of schemes which is faithfully
flat and quasi-compact.
Then, it is equivalent to give
a) an Azumaya algebra A over Y,
b) an Azumaya algebra B over X together with an isomorphism
Φ: pr ∗ 1 B → pr∗ 2 B
□

Sec. 2] THE BRAUER GROUP 65
of Azumaya algebras on X × Y X which satisfies
on X × Y X × Y X.
pr ∗ 31 (Φ) = pr∗ 32 (Φ) ◦ pr∗ 21 (Φ)
Sketch of Proof. “a) =⇒ b)” Put B := f ∗ A .
“b) =⇒ a)” This is what is called faithful flat descent.
The existence of A as a quasi-coherent O Y -module satisfying B := f ∗ A follows
directly from [K/O74, Theorem II.3.2]. The additional algebra structure descends,
too. This may easily be seen from the construction of the descent module
given in the proof. It is shown in [K/O74, Theorem II.3.4].
A is an Azumaya algebra over Y by virtue of Proposition 1.8.
1.10. Remark. –––– This means that an Azumaya algebra over a scheme X
may be described by local gluing data with respect to the fpqc-topology or any
weaker topology.
Intuitively, B describes the desired Azumaya algebra on each set of a cover.
Φ collects the gluing isomorphisms on intersections of two sets of the cover.
The condition on X × Y X × Y X is a cocycle condition encoding that the gluing
maps be compatible.
We will use this approach for constructing Azumaya algebras from data local in
the étale topology.
□
2. The Brauer group
2.1. Definition. –––– Let X be any scheme.
Two Azumaya algebras A and B on X are said to be similar if there exist two
locally free O X -modules E and F, both everywhere of positive rank, such that
A ⊗ OX End(E ) ∼ = B ⊗ OX End(E ′ ) .
2.2. Remarks. –––– i) Similarity is an equivalence relation.
For that, the only point which is not entirely obvious is transitivity. To achieve
this, assume
A ⊗ OX End(E ) ∼ = B ⊗ OX End(E ′ ) and B ⊗ OX End(E ′′ ) ∼ = C ⊗ OX End(E ′′′ ) .

66 ON THE BRAUER GROUP OF A SCHEME [Chap. II
Then,
A ⊗ OX End(E ⊗ OX E ′′ ) ∼ = A ⊗ OX End(E ) ⊗ OX End(E ′′ )
∼= B ⊗ OX End(E ′ ) ⊗ OX End(E ′′ )
∼= B ⊗ OX End(E ′′ ) ⊗ OX End(E ′ )
∼= C ⊗ OX End(E ′′′ ) ⊗ OX End(E ′ )
∼= C ⊗ OX End(E ′′′ ⊗ OX E ′′′ ) .
ii) Similarity is compatible with the tensor product of Azumaya algebras and
with pull-back.
2.3. Definition. –––– Let X be any scheme.
The set of all similarity classes of Azumaya algebras on X is called the Brauer
group of X and denoted by Br(X ).
2.4. Remarks. –––– i) The similarity class of an Azumaya algebra A will be
denoted by [A ].
ii) A binary operation on Br(X ) is given by [A ] ∗ [A ′ ] := [A ⊗ OX A ′ ] .
iii) For this binary operation, [End(E )] is the neutral element. Here, E is an
arbitrary locally free sheaf, nowhere of rank zero.
iv) The inverse element of [A ] is given by [A op ]. This shows that Br(X ) is
indeed a group.
3. The cohomological Brauer group
3.1. Lemma. –––– Let (R, m) be a Henselian local ring and A be an Azumaya
algebra over SpecR. Assume
A ⊗ OSpecR O SpecR/m
∼ = End(O
n
SpecR/m )
for a certain n ∈Æ. Then, A ∼ = End(O n SpecR ).
Proof. We denote by A the R-algebra corresponding to A . In this notation,
we, therefore, have A ⊗ R R/m ∼ = M n (R/m). On the right hand side, choose an
idempotent matrix ǫ of rank one.
Let a ∈ A be such that a := (a mod mA) = ǫ. Then, R[a] is a finite
commutative R-algebra. As R is Henselian, [SGA4, Exp. VIII, 4.1] shows that
R[a] is a direct product of local rings. From R[a]⊗ R k = k[ǫ] ∼ = k ⊕ k, we see
that R[a] ∼ = R/I ⊕ R/J is actually composed of two local R-algebras. In k ⊕ k,
let ǫ correspond to the element (1, 0). Thus, the element e ∈ R[a] mapped
to (1, 0) under the isomorphism R[a] → R/I ⊕ R/J is an idempotent lifting ǫ.

Sec. 3] THE COHOMOLOGICAL BRAUER GROUP 67
Consider the homomorphism of R-algebras
φ: A −→ End R (Ae) ,
a ↦→ (be ↦→ abe) .
Both R-modules are free of the same rank. It is classically known that φ⊗ R k
is an isomorphism. Nakayama’s lemma ensures that φ is an isomorphism itself.
□
3.2. Corollary. –––– Let (R, m) be a strictly Henselian local ring and A be an
Azumaya algebra over SpecR.
Then, A ∼ = End(O n SpecR ).
Proof. k = R/m is a separably closed field. By virtue of Remark I.4.2.b), we
know that
A ⊗ OSpecR O SpecR/m
∼ = End(O
n
SpecR/m )
is a full matrix algebra over R/m.
□
3.3. Lemma (Theorem of Skolem-Noether, version over local rings). —–
Let R be a commutative ring with unit. Then, GL n (R) operates on M n (R) by
conjugation,
(g, m) ↦→ gmg −1 .
If R = (R, m) is a local ring then this defines an isomorphism
PGL n (R) := GL n (R)/R ∗ ∼ =
−→ Aut R (M n (R)) .
Proof. (Compare Lemma I.4.4 where the special case R is a field was treated.)
One has R = Zent(M n (R)). Therefore, the mapping is well-defined and injective.
Surjectivity. Let j : M n (R) → M n (R) be an automorphism. We consider the
algebra
M := M n (R) ⊗ R M n (R) op ( ∼ = End R-mod (M n (R)) ∼ = M n 2(R)) .
M n (R) gets equipped with the structure of a left M-module in two ways.
(A ⊗ B) • 1 C := A · C · B
(A ⊗ B) • 2 C := j(A) · C · B
We will denote the resulting M-modules by N 1 and N 2 , respectively.

68 ON THE BRAUER GROUP OF A SCHEME [Chap. II
Our first claim is that N 1 is a projective M-module. This means, we have to
show that M n (R) is projective when equipped with its canonical structure as
an End R-mod (M n (R))-module.
For that, consider a morphism i : M n (R) → R of R-modules which is a section
of the canonical inclusion mapping r to r·E. Note such a section exists because
M n (R) is a projective R-module.
Then, the surjection of End R-mod (M n (R))-modules End R-mod (M n (R)) → M n (R),
given by f ↦→ f (1), admits the section
m ↦→ (m ′ ↦→ i(m ′ )m) .
This shows M n (R) is a projective End R-mod (M n (R))-module.
Over the field R/m, it is known that two M n 2(R/m)-modules of the same
R/m-dimension are isomorphic. Thus, there is an isomorphism
As M 1 is projective, the map
M 1
g : M 1 ⊗ R R/m → M 2 ⊗ R R/m .
π
−→ M 1 ⊗ R R/m −→ g
M 2 ⊗ R R/m
may be lifted to a homomorphism h : M 1 → M 2 of End R-mod (M n (R))-modules.
h⊗ R R/m is an isomorphism. Nakayama’s lemma implies that h is an isomorphism
itself.
Let us put I := h(E) to be the image of the identity matrix.
M ∈ M n (R) we have
h(M ) = h((E ⊗ M ) • 1 E) = (E ⊗ M ) • 2 h(E) = h(E) · M = I · M.
In particular, I ∈ GL n (R). Therefore,
I · M = h(M ) = h((M ⊗ E) • 1 E) = (M ⊗ E) • 2 h(E) = j(M ) · I
For every
for each M ∈ M n (R) and j(M ) = IMI −1 .
□
3.4. Proposition –––– Let X be any quasi-compact scheme. Then, the set of all isomorphismclassesofrankn
2 AzumayaalgebrasonX isgivenbytheČechcohomology
set Ȟ 1 ét (X, PGL n).
Proof. We know by Proposition 1.9 that Azumaya algebras may be described
by gluing data local in the étale topology.
Let A be any Azumaya algebra over X. For every point x ∈ X, Corollary
3.2 yields that A is isomorphic to matrix algebra after strict Henselization,
A ⊗ OX OX,x sh ∼ = M n (OX,x sh ). To describe the isomorphism, only finitely many

Sec. 3] THE COHOMOLOGICAL BRAUER GROUP 69
data on O sh
X,x are necessary. For this reason, there exists an affine scheme U x,
which is an étale neighbourhood of x, such that A | Ux
∼ = End(O
n
Ux
). By virtue of
quasi-compactness, one may select finitely many of the U i covering the whole
of X.
All these étale neighbourhoods together form gluing data for A since the morphism
U x1
F . . .
F
Uxl → X is faithfully flat and quasi-compact.
The conditions for gluing data formulated in Proposition 1.9 are equivalent to
giving a Čech cocycle for Ȟ 1 ét (X, Aut(M n)). Different gluing data describing the
same algebra differ by a coboundary.
The Theorem of Skolem-Noether implies that the sheaf Aut(M n ) is the same
as PGL n .
□
3.5. Remark. –––– On a quasi-compact scheme X, the set of all isomorphism
classes of rank n locally free sheaves is given by the Čech cohomology set
Ȟ 1 ét (X, GL n).
This is proven in the same way as Proposition 3.4.
3.6. Notation. –––– Let X be any scheme.
i) We will denote the cohomology class corresponding to an Azumaya algebra
A over X by c(A ) ∈ Ȟ 1 ét (X, PGL n).
ii) We will denote the cohomology class corresponding to a locally free sheaf E
over X by cl(E ) ∈ Ȟ 1 ét (X, GL n).
iii) Recall that there is a short exact sequence
0 −→m −→ GL n −→ PGL n −→ 0
of sheaves. In cohomology, it induces a long exact sequence of pointed sets
Ȟ 1 ét (X, GL n)
ι
−→ Ȟ 1 ét (X, PGL d
n) −→ H 2 ét (X,m).
3.7. Facts. –––– Let X be a quasi-compact scheme.
i) For a locally free sheaf E on X, one has that
ι(cl(E )) = c(End(E )) .
ii) For two Azumaya algebras A and B on X, of ranks n and m, respectively,
one has
c(A ⊗ OX B) = c(A )∗c(B) .
Here, the operation ∗ is induced by the canonical mapping
PGL n × PGL m −→ PGL nm

70 ON THE BRAUER GROUP OF A SCHEME [Chap. II
given by
GL n ×GL m = Aut(n a )×Aut(m a ) −→ Aut(n a ⊗m a ) = Aut(nm
a ) = GL nm .
iii) For α ∈ Ȟ 1 ét (X, PGL n) and β ∈ Ȟ 1 ét (X, PGL n), we have
d(α∗β) = d(α) + d(β) .
Proof. The verification of these compatibilities is purely technical and hence
omitted.
□
3.8. Proposition. –––– Let X be any quasi-compact scheme.
Then, the boundary maps d induce an injective group homomorphism
Proof. First step. i X is well-defined.
i X : Br(X ) −→ H 2 ét (X,m) .
Suppose we are given two Azumaya algebras A and B over X which are
similar. This means, we have two locally free sheaves, E and F, such that
A ⊗ OX End(E ) ∼ = B ⊗ OX End(E ′ ). In particular,
c(A ⊗ OX End(E )) = c(B ⊗ OX End(E ′ )) .
(∗)
The compatibilities above yield
hence
c(A ⊗ OX End(E )) = c(A )∗c(End(E )) = c(A )∗ι(cl(E )) ,
d(c(A )) = d(c(A ))+d(ι(cl(E ))) = d(c(A )∗ι(cl(E ))) = d(c(A ⊗ OX End(E ))) .
Completely analogously, one shows that d(c(B)) = d(c(B ⊗ OX End(E ′ ))) .
Formula (∗), therefore, yields the claim.
Second step. i X is a group homomorphism.
We have d(c(A ⊗ OX B)) = d(c(A )∗c(B)) = d(c(A )) + d(c(B)) .
Third step. i X is an injection.
Suppose, we have an Azumaya algebra A such that d(c(A )) = 0. Then, according
to the exactness of the long cohomology sequence, c(A ) = ι(cl(E )) for
a locally free sheaf E on X. Thus, A ∼ = End(E ) and [A ] = 0 ∈ Br(X ). □
3.9. Definition. –––– Let X be any scheme. Then, H 2 ét (X,m) is called the
cohomological Brauer group of X. It will be denoted by Br ′ (X ).

Sec. 4] THE RELATION TO THE BRAUER GROUP OF THE FUNCTION FIELD 71
3.10. Remark. –––– i X is, in general, not an isomorphism of groups. The first
counterexample to surjectivity has been constructed by A. Grothendieck
in [GrBrII, Remarque 1.11.b].
i X is, however, an isomorphism in a number of particular cases. One such case
are smooth algebraic surfaces. This was essentially known to M. Auslander
and O. Goldman [A/G] before Grothendieck’s invention of the general Brauer
group for schemes, may be even before the actual invention of the concept of
a scheme.
In Section 7, we will present the result of Auslander and Goldman in a more
up-to-date formulation. Although this is not the most general result known
today, we hope it may serve as a good illustration for what Br(X ) and Br ′ (X )
are and which methods might be used in order to compare them.
4. The relation to the Brauer group of the function field
4.1. –––– In this section, we always assume that X is an integral scheme.
Denote by g : SpecK = η → X the inclusion of the generic point.
4.2. Fact. –––– Let X be an integral scheme which is regular, separated, and quasicompact.
Then,
Br(X ) ⊆ Br ′ (X ) ⊆ Br ′ (K )
where K := Q(X ) denotes the function field of X.
Proof. Denote by g : η → X the inclusion of the generic point. We have the
short exact sequence
0 −→m,X −→ g ∗m,K −→ Div X −→ 0
of sheaves in the étale topology. Here, Div X = L
H 1 ét (X, Div X ) = M
It follows, at first, that
codim x=1
codimx=1
H 1 ét ({x},) = M
codimx=1
H 2 ét (X,m) ⊆ H 2 ét (X, g ∗m,K ).
Second, we consider the Leray spectral sequence
i x∗x which implies
Hom(G x ,) = 0 .
E p,q
2
:= H p
ét (X, Rq g ∗m,K ) =⇒ H n ét (SpecK,m,K).
By virtue of Hilbert’s Theorem 90, for the first higher direct image, we
have R 1 g ∗m,K = 0. This implies there is a short exact sequence

72 ON THE BRAUER GROUP OF A SCHEME [Chap. II
0 −→ H 2 ét (X, g ∗m,K ) −→ H 2 ét (SpecK,m,K) −→ H 0 ét (X, R2 g ∗m,K ) −→ 0 .
(†)
of terms of lower order.
□
4.3. Remark. –––– One may therefore think of Br(X ) as follows. It consists of
those classes of the usual Br(Q(X )) which allow an extension over the whole
of X.
4.4. Lemma. –––– Let X be an integral scheme which is separated and quasicompact.
Then, for each q > 0, the sheaf R q g ∗m,K is uniquely l-divisible for every
l ∈prime to the residue characteristics of X.
Proof. The inclusion g : η → X is the inverse limit, in the category of all
X-schemes, of the open embeddings g U : U → X for U ⊆ X affine open.
The assumption that X is separated makes sure that all the transition maps
g U ′ ,U : U ′ ⊆
−→ U are affine. In this situation, from [SGA4, Exp. VII, Corollaire
5.11], we see that
R q g ∗ µ l = lim R q g U∗ −→U
µ l .
As the g U are open embeddings, and hence smooth, they are acyclic [SGA4,
Exp. XV, Théorème 2.1]. In particular, for q > 0 and l prime to the residue
characteristics of X, we have R q g U∗ µ l = R q g U∗ g U ∗ µ l,X = 0.
Altogether,
R q g ∗ µ l = 0 .
The Kummer sequence implies that the multiplication map
l : R q g ∗m,K → R q g ∗m,K
is an isomorphism for every q > 0.
□
4.5. Proposition. –––– Let X be an integral scheme which is regular, separated, and
quasi-compact.
a) If char(Q(X )) = p > 0 then H 0 ét (X, R2 g ∗m,K ) is a p-power torsion group.
b) If all residue characteristics of X are equal to zero then H 0 ét (X, R2 g ∗m,K ) = 0.
Proof. H 2 ét (SpecK,m,K) = H 2( Gal(K/K ), K ∗) is a torsion group directly by
its definition. As we have a surjection H 2 ét (SpecK,m,K) → H 0 ét (X, R2 g ∗m,K ),
the latter is a torsion group, too.

Sec. 4] THE RELATION TO THE BRAUER GROUP OF THE FUNCTION FIELD 73
b) For every l > 0, the sheaf R 2 g ∗m,K is uniquely l-divisible. Therefore, the
multiplication map
l : H 0 ét (X, R2 g ∗m,K ) → H 0 ét (X, R2 g ∗m,K )
is an isomorphism. As H 0 ét (X, R2 g ∗m,K ) is a torsion group, this implies
H 0 ét (X, R2 g ∗m,K ) = 0.
a) Here, the same argument still works for l prime to p. This shows
H 0 ét (X, R2 g ∗m,K ) has no prime-to-p torsion.
□
4.6. Corollary. –––– Let X be an integral scheme which is regular, separated, and
quasi-compact. Assume that the residue characteristics of X are equal to zero.
Then, for g : η → X the inclusion of the generic point, one has
H 2 ét (X, g ∗m,K ) ∼ = H 2 ét (SpecK,m,K ) .
Proof. This follows from the exact sequence (†) together with Proposition 4.5.b).
□
4.7. Remark. –––– The same result is true for schemes of dimension at most
one in any characteristic as is often shown in the literature. See, e.g., [GrBrIII,
formule (2.2)].
4.8. Proposition. –––– Let X be an integral scheme which is regular, separated, and
quasi-compact. Assume that
i) dim X ≤ 1 or
ii) the residue characteristics of X are equal to zero.
Then, there is a short exact sequence
0 −→ Br ′ (X ) −→ Br ′ (SpecQ(X )) −→ M
Hom(G x ,É/) .
codim x=1
Proof. Consider once more the short exact sequence
0 −→m,X −→ g ∗m,K −→ Div X −→ 0
of sheaves in the étale topology. Since H 1 ét (X, Div X ) = 0, as shown above, we
have the following fragment of the long exact sequence
0 −→ H 2 ét (X,m,X ) −→ H 2 ét (X, g ∗m,K ) −→ H 2 ét (X, Div X )
in cohomology. Here, the entry on the left hand side is Br ′ (X ), according to
its very definition. The term in the middle is isomorphic to Br ′ (Q(X )) as was

74 ON THE BRAUER GROUP OF A SCHEME [Chap. II
shown in Corollary 4.6. For the term on the right hand side, we have
H 2 ét (X, Div X ) = M
H 2 ét ({x},)
codim x=1
= M
H 2 (G x ,)
codim x=1
= M
Hom(G x ,É/) .
codim x=1
□
4.9. Corollary. –––– Let X be an integral scheme which is regular, separated, and
quasi-compact. Assume that
i) dim X ≤ 1 or
ii) the residue characteristics of X are equal to zero.
Then, one has
Br ′ (X ) = \
codimx=1
where the intersection takes place in Br ′ (Q(X )).
Proof. “⊆” is clear.
Br ′ (Spec O X,x )
“⊇” From Proposition 4.8, we see that all obstructions against an extension of
a central simple algebra over Q(X ) to the whole of X are given by the points
of codimension one. For SpecO X,x instead of X, the same is true. □
4.10. Remark. –––– Under certain hypotheses on X, there is the same formula
for the Brauer group instead of the cohomological Brauer group. Compare Theorem
7.1.i), below.
5. The Brauer group and the cohomological Brauer group
5.1. –––– In order to compare the two Brauer groups in relatively simple cases,
the following lemma is helpful.
5.2. Lemma. –––– Let X = SpecR for a local ring R and γ ∈ Br ′ (X ) an element
in the cohomological Brauer group. Assume there exists a morphism π : Y → X of
schemes which is finite and faithfully flat such that γ ∈ ker(Br ′ (X ) → Br ′ (Y )).
Then, one has even γ ∈ Br(X ).
Proof. We have Y = SpecS. Nakayama’s lemma implies that S ∼ = R n is free
as an R-module. As π is finite, it has no higher direct images and the Leray
spectral sequence collapses to
H 2 ét (Y,m) = H 2 ét (X, π ∗m) .

Sec. 5] THE BRAUER GROUP AND THE COHOMOLOGICAL BRAUER GROUP 75
We also have the long exact sequence
H 1 ét (X, π ∗m/m) −→ H 2 ét (X,m) −→ H 2 ét (X, π ∗m)
at our disposal. γ vanishes in H 2 ét (X, π ∗m). It belongs, therefore, to the image
of H 1 ét (X, π ∗m/m).
The canonical map S ∗
of sheaves
֒→ Aut(R n ) = GL n (R) induces a homomorphism
π ∗m ֒→ GL n .
Finally, we note that the diagram
H 1 ét (X, π ∗m/m)
H 2 ét (X,m)
Ȟ 1 ét (X, GL n/m)
Ȟ 2 ét (X,m)
commutes. Hence, γ is in the image of Ȟ 1 ét (X, GL n/m) = Ȟ 1 ét (X, PGL n).
□
5.3. Corollary. –––– One has Br(SpecK ) = Br ′ (SpecK ) for every field K.
Proof. Let γ ∈ Br ′ (SpecK ) = H 2( Gal(K/K ), K ∗) . Then, there is a finite Galois
extension L/K such that γ ∈ H 2( Gal(L/K ), L ∗) , already. When restricted
to SpecL, the cohomology class γ vanishes.
□
5.4. Corollary. –––– One has Br(SpecR) = Br ′ (SpecR) = Br(SpecR/m) for
every Henselian local ring (R, m). In particular, Br(SpecR) = 0 for R strictly local.
Proof. The equality Br ′ (SpecR) = Br ′ (Spec R/m) follows directly from a
general result on étale cohomology [SGA4, Exp. VIII, Corollaire 8.6].
Thus, let γ ∈ Br ′ (SpecR). For its image in Br ′ (Spec R/m), choose a splitting
field l which is finite over k := R/m. Let a be a primitive element and f ∈ k[X ]
be its minimal polynomial. Then, l ∼ = k[X ]/( f ). Choose a lift F ∈ R[X ] of f .
The local ring (S, n) := R[X ]/(F ) is finite and flat over R. S is Henselian
by [SGA4, Exp. VIII, 4.1.ii)]. Under pull-back to SpecS, the cohomology
class γ vanishes since Br ′ (SpecS) = Br ′ (Spec S/n).
□
5.5. Corollary. –––– One has Br(SpecR) = Br ′ (SpecR) for every discrete valuation
ring R.
Proof. Let γ ∈ Br ′ (SpecR) ⊆ Br ′ (SpecQ(R)). We know that γ vanishes after
a finite, separable field extension L/Q(R). Take for S the integral closure of R
in L. Then, γ vanishes in Br ′ (SpecS) ⊆ Br ′ (L) and S is finite and flat over R.
□

76 ON THE BRAUER GROUP OF A SCHEME [Chap. II
5.6. Remark. –––– For X an integral scheme which is regular, separated, and
quasi-compact in characteristic zero, one therefore has
Br ′ (X ) = \
Br(SpecO X,x )
in Br(Q(X )).
codimx=1
6. Orders in central simple algebras
i. Orders over a general scheme. —
6.1. Definition. –––– Let X be an integral scheme and A be a central simple
algebra over its quotient field Q(X ).
By an order in A over X, one means an O X -algebra A which is coherent as
an O X -module and has A as its stalk at the generic point.
6.2. Remark. –––– Assume the integral scheme X is locally Noetherian.
Then, for every central simple algebra A over Q(X ) there exists an order
over X.
Indeed, an order may be constructed as follows. Choose a set {a 1 , . . . , a l } of
generators of A as a Q(X )-vector space. In the sheaf g ∗ A on X, these generate
a coherent subsheaf G . Then, the sheaf (G : G ) ⊆ g ∗ A which is defined by
(G : G )(U ) := {s ∈ g ∗ A(U ) | G (U )·s ⊆ G (U ) }
for every open U ⊆ X is, by definition, an O X -algebra. Clearly, it is coherent
and has A as its stalk at the generic point. It is, therefore, an order.
6.3. Remarks. –––– i) If X = SpecR is the spectrum of a Noetherian, integrally
closed domain then every global section s ∈ Γ(X, A ) ⊆ A of every order A is
integral over R.
ii) On a central simple algebra A over a field K, there is the reduced trace
tr: A → K. It admits the property that
(x, y) ↦→ tr(xy)
is a non-degenerate bilinear form [Re, Theorem (9.26)].
Further, if R ⊆ K is integrally closed and s a section of an order over SpecR
then tr(s) ∈ R.

Sec. 6] ORDERS IN CENTRAL SIMPLE ALGEBRAS 77
6.4. Lemma. –––– Let X be an integral scheme which is Noetherian and normal.
Then, every order A over X in a central simple algebra A over Q(X ) is contained
in a maximal order.
Proof. Let SpecR ⊆ X be affine open. Then, A | SpecR is generated by finitely
many global sections s 1 , . . . , s l . Clearly, s 1 , . . . , s l generate A over K.
If A ′ ⊃ A | SpecR is any order and t any global section then ts 1 , . . . , ts l are
global sections and tr(ts 1 ), . . . , tr(ts l ) ∈ R. This list of conditions defines a
finite R-module, i.e. a coherent sheaf à over SpecR containing every order
which contains A.
Since X is covered by finitely many open affine sets, every ascending chain of
orders in A over X stops after finitely many steps.
□
6.5. Lemma. –––– Let X be an integral scheme which is Noetherian and normal.
Let A be a maximal order over X in a central simple algebra A over Q(X ).
Then, for every (not necessarily closed) point x ∈ X, A | SpecOX,x is a maximal order
in A over Spec O X,x .
Proof. For X affine, this follows from [Re, Theorem (1.11)].
For the general case, it remains to verify that A | V is maximal for every affine
open subscheme SpecR = V ⊆ X. We shall do this by contradiction.
Assume, there would be a larger order B over V . Then, there exists some
non-zero r ∈ R such that r·B ⊆ A . Extend r·B to a coherent subsheaf E ⊆ A
on X in the obvious way. I.e., a local section s ∈ E (U ) is, by definition, a
section of A such that s| U ∩V ∈ (r·B)(U ∩ V ).
Now, consider the order (E : E ) over X. Its restriction to V is easily understood.
One has (E : E )| V = ((r ·B) : (r ·B)) = (B : B) = B as r
commutes with B and B is an order. In particular, A | V ⊂ B has the property
that E | V ·A | V ⊆ E | V . Every local section s ∈ (E ·A )(U ) is, therefore, a local
section of A with the additional property that s| U ∩V ∈ E (U ∩ V ). By construction
of E , this means s ∈ E (U ). Hence, E ·A ⊆ E and A ⊆ (E : E ).
Since A is a maximal order, this implies A = (E : E ).
Together with (E : E )| V = B, this shows A | V = B in contradiction to our
assumption, B would be a larger order.
(This is nothing but a geometrization of the proof for [Re, Theorem (1.11)].)
□
6.6. –––– Our interest in maximal orders comes from the following proposition.

78 ON THE BRAUER GROUP OF A SCHEME [Chap. II
6.7. Proposition. –––– Let X be a Noetherian, normal, integral scheme with
generic point η and A be an Azumaya algebra over X.
Then, A is a maximal order in A η over X.
Proof. It is clear from the definition that A is an order.
If it were not maximal then there would be a larger order B. B admits a local
section s ∉ A (U ) over a certain subscheme U. We may therefore assume that
X = SpecR is affine, A = Ã, and B = ˜B for B A.
We have A⊗ R A op = M n (R) for a certain n ∈Æ. Then, O := B⊗ R A op M n (R)
would be a larger order. I.e., the order M n (R) would not be maximal.
Choose M ∈ O \M n (R). There is one entry x not in R. Elementary matrix
operations produce a diagonal matrix
⎛ ⎞
x 0 · · · 0
0 x · · · 0
⎜
⎝
.
. . ..
⎟ . ⎠ ∈ O .
0 0 · · · x
This element alone generates the commutative R-algebra R[x] which can not be
finite since R is integrally closed. This is a contradiction.
□
6.8. Remark. –––– The same approach yields that M n (A ) is a maximal order
as soon as A is.
ii. Orders over a discrete valuation ring. —
6.9. Notation. –––– In this subsection, we will consider the following situation.
– (R, m) is a discrete valuation ring. By π, we denote a generator of m.
– K := Q(R) is its quotient field,
– A is a central simple algebra over K.
As usual, ̂R and ̂K are the completions of R and K.
For x ∈ A, we will denote by N A/K (x) := det(·x : A → A) the (non-reduced)
norm of x.
Finally, we will simply formulate that O is an Azumaya algebra (an order)
over R when Õ is an Azumaya algebra (an order) over SpecR.
6.10. Lemma. –––– Assume R to be complete and let D = A be a (finite) division
algebra over K. Then,
x ∈ D is integral over R ⇐⇒ N D/K (x) ∈ R .

Sec. 6] ORDERS IN CENTRAL SIMPLE ALGEBRAS 79
Proof. K (x) is a finite field extension of K. In D, two K (x)-modules K (x)·y 1
and K (x)·y 2 are either equal or disjoint. Thus, there is a decomposition
rM
D = K (x) · y i
i=1
of D into a direct sum. The multiplication map ·x : D → D is a block matrix.
Consequently,
N D/K (x) = [N K (x)/K (x)] r .
It is, therefore, sufficient to show the assertion for N K (x)/K instead of N D/K .
As K is complete, there is a unique extension of the discrete valuation from K
to K (x). It is given by
ν K (x) (y) :=
1
[K (x) : K ] ν K (N K (x)/K (y)) .
The elements y of valuation ν K (x) (y) ≥ 0 form a finite R-algebra and are,
therefore, integral. If ν K (x) (y) < 0 then a relation y n +a n−1 y n−1 + . . . +a 0 = 0
would contradict the ultrametric triangle inequality.
□
6.11. Proposition. –––– Assume R to be complete. If D = A is a (finite) division
algebra over K then, in D, there is exactly one maximal order over R. This is the set
of all elements which are integral over R.
O = {x ∈ D | N D/K (x) ∈ R}
Proof. The only thing not yet proven is that O is actually an order. For this, in
turn, we still have to verify that O is a ring.
Recall that the valuation on D is given by
ν D (x) :=
1
[D : K ] ν K (N D/K (x)) .
The assertion would follow if we could verify the usual formulas
ν D (xy) = ν D (x) + ν D (y) and ν D (x + y) ≥ min{ν D (x), ν D (y)}
in the non-commutative situation, too.
For this, note that det(·xy) = det((·x) ◦ (·y)) = det(·x) det(·y) implies
N D/K (xy) = N D/K (x)N D/K (y) from which the first formula follows.
For the ultrametric triangle inequality, we remark first that it is true in the
case x and y commute. The point is, K (x, y) is then a finite field extension
of K. One has ν D (·) = ν K (x,y) (·) and the inequality carries over from the
commutative case.

80 ON THE BRAUER GROUP OF A SCHEME [Chap. II
In general, we have x −1 y ∈ O or y −1 x ∈ O. Assume without restriction
that x −1 y ∈ O. Then, since 1 commutes with x −1 y,
and
ν D (1 + x −1 y) ≥ 0 = min{ν D (1), ν D (x −1 y)}
ν D (x + y) = ν D (x·(1 + x −1 y)) = ν D (x) + ν D (1 + x −1 y) ≥ ν D (x) .
x −1 y ∈ O is equivalent to ν D (x) = min{ν D (x), ν D (y)}.
□
6.12. Remark. –––– For I ⊂ O the maximal ideal, O/I is a division algebra.
O is Azumaya if and only if I = mO.
6.13. Proposition. –––– Assume R to be complete. LetD be a (finite) division
algebra over K and consider the Azumaya algebra A = M n (D) for a certainn ∈Æ.
Then,
i) ∆ := M n (O) is a maximal order in A.
ii) In ∆, every left ideal is projective as a ∆-module.
Proof. i) is already proven. Cf. Remark 6.8.
ii) Let I ⊆ ∆ be any left ideal.
We observe first that ∆/m∆ ∼ = M n (O/m) is a matrix algebra over a skew field.
In particular, every ∆/m∆-module is projective as a ∆/m∆-module. Consequently,
every ∆/m∆-module is of cohomological dimension ≤ 1 as a ∆-module.
This is true, in particular, for I/mI.
For every i ≥ 1 and every ∆-module M, we have the fragments
Ext i ∆ (I, M ) ·π
−→ Ext i ∆ (I, M )
d
−→ Ext i+1
∆ (I/πI, M )
of the Ext-long exact sequence. Here, as i +1 ≥ 2, the rightmost term vanishes.
This means, E := Ext i ∆
(I, M ) has the property that E/mE = 0.
If M is a finite ∆-module then E is a finite ∆-module and hence also a finite R-
module. Nakayama’s lemma implies that Ext i ∆ (I, M ) = E = 0. For the general
case, we note that Ext commutes with filtered direct limits in the second argument.
This yields Ext i ∆ (I, M ) = 0 for i ≥ 1, unconditionally. I is projective.
□
6.14. Corollary. –––– Let R be a complete discrete valuation ring and A be an
arbitrary Azumaya algebra over K = Q(R). Then, there exists a maximal order
∆ ⊂ A over R such that every left ideal in ∆ is projective.
Proof. There exists a division algebra D such that A = M n (D).
□

Sec. 6] ORDERS IN CENTRAL SIMPLE ALGEBRAS 81
6.15. Remark. –––– We want to abolish the completeness assumption. It will
turn out that Corollary 6.14 is true for arbitrary discrete valuation rings.
6.16. Example. –––– It is, however, not true that every division algebra over a
non-complete discrete valuation ring has a unique maximal order.
Consider, for example, the quaternion algebraÀ:=É·1 ⊕É·i ⊕É·j ⊕É·k
with i 2 = j 2 = k 2 = −1 and ij = −ji = k, etc. This is a division algebra
over K =É. InÉ, consider the discrete valuation ring R =(5) corresponding
to the 5-adic valuation. This leads to ̂É=É5.
We have thatÀ⊗ÉÉ5 ∼ = M 2 (É5) splits since (−1) is a square inÉ5. Lemma 6.18
will show that there is a one-to-one correspondence between maximal orders
inÀover(5) and maximal orders in M 2 (É5) over5. In particular, there are
several maximal orders.
6.17. Sublemma. –––– Let V be a finite-dimensional K-vector space and put
̂V := V ⊗ K ̂K.
Then, there is a bijection
{R-lattices of full rank in V } −→ { ̂R-lattices of full rank in ̂V } ,
Γ ↦→ i
Γ ⊗ R ̂R ,
Proof. We put n := dim V .
Γ ∩ V ι ← Γ .
“ι◦i = id” This follows from the fact that ̂R is faithfully flat over R.
“i◦ι = id” Choose a basis for V .
Then, an ̂R-lattice of full rank in ̂V is given by n linearly independent vectors
in ̂K n , i.e. by a matrix M ∈ GL n (̂K ).
The lattice is not altered by column operations. I.e., by the interchange of two
columns or by adding λ times a column to another column for λ ∈ ̂R. It is not
changed either by the multiplication of a column with a unit of ̂R.
The goal is to verify that these operations allow to transform M into a matrix
in GL n (K ).
Column operations alone may bring M into upper triangular form. Multiplying
the columns by units, we obtain a matrix as follows,
⎛
⎞
π a 1
0 0 . . . 0
c 21 π a 2
0 . . . 0
c 31 c 32 π a 3
. . . 0
.
⎜
⎝
.
. . . ..
⎟
. ⎠
c n1 c n2 c n3 . . . π a n

82 ON THE BRAUER GROUP OF A SCHEME [Chap. II
The entries under the diagonal are not yet in K. We transform them inductively,
row-by-row. For the first row, there is nothing left to do. So assume, the
entries in rows 1, . . . , k − 1 are already in K. Write c kj = u kj π e j
with a
unit u kj . The operations allowed may alter c kj by any element from ̂Rπ a k .
Since ̂Rπ e j /̂Rπ a k ∼ = Rπ
e j/Rπ a k, the assertion follows.
□
6.18. Lemma. –––– There is a bijection
{ (Maximal) orders in A } −→ { (Maximal) orders in A ⊗ K ̂K } ,
∆ ↦→ ∆ ⊗ R ̂R .
Proof. In our situation, an order is the same as a lattice of full rank which
inherits a ring structure from the central simple algebra. It is, therefore, to
be shown that the operations described in the sublemma above preserve such
ring structures.
This is obvious for ∆ ↦→ ∆ ⊗ R ̂R.
For the other direction, the map is given by Λ ↦→ Λ ∩ A. We note that
to have a ring structure means to contain the unit element and closedness
under multiplication. The intersection clearly preserves the unit element.
Closedness under multiplication carries over to ∆ := Λ ∩A, too. Indeed, by the
sublemma, Λ = ∆⊗ R ̂R. If we had ∆ ⊗ R ∆ ։ I and I/∆ ≠ 0 then this would
remain true after the faithful flat base change from R to ̂R.
□
6.19. Proposition. –––– LetR be any discretevaluation ring and A bean arbitrary
Azumaya algebra over K = Q(R). Then, there exists a maximal order ∆ ⊂ A
over R such that every left ideal in ∆ is projective.
Proof. If R is complete then this is Corollary 6.14.
In general, we have a maximal order Λ ⊂ A⊗ R ̂R over ̂R such that every left
ideal in Λ is projective. Lemma 6.18 provides us with a maximal order ∆ ⊂ A
such that ∆⊗ R ̂R = Λ. We claim, in ∆, every left ideal I is projective.
Indeed, I⊗ R̂R is an ideal in ∆⊗ R̂R and therefore projective. Projectivity descends
under this faithful flat map as always
Ext i R (M,N)⊗ R ̂R = Ext îR (M ⊗ R ̂R, N ⊗ R ̂R) .
□
6.20. Lemma. –––– In ∆, every left ideal is principal.
Proof. Let I ⊆ ∆ be any left ideal. Then, KI is a left ideal in A.
Hence, KI = Ae = K ∆e for some idempotent e. In particular, I and ∆e
are free R-modules of the same rank.

Sec. 6] ORDERS IN CENTRAL SIMPLE ALGEBRAS 83
From this, it follows that the ideals I/mI and (∆e)/m(∆e) in the central simple
algebra ∆/m∆ are isomorphic.
As I and ∆e are both projective, we may lift a pair of mutually inverse isomorphisms
to homomorphisms f : I → ∆e and g : ∆e → I. f ◦g and g ◦ f are
automorphisms by virtue of Nakayama’s lemma. The assertion follows. □
6.21. Proposition. –––– Let R be a discrete valuation ring, K = Q(R), and A be
an Azumaya algebra over K.
Then, all maximal orders in A over R are conjugate to each other.
Proof. Let ∆ and ∆ ′ be two such orders. We may assume that, in ∆, every left
ideal is principal. Put
F := {x ∈ A | x∆ ′ ⊆ ∆} .
This is a left ∆-module in A, hence a broken ideal. It follows that F = ∆t for a
suitable t ∈ A.
∆ and ∆ ′ are orders. Hence, they are full lattices. This implies, there exists
r ∈ R\{0} such that r∆ ′ ⊆ ∆. We have r ∈ F. Consequently, t ∈ A is an
invertible element.
F is, by definition, a right ∆ ′ -module. This shows
t∆ ′ ⊆ ∆t ,
hence ∆ ′ t −1 ⊆ ∆t. Maximality yields the assertion.
□
6.22. Corollary. –––– Let R be a discrete valuation ring, K = Q(R), A an
Azumaya algebra over K, and Λ ⊂ A be a maximal order over R.
Then, in Λ, every left ideal is principal.
Proof. Combine Lemma 6.20 with Proposition 6.21.
□
6.23. Remark. –––– There are analogous results for right ideals.
6.24. Remarks. –––– i) If A allows an extension A which is an Azumaya algebra
over SpecR then A is necessarily a maximal order (Proposition 6.7).
Proposition 6.21 shows that every maximal order is Azumaya in this case.
ii) If A does not allow an extension as an Azumaya algebra then the same is true
for M n (A). Indeed, for Λ a maximal order in A, we have that Λ⊗Λ op → End(Λ)
is not surjective. This remains true when going over from Λ to the full matrix
algebra M n (Λ).

84 ON THE BRAUER GROUP OF A SCHEME [Chap. II
7. The Theorem of Auslander and Goldman
7.1. Theorem (M. Auslander and O. Goldman). —–
Let X be an integral scheme which is regular, separated, and quasi-compact. Assume
that the dimension of X is at most two.
i) Then, one has
Br(X ) = \
codimx=1
where the intersection takes place in Br(Q(X )).
ii) In particular, Br(X ) = Br ′ (X ).
Br(Spec O X,x )
Proof. Recall that, by Corollary 5.5, Br(Spec O X,x ) = Br ′ (Spec O X,x ). Further,
Br(Q(X )) = Br ′ (Q(X )).
By virtue of the general Fact 4.2, we have that Br(X ) ⊆ Br(Q(X ))
and Br(Spec O X,x ) ⊆ Br(Q(X )). Thus, the natural maps Br(X ) → Br(Spec O X,x )
are all injective. This shows the “⊆”-part of i).
The same argument is true for the cohomological Brauer group. We obtain that
Br ′ (X ) ⊆ \
Br(Spec O X,x ) . (‡)
codimx=1
This, in turn, makes sure that the second assertion immediately follows
from the first. Indeed, the inclusion (‡) together with i) immediately yields
Br ′ (X ) ⊆ Br(X ). The inclusion the other way round was established in Fact 4.2.
The main part is to prove “⊇”. For that, let α ∈ Br ′ (Q(X )) such that
α ∈ \
Br(Spec O X,x ) .
codim x=1
We choose an Azumaya algebra A over Q(X ) corresponding to the class α.
By Lemma 6.4, there exists a maximal O X -order A in A. We claim, A is a
locally free sheaf.
This may be tested locally. Let x ∈ X be any point on X. We write R := O X,x
and denote the maximal ideal in R by m. The maximality property of A implies
that A is equal to its bidual. In particular, A x = A ∨∨
x .
We consider the short exact sequence
0 −→ M −→ F 0 −→ A ∨
x −→ 0
of O X,x -modules with F 0 free. Dualizing yields
0 A ∨∨
x
F ∨ 0
M ∨ . . . .
A

Sec. 7] THE THEOREM OF AUSLANDER AND GOLDMAN 85
M ∨ := Hom R (M,R) is a torsion-free R-module. Thus, M ∨ → M ∨ ⊗ R K
is injective. We see, M ∨ may be embedded into a K-vector space. It, therefore,
allows an embedding into a free R-module F 1 , too,
0 −→ A x −→ F ∨ 0 −→ F 1 −→ Q −→ 0 .
R is of cohomological dimension ≤ 2 [Mat, Theorem 19.2]. To establish
that A x is free, it is sufficient to verify that Tor R i (A x , R/m) = 0 for i = 1
and i = 2 [Mat, §19, Lemma 1].
For this, writing Q 1 := ker(F 1 → Q), we see
Tor R 2 (A x, R/m) = Tor R 3 (Q 1, R/m) = 0
and
Tor R 1 (A x, R/m) = Tor R 2 (Q 1, R/m) = Tor R 3 (Q, R/m) = 0 .
A is a locally free sheaf.
Corollary 1.5 shows that the non-Azumaya locus of A is a closed subset T ⊆ X,
pure of codimension one. We have to show that T = ∅.
Assume the contrary. Then, there is an irreducible divisor D = {x} ⊆ T for x
a codimension one point. By assumption, A allows an extension to Spec O X,x
as an Azumaya algebra B. Furthermore, we may consider A | SpecOX,x . This is a
maximal order in A over Spec O X,x . As maximal orders, B and A | SpecOX,x are
conjugate by Proposition 6.21. In particular, A | SpecOX,x is an Azumaya algebra.
Thus, the assumption D = {x} ⊆ T leads to a contradiction.
7.2. Remark. –––– Using more refined methods, it was shown that the two
Brauer groups coincide in several more cases.
The most general result in this direction is due to O. Gabber ([Ga, Hoo] or
[K/O80]). It shows that Br(X ) = Br ′ (X ) tors if X = U ∪ V is the union of two
affine schemes such that U ∩ V is again affine.
Quite recently, S. Schröer [Schr] showed Br(X ) = Br ′ (X ) tors for any quasicompact,
separated, geometrically normal algebraic surface.
If one assumes X to be regular and separated then the subscript may be
=/2
omitted
in view of Fact 4.2. A. Grothendieck knew that Br ′ (X ) may be non-torsion,
even for an affine normal surface over, in 1965/66, already [GrBrII, Remarque
1.11.b].
D. Edidin, B. Hassett, A. Kresch, and A. Vistoli [E/H/K/V] (see also [Bt])
constructed an example of a non-separated surface such that Br ′ (X )
but Br(X ) = 0. X is the union of two affine schemes, regular in codimension
one, but not normal.
□

86 ON THE BRAUER GROUP OF A SCHEME [Chap. II
8. Examples
i. Some rings and fields from number theory. —
8.1. Examples. –––– Let p be a prime and K be a finite field extension ofÉp.
i) Then, Br(SpecK ) =É/.
This is shown by local class field theory [Se67, Sec. 1, Theorems 1 and 2].
Classically, the isomorphism is given as follows. It turns out that
Br(SpecK ) = H 2( Gal(K nr /K ), (K nr ) ∗)
for K nr the maximal unramified extension. The valuation ν : K nr →induces
H 2( Gal(K nr /K ), (K nr ) ∗) ∼ =
−→ H 2( Gal(K nr /K ),) = H 2 (̂,)
= Hom(̂,É/) =É/.
ii) For O K , the ring of integers, one has Br(Spec O K ) = 0.
Indeed, by the description of Br(SpecK ), the canonical homomorphism
Br ′ (SpecK ) −→ Hom ( Gal(p/p),É/)
from Proposition 4.8 is bijective.
8.2. Examples. –––– i) One has Br(SpecÊ) = 1 2/.
ii) One has Br(Spec) = 0.
8.3. Examples. –––– Let K be a number field.
i) Then,
Br(SpecK ) = ker ( M
s :
pprimeÉ/⊕ M
2/−→É/) .
σ : K→Ê1
Here, s is just the summation.
→É/
This is shown by global class field theory [Ta67, 11.2]. The isomorphism is
induced by the canonical maps
i ν : Br(SpecK ) → Br(SpecK ν )
for ν ∈ Val(K ) composed with the invariant map inv ν : Br(SpecK ν )
(or to 1 2/or 0, respectively) from Example 8.1.i).

Sec. 8] EXAMPLES 87
ii) Denote by r the number of real embeddings of K. Then, for the integer
ring O K , one has
{ (
1
Br(Spec O K ) = 2/) r−1 if r ≥ 1 ,
0 otherwise .
In particular, Br(Spec) = 0.
Here, the canonical homomorphism
Br ′ (SpecK ) −→
M
codimx=1
Hom(G x ,É/)
from Proposition 4.8 is nothing but the projection
ker ( M
M
s :
2/−→É/) −→
ii. Geometric examples. —
pprimeÉ/⊕ M
σ : K→Ê1
p primeÉ/.
8.4. Lemma. –––– Let n ∈Æand X be any scheme such that n is invertible on X.
Then, there is the short exact sequence
0 −→ Pic(X )/n Pic(X )
c 1
−→ H
2
ét (X, µ n) −→ Br ′ (X ) n −→ 0 .
Proof. This follows from the long exact sequence in étale cohomology associated
to the Kummer sequence
0 −→ µ n −→m
n
−→m −→ 0 .
□
8.5. Proposition. –––– Let k ⊆be an algebraically closed field and X be a
scheme which is regular and proper over k.
Then, Br ′ (X ) = (É/) rkH 2 (X (),)−rkNS(X )
⊕ H 3 (X (),) tors .
Proof. Br ′ (X ) is torsion in view of Fact 4.2.
For the middle term in the exact sequence above, by [SGA4, Exp. XVI, Corollaire
1.6], we have H 2 ét (X, µ n) ∼ = H 2 ét (X, µ n ). Further, the comparison theorem
[SGA4, Exp. XI, Théorème 4.4] shows H 2 ét (X, µ n ) ∼ = H 2 (X (),/n).
On the other hand, by the Theorem of Murre and Oort [SGA6, Exp. XII,
Corollaires 1.2 et 1.5.a)], the Picard functor is representable by a scheme Pic X/k .
This is a group scheme, hence smooth over k. It is the disjoint union of quasiprojective
k-schemes. Among other properties, one has Pic X/k (k) = Pic(X )
and Pic X/k () = Pic(X).
The connected components of Pic X/k are in bijection with NS(X ).
Base change from k todoes not change these components. In particular,
NS(X ) = NS(X).

88 ON THE BRAUER GROUP OF A SCHEME [Chap. II
Corresponding to this, Pic(X ) is an extension of the Néron-Severi group NS(X )
which is finitely generated by a divisible group,
0 −→ Pic 0 (X ) −→ Pic(X ) −→ NS(X ) −→ 0 .
This shows Pic(X )/n Pic(X ) ∼ = NS(X )/nNS(X ) ∼ = NS(X)/nNS(X).
By consequence, we may assume from now on that k =.
Doing this, we first observe that the homomorphism
Pic(X )/n Pic(X ) −→ H 2 ét (X, µ n)
induced by the Kummer sequence is compatible with the usual first Chern class
homomorphism coming from the exponential sequence. Indeed, there is the
commutative diagram
0
2πi O X
exp O ∗ X
0
exp( 2πi
n ·)
0 µ n O ∗ X
exp( 1 n ·)
(·) n O ∗ X
of exact sequences of sheaves which induces the commutative diagram
Pic(X )
Pic(X )
=
0
= H 1 (X, OX ∗ ) H 2 (X,)
=
= H 1 (X, O ∗ X ) H 2 (X, µ n ) .
By the Lefschetz theorem on (1, 1)-classes [G/H, p. 163], the cokernel of the
first Chern class homomorphism
c 1 : Pic(X ) −→ H 2 (X,)
is always torsion-free. Writing r := rkH 2 (X (),) and ρ := rkNS(X ), we
therefore have coker c 1
∼ =r−ρ . Thus, the cokernel of the induced homomorphism
c 1 ⊗/n: Pic(X )/n Pic(X ) → H (X,)⊗/n
2
is isomorphic to (/n) r−ρ .
Further, the universal coefficient theorem [Sp, Chap. 5, Sec. 5, Theorem 10]
yields the short exact sequence
0 → H 2 (X (),) ⊗/n−→ H 2 (X (),/n) −→ H 3 (X (),) n → 0 .
Thus, from Lemma 8.4 we see that there is a surjective canonical homomorphism
Br ′ (X ) n → H 3 (X (),) n the kernel of which is isomorphic to
coker(c 1 ⊗/n) ∼ = (/n) r−ρ .

Sec. 8] EXAMPLES 89
Altogether, there is a short exact sequence
0 −→ H −→ Br ′ (X ) −→ H 3 (X (),) tors −→ 0
where #H n = n r−ρ for each n ∈Æ. A simple application of the structure
theorem for finite abelian groups shows that (É/) r−ρ is the only torsion
group having this property.
Finally, there are no non-trivial extensions of a finite torsion group by (É/) r−ρ .
□
8.6. Remark. –––– A generalization to an arbitrary algebraically closed field k
of characteristic zero is not difficult. Indeed, to describe X only finitely many
data are needed. Thus, X ∼ = X 0 × Speck0 Speck for a certain scheme X 0 over a
field k 0 which is finitely generated overÉ. k 0 allows an embedding into.
8.7. Examples. –––– Let k ⊆be an algebraically closed field.
i) Let X be a smooth proper curve over k. Then, Br(X ) = 0.
(Actually, in this case, one has even Br(Q(X )) = 0 by the Theorem of Tsen
[SGA4 1 , Arcata, III, Théorème (2.3)].)
2
ii) Let X be a rational surface over k (proper, not necessarily minimal).
Then, Br(X ) = 0.
iii) Let X be a K3 surface over k. Then, Br(X ) = (É/) 22−rkPic(X ) .
iv) Let X be an Enriques surface over k. Then, Br(X ) =/2.
8.8. Fact. –––– Let k ⊆be an algebraically closed field and X be a smooth
complete intersection of dimension ≥ 3 in P n k .
Then, Br ′ (X ) = 0.
Proof. The Lefschetz hyperplane theorem [Bot, Corollary of Theorem 1]
implies that H 1 (X (),) = 0 and H 2 (X (),) =. From the universalcoefficient
theorem for cohomology [Sp, Chap. 5, Sec. 5, Theorem 3], we
deduce H 2 (X (),) =and
H 3 (X (),) ∼ = Hom(H 3 (X (),),) .
In particular, H 3 (X (),) is torsion-free.
□
iii. Varieties over a number field or local field. —
8.9. –––– In this subsection, we deal with the case that X is a scheme over an
algebraically non-closed field K. The relevant cases for us are that K is either a
number field or a local field.

90 ON THE BRAUER GROUP OF A SCHEME [Chap. II
It is not true that the homomorphism Br(SpecK ) → Br(X ), induced by the
structural map, is always injective. For this, an additional hypothesis is needed.
8.10. Proposition. –––– Let K be a field and π : X → SpecK be any scheme
over K. Assume that
a) either K is a local field and X (K ) ≠ ∅
b) or K is a number field and ∏ X (K ν ) ≠ ∅.
ν∈Val(K )
Then, the canonical map π ∗ : Br(SpecK ) → Br(X ) is injective.
Proof. a) A K-valued point x : SpecK → X induces a section of π ∗ .
b) Put S := F ν SpecK ν . The assumption says that X has an S-valued point.
Therefore, the natural homomorphism
i : Br(K ) → Br(S) = ∏ Br(K ν )
ν
factors via π ∗ . i is injective by Example 8.3.i).
8.11. Proposition. –––– Let K be a number field or a local field and
π : X → SpecK be a geometrically integral scheme which is proper over K.
i) Then, there is an exact sequence
0 −→ Pic(X )
i
−→ Pic(X K
) Gal(K/K ) π
−→ Br(SpecK ) −→
∗
π ∗
−→ ker(Br ′ (X ) i′
→ Br ′ (X K
)) −→ H 1( Gal(K/K ), Pic(X K
) ) −→ 0 .
Here, i and i ′ are induced by the natural morphism X K
→ X.
ii) If π ∗ : Br(SpecK ) → Br ′ (X ) is injective then there is the short exact sequence
0 → Br(SpecK) → π∗
ker(Br ′ (X ) → i′
Br ′ (X K
)) → H 1( Gal(K/K ), Pic(X K
) ) → 0 .
Further, Pic(X ) ∼ = Pic(X K
) Gal(K/K ) .
Proof. Consider the Hochschild-Serre spectral sequence
in étale cohomology.
E p,q
2
:= H p( Gal(K/K ), H q ét (X K ,m) ) =⇒ H n ét (X,m)
As X is integral and proper over K, we have that H 0 ét (X K ,m) = K ∗ . Further,
H 1 ét (X K ,m) = Pic(X K
) and H 2 ét (X K ,m) = Br ′ (X K
).
This yields E 0,1
2
= Pic(X K
) Gal(K/K ) and E 2,0
2
= Br(SpecK ). Furthermore,
E 1,1
2
= H 1( Gal(K/K ), Pic(X K
) ) and E 0,2
2
= Br ′ (X K
) Gal(K/K ) .
The classical Theorem Hilbert 90 shows that E 1,0
2
= H 1( Gal(K/K ), K ∗) = 0.
□

Sec. 8] EXAMPLES 91
In addition, if K is a number field then one has E 3,0
2
= H 3( Gal(K/K ), K ∗) = 0
by [Ta67, Section 11.4]. Otherwise, if K is a local field then, by class field theory
H 3( Gal(L/K ), L ∗) ∼ = H
1 ( Gal(L/K ),) = 0 for every finite extension of K.
Thus, H 3( Gal(K/K ), K ∗) = 0.
Finally, E 1 = H 1 ét (X,m) = Pic(X ) and E 2 = H 2 ét (X,m) = Br ′ (X ).
The sequence in i) is nothing but a sequence of lower order terms in the spectral
sequence E.
ii) directly follows from i).
8.12. Corollary. –––– Let K be a number field or local field and π : X → SpecK
be a geometrically integral scheme which is proper over K.
i) Then, Br(X ) is a countable group.
ii) Suppose that rk NS(X ) = rkH 2 (X (),) and rkH 1 (X (),) = 0. Then,
Br(X )/π ∗ Br(SpecK ) is finite.
Proof. i) Br(X K
) is a countable group by Proposition 8.5. It is therefore sufficient
to verify that ker(Br(X ) → Br(X K
)) is a countable group. We will even show
that ker(Br ′ (X ) → Br ′ (X K
)) is countable.
It is known from Example 8.3.i) that Br(SpecK ) is countable. We are left with
proving H 1( Gal(K/K ), Pic(X K
) ) is a countable group.
There is a short exact sequence of Gal(K/K )-modules
0 −→ Pic 0 (X K
) −→ Pic(X K
) −→ NS(X K
) −→ 0 .
Here, H 1( Gal(K/K ), NS(X K
) ) is finite since NS(X K
) is a finitely generated
abelian group acted upon by a finite quotient of Gal(K/K ).
On the other hand, Pic 0 (X K
) is divisible and all the groups Pic 0 (X K
) n are finite.
The Kummer sequence
0 −→ Pic 0 (X K
) n −→ Pic(X K
)
n
−→ Pic(X K
) −→ 0
induces a surjection H 1( Gal(K/K ), Pic 0 (X K
) n
) → H
1 ( Gal(K/K ), Pic 0 (X K
) ) n .
In particular, H 1( Gal(K/K ), Pic 0 (X K
) ) n is finite and H 1( Gal(K/K ), Pic 0 (X K
) )
is countable.
ii) Here, the assumption rk NS(X ) = rkH 2 (X (),) makes sure that Br(X K
)
is finite. It is therefore sufficient to verify that
is finite. Again, we will even show that
ker(Br(X ) → Br(X K
))/π ∗ Br(SpecK )
ker(Br ′ (X ) → Br ′ (X K
))/π ∗ Br(SpecK ) ∼ = H 1( Gal(K/K ), Pic(X K
) )
□

92 ON THE BRAUER GROUP OF A SCHEME [Chap. II
is finite.
For this, the assumption rkH 1 (X (),) = 0 implies that Pic(X K
) = NS(X K
).
The assertion follows.
□
8.13. Remark. –––– The case that Br ′ (X K
) = 0 is of particular interest. For
example, this happens if X K
is a rational surface, a K3 surface, or a smooth
complete intersection of dimension ≥ 3 in P n (cf. Subsection ii).
K
In this case, the short exact sequence in assertion ii) simply means
Br ′ (X )/π ∗ Br(SpecK) ∼ = H 1( Gal(K/K ), Pic(X K
) ) .
8.14. Fact. –––– LetK beanumber field or alocalfield andX beasmoothcomplete
intersection of dimension ≥ 3 in P n K .
i) Then, Br(X ) = Br ′ (X ).
ii) The homomorphism π ∗ : Br(SpecK ) → Br(X ) induced by the structural map
is surjective.
iii) If X (K ) ≠ ∅ for K a local field or ∏ X (K ν ) ≠ ∅ for K a number field
ν∈Val(K )
then Br(X ) = Br(SpecK ).
Proof. The Lefschetz hyperplane theorem implies that Pic(X K
) =. Therefore,
H 1( Gal(K/K ), Pic(X K
) ) = H 1( Gal(K/K ),) = 0 .
Since Br ′ (X K
) = 0, Proposition 8.11 shows that π ∗ : Br(SpecK ) → Br ′ (X )
is surjective.
In particular, π ∗ does not factorize via any proper subgroup of Br ′ (X ).
Hence, Br(X ) = Br ′ (X ). This proves i) and ii).
iii) follows now from Proposition 8.10.
8.15. –––– In the situation of a curve over a local field, Br(X )/π ∗ Br(SpecK )
has been computed by S. Lichtenbaum. The following result is often referred
to as Lichtenbaum duality.
8.16. Theorem (Lichtenbaum). —– Let p beaprimenumber,K afiniteextension
ofÉp, and X be a geometrically integral curve which is proper and smooth over K.
Then, the canonical pairing
Br(X ) × X (K ) −→É/,
(α, x) ↦→ inv νp (α| x )
□
induces an isomorphism Br(X )/π ∗ ∼=
Br(SpecK ) −→ Hom(Pic 0 (X ),É/).
Proof. This is [Lic, Corollary 1].
□

Sec. 8] EXAMPLES 93
iv. Manin’s formula. —
8.17. –––– If X K
is a rational surface then H 1( Gal(K/K ), Pic(X K
) ) is often
effectively computable using the following result due to Yu. I. Manin.
8.18. Proposition (Manin, [Man,кIV,ÈÖÐÓÒº¿]). —–
Let X be a regular, integral scheme which is of finite type over a field k. Suppose
further, we are given a finite, G := Gal(k sep /k)-invariant set {D i } of divisors
on X := X × Speck Speck sep generating Pic(X ).
Denote by S ⊆ Div(X ) the group generated by {D i } and by S 0 ⊆ S the subgroup
of all principal divisors. Finally, let H ⊆ G be a normal subgroup acting trivially
on {D i }.
Assume,
i) Pic(X ) is a free abelian group and
ii) there is a perfect pairing
Then, there is a canonical isomorphism
H 1 (G, Pic(X ))
Pic(X ) × Pic(X ) −→.
i
−→ Hom((NS ∩ S 0 )/NS 0 ,É/) .
Here, N : S → S denotes the norm map on S as a G/H -module.
Proof. By Lemma 8.19, we have
Ĥ −1 (G/H , Pic(X )) ∼ = (NS ∩ S 0 )/NS 0 .
To this relation, we apply the duality theorem [C/E, Chap. XII, Corollary 6.5].
It shows that
Ĥ 0 (G/H , Hom(Pic(X ),É/)) ∼ = Hom(Ĥ −1 (G/H , Pic(X )),É/)
It remains to construct a canonical isomorphism
= Hom((NS ∩ S 0 )/NS 0 ,É/) .
Ĥ 0 (G/H , Hom(Pic(X ),É/)) ∼ = H 1 (G, Pic(X )) .
For this, since Pic(X ) is free, we have a short exact sequence of G/H -modules
0 −→ Hom(Pic(X ),) −→ Hom(Pic(X ),É) −→ Hom(Pic(X ),É/) −→ 0 .
As Hom(Pic(X ),É) is uniquely divisible, this shows
Ĥ 0 (G/H , Hom(Pic(X ),É/)) ∼ = H 1 (G/H , Hom(Pic(X ),)) .

94 ON THE BRAUER GROUP OF A SCHEME [Chap. II
We finally apply the perfect pairing on Pic(X ).
□
8.19. Lemma. –––– Let G be a finite group and D be a finite G-set. Consider the
free abelian group S over D as a G-module. Then, for every G-submodule S 0 ⊆ S,
there is a canonical isomorphism
Ĥ −1 ∼=
(G, P) −→ (NS ∩
=
S 0 )/NS 0 .
Proof. We first calculate Ĥ −1 (G, S) directly, according to the definition. Observe
that S is a direct sum of G-modules of the form[G/H ] for normal
subgroups H ⊆ G. For these, one has
H 0 (G,[G/H ]) ∼
and the isomorphism is given by the map[G/H ] →sending a formal
sum to the sum of all coefficients. Further, the homomorphism→[G/H ]
mapping 1 to ∑ g∈G/H g has no kernel. Thus, Ĥ −1 (G, S) = 0.
The short exact sequence
0 −→ S 0 −→ S −→ P −→ 0
of G-modules therefore induces a long exact sequence
0 −→ Ĥ −1 (G, P) −→ Ĥ 0 (G, S 0 ) −→ Ĥ 0 (G, S) .
on Tate cohomology. We may consequently write
Ĥ −1 (G, P) = ker(S G 0 /NS 0 → S G /NS) ∼ = (NS ∩ S 0 )/NS 0 .
□
8.20. Remark. –––– The assumptions are fulfilled if X is a proper surface such
that H 1 (X (),) = 0 and H 2 (X, O X ) = 0. For example, X may be a (not
necessarily minimal) rational surface or a K3 surface.
The first condition implies H 1 (X (),) = 0 and H 2 (X (),) is torsion-free.
In particular, the first Chern class
c 1 : Pic(X) → H 2 (X (),)
is injective. The second condition makes sure it is a surjection. As H 2 (X (),)
is torsion-free, Poincaré duality yields that the intersection pairing is perfect.
8.21. –––– Let us consider the case that X is a smooth cubic surface in a bit
more detail.

Sec. 8] EXAMPLES 95
If is well known that, on a cubic surface X over an algebraically closed field,
there are exactly 27 lines D 1 , . . . , D 27 . Further, Pic(X ) ∼ =7 is generated by
the classes of these lines.
The set L := {D 1 , . . . , D 27 } is equipped with the intersection product
〈 , 〉: L ×L → {−1, 0, 1}. The pair (L , 〈 , 〉) is the same for all smooth
cubic surfaces. It is well known [Man,кIV,ÌÓÖѽººµ] that the group
of permutations of L respecting 〈 , 〉 is isomorphic to W (E 6 ). We fix such
an isomorphism.
Further, we put S ∼ =27 to be the free group over L and
{ ∣ }
∣∣
S 0 := ∑ a i D i ∑a i 〈D i , D j 〉 = 0 for j = 1, . . . , 27 .
i i
Let X be a smooth cubic surface over a number field K. Then, Gal(K/K )
operates canonically on the set L X of the 27 lines on X K
. Fix a bijection
∼ =
i X : L X −→ L respecting the intersection pairing. This induces a group homomorphism
ι X : Gal(K/K ) → W (E 6 ). We denote its image by G ⊂ W (E 6 ).
As L X is clearly Gal(K/K )-invariant, we may apply Proposition 8.18. It shows
H 1( Gal(K/K ), Pic(X K
) ) = H 1 (G, S/S 0 ) = Hom ( (NS ∩ S 0 )/NS 0 ,É/) .
The right hand side depends only on the conjugacy class of the subgroup
G ⊆ W (E 6 ). Actually, it depends only on the decomposition of L
into G-orbits.
8.22. Remark. –––– Almost as a byproduct, we may write down a similar
formula for another arithmetic invariant of the cubic surface X, namely for the
Picard rank. Indeed, Proposition 8.11.i) shows
rkPic(X ) = rkH 0( Gal(K/K ), Pic(X K
) )
and, therefore, rk Pic(X ) = rk(S/S 0 ) G = rkN(S/S 0 ) = rk[(NS + S 0 )/S 0 ]. I.e.,
rkPic(X ) = rkNS/(NS ∩ S 0 )
= rkNS − rk(NS ∩ S 0 ) .
Observe that rkNS is the number of Gal(K/K ) orbits into which the
27 lines are decomposed. The group NS ∩ S 0 has to be computed anyway
for H 1( Gal(K/K ), Pic(X K
) ) .
8.23. Explicit computation. –––– There are exactly 350 conjugacy classes of
subgroups in W (E 6 ). UsingGAP, we computed the right hand side in each case.
The result is the following list.

96 ON THE BRAUER GROUP OF A SCHEME [Chap. II
TABLE 1. H 1 (G,Pic) and rk Pic(X ) for smooth cubic surfaces
1 #U = 1 [ ], #H^1 = 1, Rk(Pic) = 7, Orbits [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
2 #U = 2 [ 2 ], #H^1 = 1, Rk(Pic) = 6, Orbits [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 2, 2, 2, 2, 2, 2 ]
3 #U = 2 [ 2 ], #H^1 = 4 [ 2, 2 ], Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2 ]
...........................................
347 #U = 1440 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 12, 15 ]
348 #U = 1920 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 10, 16 ]
349 #U = 25920 [ ], transitive
/2
350 #U = 51840 [ 2 ], transitive
/3
0 for 257 classes,
for 65 classes,
for 16 classes,
(/2) 2 for 11 classes,
The code written in GAP as well as the full list are reproduced in the appendix.
8.24. –––– It turns out that H 1( Gal(K/K ), Pic(X K
) ) is isomorphic to
(/3) 2
for one class.
The computations took approximately 28 seconds of CPU time.
In 25 of the 257 classes, we have H 1( Gal(K/K ), Pic(X K
) ) = 0 for the trivial
reason that the operation of G on the 27 lines is transitive. Among these classes,
there are the general cubic surfaces. I.e., the case that G = W (E 6 ). We will
consider general cubic surfaces in more detail in Chapter IX.
A different case where H 1( Gal(K/K ), Pic(X K
) ) = 0 is that when G = 0.
This means that all 27 lines are defined over K.
The case where H 1( Gal(K/K ), Pic(X K
) ) = (/3) 2 was described already
by Yu. I. Manin [Man,кVI,ËÐ×ØÚº]. It occurs in a situation
when G ∼ =/3splits the 27 lines into nine orbits of three lines each. This is
realized, for example, by the diagonal cubic surface “x 3 + y 3 + z 3 + dw 3 = 0”
in P 3 K when K ( √ 3 d)/K is a cubic Galois extension.
8.25. Remark. –––– That only these five groups may appear was known to
Sir P. Swinnerton-Dyer in 1993, already. Swinnerton-Dyer’s proof fills almost
the entire article [S-D93]. A different proof is given in the Ph.D. thesis of
P. K. Corn [Cor]. That proof consists of a mixture of mathematical arguments
with Computer work.
A purely computational approach was, seemingly, considered rather hopeless at
that time (cf. [Cor, Proposition 1.3.11] and the remarks before).

Sec. 8] EXAMPLES 97
8.26. Remark. –––– We find a Picard rank of
1 for 137 classes,
2 for 133 classes,
3 for 62 classes,
4 for twelve classes,
5 for four classes,
6 for one class,
7 for one class.
8.27. Remark. –––– In the next chapter, we will use the list given in Table 1
in order to describe the Brauer-Manin obstruction for a large class of diagonal
cubic surfaces.

CHAPTER III
AN APPLICATION:
THE BRAUER-MANIN OBSTRUCTION
I was at first almost frightened when I saw such mathematical force made
to bear upon the subject, and then wondered to see that the subject stood it
so well.
MICHAEL FARADAY (1857, in a letter to J. C. Maxwell)
1. Adelic points
i. The concept of an adelic point. —
1.1. Definition. –––– Let K be a number field and X be a scheme over K.
Then, an adelic point on X is a morphism SpecK → X of K-schemes from
the spectrum of the adele ring [Cas67, Sec. 14]. The set of all adelic points on X
is denoted by X (K).
1.2. Remark. –––– The scheme SpecK is not Noetherian.
1.3. Remark. –––– K is a subring ofK via the diagonal homomorphism.
Thus, every K-valued point on X induces an adelic point.
1.4. Remarks. –––– i)K is canonically a sub-K-algebra of ∏ ν K ν . Therefore,
every adelic point on X gives rise to a ∏ ν K ν -valued point.
ii) A ∏ ν K ν -valued point induces a K ν -valued point for every ν ∈ Val(K ).
If the scheme X is separated and quasi-compact then the ∏ ν K ν -valued points
are in a canonical bijection with ∏ ν X (K ν ). This is a technical point which we
defer to Lemma 3.2.

100 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
1.5. Lemma. –––– Let K be a number field and X be a separated K-scheme.
Then, the sequence x = (x ν ) ν ∈ ∏ ν X (K ν ) determines the corresponding adelic
point, if it exists, uniquely.
Proof. We first observe that the set
[
SpecK ν ⊂ SpecK
ν∈Val(K )
is Zariski dense. Indeed, a base of the open subsets is provided by the sets D( f )
for 0 ≠ f = ( . . . , f ν , . . . ) ∈K. If f ν ≠ 0 then SpecK ν is contained
in D( f ).
Now suppose that two morphisms x 1 , x 2 : SpecK → X induce the
same element in ∏ ν X (K ν ). This means simply that x 1 and x 2 coincide
on S ν∈Val(K ) SpecK ν .
In the category of schemes, there exists a universal morphism i : V → SpecK
such that x 1 ◦i = x 2 ◦i. According to [EGA, Chapitre I, Proposition (5.2.5)],
i is a closed embedding.
Therefore, x 1 and x 2 coincide on the Zariski closure of S ν∈Val(K ) SpecK ν . This is
the whole of SpecK.
□
1.6. Lemma. –––– Let K be a number field, O K its ring of integers, and X be a
separated O K -scheme of finite type. Denote by X its generic fiber.
Then, a ∏ ν K ν -valued point x = (x ν ) ν ∈ ∏ ν X (K ν ) is induced by an adelic point if
and only if all but finitely many of the x ν extend to O Kν -valued points on X .
Proof. “⇐=” We are given a finite set S of valuations, including all archimedean
ones, and morphisms SpecO Kν → X for ν ∉ S and SpecK ν → X for ν ∈ S.
To ease notation, we write R ν := K ν for ν ∈ S and R ν := O Kν for ν ∉ S.
Thus, we have morphisms SpecR ν → X . We claim that these may be put
together to give a morphism SpecR → X for R := ∏ ν∈Val(K ) R ν . The assertion
then follows as there is a canonical morphism SpecK → SpecR.
For that, we first note that the assertion is true when X = SpecA is an
affine scheme. Indeed, to give a ring homomorphism A → ∏ ν∈Val(K ) R ν is the
same as giving a system of ring homomorphisms {A → R ν } ν∈Val(K ) .
We cover X by finitely many affine schemes X j
∼ = SpecAj , j ∈ {1, . . . , n}.
Given a system of morphisms {SpecR ν → X } ν∈Val(K ) , we may find a decomposition
Val(K ) = S 1 ∪ . . . ∪ S n
into mutually disjoint subsets such that SpecR ν maps to X j if ν ∈ S j . For this,
observe that all the rings R ν are local.

Sec. 1] ADELIC POINTS 101
Now we use the assumption that the X j are affine. For j fixed, the morphisms
SpecR ν → X j for ν ∈ S j give rise to a morphism
Finally, observe that
nG
j=1
since the index set is finite.
Spec ( ∏
ν∈S j
R ν
) → Xj ⊆ X .
Spec ( )
∏ R ν ∼=
( )
Spec ∏ R ν
i∈S j ν∈Val(K )
“=⇒” Cover X by affine schemes X 1 , . . . , X n and write
X j = Spec O K [T j1 , . . . , T jlj ]/I j .
The adelic point given induces homomorphisms of O K -algebras
ϕ j : O K [T j1 , . . . , T jlj ]/I j −→ (K ) fj
where ( f 1 , . . . , f n ) = (1). Fix adeles g 1 , . . . , g n such that g 1 f 1 + . . . +g n f n = 1
and write
h jk
:= ϕ j (T jk )
where h jk ∈K and e jk ≥ 0.
f e jk
j
We let S be the set of all valuations ν ∈ Val(K ) which are either archimedean
or such that not all of the adeles g j and h jk are integral at ν. This is a finite set
of valuations.
Assume ν ∉ S. Then, for one adele f i , we certainly have that ‖( f i ) ν ‖ ν ≥ 1.
Since (h jk ) ν is integral, we see that [ϕ j (T jk )] ν ∈ O Kν for every k. ϕ j induces a
morphism SpecO K → X j as claimed.
□
1.7. Definition (Topology on X (K)). —– Let K be a number field and X be
a separated K-scheme of finite type.
Then, X allows a model X which is separated and of finite type over the integer
ring O K . The sets X (K ν ) carry natural topologies as ν-adic analytic spaces.
For ν non-archimedean we have the subsets X (O Kν ) ⊆ X (K ν ) carrying the
subspace topology.
We equip the set X (K) of all adelic points on X with the restricted topological
product topology [Cas67, Sec. 13]. A basis is provided by all direct products
∏
ν
Γ ν
where Γ ν ⊆ X (K ν ) is open for all ν and Γ ν = X (O Kν ) for almost all ν.

)É
102 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
1.8. Remarks. –––– a) For two different models X 1 and X 2 , the isomorphism
=
(X 1 )É∼ −→ X ∼ =
−→ (X 2
may be extended to an open neighbourhood of the generic fiber. This implies
X 1 (O Kν ) ∼ = X 2 (O Kν ) for all but finitely many valuations ν. Hence, the definition
is independent on the choice of the model. The existence of a model is
provided by Lemma 3.4.a).
b) X (K) is a Hausdorff topological space satisfying the second axiom of countability.
i) If X = A 1 then there is a bijection between A 1 (K ) and the adele ringK.
The topology we defined on A 1 (K) coincides with the adele topology onK
defined in [Cas67, Sec. 14].
ii) If X = A 1 \{0} then X (K) corresponds to the idele group. It is, however,
equipped with the restriction of the adele topology, not with the idele topology.
1.9. Lemma. –––– Let K be a number field and X be a scheme proper over K.
Then,
X (K) = ∏ X (K ν )
ν∈Val(K )
equipped with the Tychonov topology.
Proof. Lemma 3.4.b) guarantees the existence of a model X which is proper
over O K except for a finite set S of special fibers. The valuative criterion for
properness implies that X (O Kν ) = X (K ν ) for ν ∉ S.
□
ii. Weak approximation and the Hasse principle. —
1.10. Definition. –––– Let X be a separated scheme which is of finite type over
a number field K.
a) One says that X satisfies the Hasse principle if the condition X (K) ≠ ∅
implies X (K ) ≠ ∅.
b) X is said to satisfy weak approximation if X (K ) is dense in X (K).
1.11. Theorem. –––– a) The following classes of projective schemes satisfy the
Hasse principle.
i) Smooth projective quadrics,
ii) Brauer-Severi varieties,
iii) smooth projective cubics in P nÉfor n ≥ 9,
iv) smooth complete intersections of two quadrics in P n K
for n ≥ 8.

Sec. 2] THE BRAUER-MANIN OBSTRUCTION 103
b) The following projective schemes satisfy weak approximation.
i) Projective varieties rational over K,
for n ≥ 6 satisfy-
ii) smooth complete intersections X of two quadrics in P n K
ing X (K ) ≠ ∅.
1.12. Remark. –––– These lists are not meant to be exhaustive. Further notes
and references to the literature may be found in [Sk, Sec. 5.1].
2. The Brauer-Manin obstruction
2.1. –––– Weak approximation and even the Hasse principle are not always satisfied.
Isolated counter examples have been known for a long time.
Genus one curves violating the Hasse principle have been constructed by
C.-E. Lind [Lin] as early as 1940 and by E. S. Selmer [Sel] in 1951.
The first example of a cubic surface not fulfilling the Hasse principle is due to
Sir P. Swinnerton-Dyer [S-D62]. A series of examples generalizing Swinnerton-Dyer’s
work has been given by L. J. Mordell [Mord]. We will generalize
Mordell’s examples even further in Section 5.
Diagonal cubic surfaces which are counterexamples to the Hasse principle have
been constructed by J. W. S. Cassels and M. J. T. Guy [Ca/G] as well as A. Bremner
[Bre]. These investigations were systematized by J.-L. Colliot-Thélène,
D. Kanevsky, and J.J. Sansuc [CT/K/S]. We will discuss this topic in more
detail in Section 6.
A singular quartic surface which is a counterexample to the Hasse principle has
been given by V. A. Iskovskikh [Is].
A method to explain all these examples in a unified manner was provided by
Yu. I. Manin in his book on cubic forms [Man]. This is what is nowadays called
the Brauer-Manin obstruction.
2.2. Definition. –––– Let K be a number field and X be a scheme separated and
of finite type over K.
i) Then, for each ν ∈ Val(K ), there is the local evaluation map, given by
ev ν : Br(X ) × X (K ν ) −→É/,
(α, x) ↦→ inv ν (α| x ) .
ii) Further, there is the global evaluation map or Manin map
ev: Br(X ) × X (K) −→É/,
( )
α, (xν ) ν ↦→ ∑ev ν (α, x ν ) .
ν

104 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
2.3. Proposition. –––– Let K be a number field and π : X → SpecK be a scheme
which is separated and of finite type over K.
Then,
a) The local evaluation map is
i) additive in the first variable and
ii) continuous in the second variable.
iii) If α = π ∗ a for a ∈ Br(SpecK ) then ev ν (α, ·) = inv ν (a) is constant.
iv) For each α ∈ Br(X ), there exists a finite set S of valuations including all archimedean
ones such that
ev ν (α, x) = 0
whenever ν ∉ S and x ∈ X (O Kν ).
b) The Manin map is well-defined. Further, it is
i) additive in the first variable and
ii) continuous in the second variable.
iii) In the first variable, ev factors via Br(X )/π ∗ Br(SpecK ).
iv) Further,
if x ∈ X (K ) ⊆ X (K).
Proof. a) i) and iii) are obvious.
ev(α, x) = 0
ii) Let α ∈ Br(X ) and x ∈ X (K ν ). We have to show that ev ν (α, ·) is constant in
a neighbourhood of x in the ν-adic topology.
By iii), we may assume that ev ν (α, x) = 0. Then, there is an Azumaya algebra A
over X such that [A ] = α and A | x
∼ = Mn (K ν ). By Corollary II.5.4, this implies
A | SpecO h
XKν ,x
∼ = Mn (O h X Kν ,x)
for the Henselization of the local ring at x ∈ X Kν . To define this isomorphism,
only finitely many data are required. Thus, there exist an étale morphism
f : X ′ → X and a K ν -valued point x ′ ∈ X ′ such that f (x ′ ) = x
and A | X ′
∼ = Mn (O X ′).
f induces a local isomorphism of ν-adic analytic spaces.
A | y
∼ = Mn (K ν ) for y in a ν-adic neighbourhood of x.
In particular,
iv) By Lemma 3.4.a), there exists a model X of X which is separated and of
finite type over O K . We may assume that X is reduced.
Let A be an Azumaya algebra over X such that [A ] = α. By Lemma 3.5,
there is an open subset X ◦ ⊆ X such that A may be extended to an Azumaya
algebra à over X ◦ .

Sec. 2] THE BRAUER-MANIN OBSTRUCTION 105
The closed subset X \X ◦ ⊂ X does not meet the generic fiber. Since X is of
finite type, X \X ◦ is contained in finitely many special fibers. We put S to be
the set of the corresponding valuations together with all archimedean valuations.
We have to prove that ev ν (α, x) = 0 for ν ∉ S and x ∈ X (K ν ). The point x
induces a morphism x : Spec O Kν → X of O K -schemes. The assumption ν ∉ S
implies that x actually maps to X ◦ .
Over X ◦ , there is the Azumaya algebra à extending A . The restriction
of [A ] ∈ Br(X ) along x : SpecK ν → X coincides with the restriction
of [Ã ] ∈ Br(X ◦ x
) along SpecK ν −→ X −→ ⊂
X . The latter factorizes
via Br(Spec O Kν ). We note, finally, that Br(Spec O Kν ) = 0 by virtue of Example
8.1.ii).
b) In order to show that ev is well-defined, one has to verify that the sum is
always finite. This, in turn, immediately follows from a.iv).
i) and iii) are direct consequences from a.i) and a.iii).
ii) This follows from a.ii) together with a.iv).
iv) is a consequence of the description of Br(K ) given in Example II.8.3.i).
2.4. Remark. –––– Let x ∈ X (K) be an adelic point. If there exists a Brauer
class α ∈ Br(X ) such that ev(α, x) ≠ 0 then x can not be approximated by a
sequence of K-valued points. The Brauer class α therefore “obstructs” the adelic
point x from being approximated by rational points. This justifies the name
Brauer-Manin obstruction.
2.5. Notation. –––– Let K be a number field and π : X → SpecK be a scheme
separated and of finite type over K.
For α ∈ Br(X ), we write
We define
X (K) α := {x ∈ X (K) | ev(α, x) = 0} .
X (K ) Br := \
X (K ) α .
α∈Br(X )
2.6. Remarks. –––– a) According to Proposition 2.3.b.ii), X (K) Br is always a
closed subset of X (K).
b) Proposition 2.3.b.iv) shows X (K ) ⊆ X (K) Br .
2.7. Remark. –––– As shown in Corollary II.8.12.ii), there are many particular
cases in which Br(X )/π ∗ Br(K ) is actually a finite group.
In this case, one might choose a finite system α 1 , . . . , α n ∈ Br(X ) generating
Br(X )/π ∗ Br(SpecK ). This leads to X (K) Br = X (K) α 1 ∩ . . . ∩ X (K) α n
.
□

106 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
By Proposition 2.3.a.iv), only finitely many places of K are involved in the
evaluation of α 1 , . . . , α n .
2.8. Definition. –––– Let K be a number field and π : X → SpecK be a scheme
separated and of finite type over K.
a) If X (K) Br ≠ X (K) then one says that, on X, there is a Brauer-Manin
obstruction to weak approximation.
b) If X (K) Br = ∅ and X (K) ≠ ∅ then one says that, on X, there is a
Brauer-Manin obstruction to rational points on X or that there is a Brauer-Manin
obstruction to the Hasse principle on X.
2.9. Remark. –––– In the case that X (K ) Br ≠ ∅, many authors use the somewhat
confusing formulation that the Brauer-Manin obstruction would be empty.
2.10. Remark. –––– There is clearly no Brauer-Manin obstruction when
Br(X )/π ∗ Br(SpecK ) = 0, neither to the Hasse principle nor to weak approximation.
We know several such cases from the computations described above. For example,
as indicated in II.8.23, we have Br(X )/π ∗ Br(SpecK ) = 0 for X a general
cubic surface or a smooth cubic surface such that all 27 lines are defined over K.
By Fact II.8.14, we have Br(X )/π ∗ Br(SpecK ) = 0 when X is a smooth complete
intersection of dimension ≥ 3 in P n K .
2.11. Definition. –––– If the statement
X (K ) Br ≠ ∅ =⇒ X (K ) ≠ ∅
is true for a certain class of separated K-schemes of finite type then one says that
the Brauer-Manin obstruction is the only obstruction to the Hasse principle for
that class.
2.12. Conjecture (Colliot-Thélène, cf. [CT/S, Conjecture C]). —–
The Brauer-Manin obstruction is the only obstruction to the Hasse principle for
smooth cubic surfaces.
3. Technical lemmata
3.1. –––– Let (K i ) i∈Æbe a sequence of fields. Then, there is the canonical
morphism of schemes
G
ι:
i −→ Spec
i∈ÆSpecK ( )
∏ i .
i∈ÆK

Sec. 3] TECHNICAL LEMMATA 107
3.2. Lemma. –––– LetX beaschemewhichisseparatedandquasi-compact. Then,ι
induces a bijection
X ( ∏
i∈ÆK i
) −→ ∏
i∈ÆX(K i ) .
Proof. We first note that the assertion is true when X = SpecA is an
affine scheme. Indeed, to give a ring homomorphism A → ∏ i∈ÆK i is the
same as giving a system of ring homomorphisms (A → K i ) i∈Æ.
Surjectivity. We cover X by finitely many affine schemes X j
∼ = SpecAj ,
j ∈ {1, . . . , n}. Given a system of morphisms (SpecK i → X ) i∈Æ, we may find
a decomposition
Æ=S 1 ∪ . . . ∪ S n
into mutually disjoint subsets such that SpecK i maps to X j if i ∈ S j .
Now we use the assumption that the X j are affine. For j fixed, the morphisms
SpecK i → X j for i ∈ S j give rise to a morphism
Finally, observe that
nG
j=1
since the index set is finite.
Spec ( ∏
i∈S j
K i
) → Xj ⊆ X.
Spec ( ∏
i∈S j
K i
) ∼= Spec
(
∏
i∈ÆK i
)
Injectivity. Suppose two morphisms g 1 , g 2 : Spec(∏ i∈ÆK i ) → X induce the
same morphism on F i∈ÆSpecK i . Proposition (5.2.5) of [EGA, Chapitre I] implies
that g 1 and g 2 coincide on the Zariski closure of F i∈ÆSpecK i .
We claim that F i∈ÆSpecK i is actually dense in Spec(∏ i∈ÆK i ). Indeed, a base of
the open subsets is provided by the sets D( f ) for
0 ≠ f = ( . . . , f ν , . . . ) ∈ ∏
i∈ÆK i .
If f ν ≠ 0 then SpecK ν is contained in D( f ). The claim follows.
□
3.3. Remark. –––– The assertion of the lemma is certainly not true in general
for X an arbitrary scheme. Indeed, Yoneda’s lemma would then imply
that ι: F i∈ÆSpecK i → Spec(∏ i∈ÆK i ) is an isomorphism.
ι induces, however, not even a bijection on closed points. In fact, there is
the ideal ⊕ i∈ÆK i ⊂ ∏ i∈ÆK i . According to the lemma of Zorn, this ideal is
contained in a maximal ideal which is different from those in F i∈ÆSpecK i .

108 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
3.4. Lemma. –––– Let K be a number field and X be a scheme which is separated
and of finite type over K.
a) Then, there exists a model X of X. This means that X is a scheme which is
separated and of finite type over O K and fulfills X × SpecOK SpecK ∼ = X.
b) If X is proper over K then X is proper over Spec(O K ) f for some 0 ≠ f ∈ O K .
Proof. First step. Existence of a model.
X is given by gluing together finitely many affine K-schemes U i = SpecR i .
The intersections U i ∩ U j are affine. Thus, gluing takes place along isomorphisms
ϕ ij : Spec(R i ) gij −→ Spec(R j ) gji .
The K-algebras may be described by generators and relations in the form
R i = K [T i1 , . . . , T iji ]/(h i1 , . . . , h iki ). In the polynomials h ik , g ij , only finitely
many denominators are occurring. We, therefore, have R i = R ′ i⊗ OK K for some
finitely generated O K -algebras R ′ i such that g ji ∈ R ′ i .
The ϕ ij induce isomorphisms ϕ # ij : (R′ j ) g ji
⊗ OK K → (R i ′) g ij
⊗ OK K. For some suitable
0 ≠ f ij ∈ O K , the isomorphism ϕ # ij
is the base extension of an isomorphism
(R ′ j ) g ji
⊗ OK (O K ) fij → (R i ′) g ij
⊗ OK (O K ) fij .
Putting R i := (R i) ′ f for f := ∏ f ij , the ϕ ij therefore extend to isomorphisms
i,j
˜ϕ ij : Spec(R i ) gij −→ Spec(R j ) gji .
The cocycle relations “˜ϕ ik = ˜ϕ jk ◦ ˜ϕ ij ” are satisfied after restriction to the
generic fiber. Since the schemes Spec(R i ) gij are of finite type, this implies
that these relations are still fulfilled when the R i are localized only by some
suitable 0 ≠ f ′ ∈ O K .
We constructed gluing data for an O K -scheme X . X is clearly of finite type.
Second step. Separatedness.
Denote by ∆ ⊂ X × SpecOK X the Zariski closure of the diagonal. Then, the
diagonal morphism δ : X → ∆ is an isomorphism on the generic fiber. Since X
and ∆ are O K -schemes of finite type, δ is an isomorphism outside finitely many
special fibers. We delete the corresponding special fibers from X .
Third step. Properness.
Assume first that X is reduced. Then, by Chow’s lemma, we have a surjective
map X ′ → X from a projective K-scheme X ′ ⊆ P n K . Let X ′ be the Zariski
closure of X ′ in P n O K
equipped with the induced reduced structure. This is a
model of X ′ which is proper over Spec O K .

Sec. 3] TECHNICAL LEMMATA 109
Let X be any model of X in the sense of the previous steps. Then, there
exists some 0 ≠ f ∈ O K such that the morphism X ′ → X extends to a morphism
of (O K ) f -schemes g : X ′ × SpecOK Spec(O K ) f → X . [EGA, Chapitre II,
Corollaire (5.4.3.ii)] shows that im g is proper over Spec(O K ) f . This implies that
im g ⊆ X × SpecOK Spec(O K ) f is a closed subscheme containing the generic fiber.
In particular, the closure ofX in X × SpecOK Spec(O K ) f is proper over Spec(O K ) f .
Return to the general case. Let X be any model of X in the sense of the
two steps above. We may assume that X is equal to the closure of X.
Then, there exists some 0 ≠ f ∈ O K such that X red × SpecOK Spec(O K ) f is
proper over Spec(O K ) f . By virtue of [EGA, Chapitre II, Corollaire (5.4.6)], this
suffices for X × SpecOK Spec(O K ) f being proper.
□
3.5. Lemma. –––– Let K be a number field, X a reduced scheme which is separated
and of finite type over the integer ring O K , and A be an Azumaya algebra over the
generic fiber X.
Then, there exist an open subset X ◦ ⊆ X containing X and an Azumaya algebra
à over X ◦ which extends A .
Proof. First step. Extending A to a coherent sheaf.
Let j : X → X be the embedding of the generic fiber. Then, j ∗ A is a quasicoherent
sheaf on X . We claim, j ∗ A contains a coherent subsheaf F such that
F | X = A .
Assume that would not be the case. We construct an increasing sequence of
subsheaves of j ∗ A recursively as follows.
Put F 1 := 〈1〉 for the section 1 ∈ Γ(X , j ∗ A ). Having F n already constructed,
we have, by our assumption, F n | X A . We observe that X is a Noetherian
scheme. Therefore, j ∗ A is the union of its coherent subsheaves [Ha77,
Chap. II, Exercise 5.15.a)]. By consequence, there exists a coherent sheaf
G ⊆ j ∗ A such that G | X ⊈ F n | X . We put F n+1 := F n + G .
This means, (F n | X ) n∈Æis a strictly increasing sequence of coherent subsheaves
of A . This is a contradiction.
Second step. Extending A to a locally free sheaf.
We may assume X to be connected. Then,
ψ(x) := dim k(x) F X ⊗ OX k(x)
is constant on the generic fiber X. As ψ is upper semicontinuous, there is
an open subset X ′ ⊆ X containing the generic fiber such that ψ is constant
on X ′ . By reducedness, this suffices for F | X ′ being locally free.
Third step. Extending A to an Azumaya algebra.
j ∗ A carries a natural structure of a sheaf of O X -algebras.

110 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
F ⊂ j ∗ A contains the constant section 1 by the construction given in the
first step. To obtain a sheaf of algebras, we must establish closedness under multiplication.
The image F ′ of the multiplication map F ×F → j ∗ A is a coherent subsheaf.
On the generic fiber, we have F ′ | X ⊆ F | X . This implies Q := (F ′ + F )/F
is a coherent sheaf on X such that Q| X = 0. Upper semicontinuity shows that
Q vanishes on an open neighbourhood X ′′ of the generic fiber. F | X ′ ∩X ′′ is a
locally free sheaf of O X ′ ∩X ′′-algebras.
By Corollary 1.5, the non-Azumaya locus is again a closed subset.
□
4. Computing the Brauer-Manin obstruction – The general strategy
4.1. –––– In this section, we want to illustrate that the Manin pairing ev(α, x)
is effectively computable in certain cases.
We will assume that X is a geometrically integral scheme which is proper over
a number field K. Further, we suppose that Br ′ (X K
) = 0 and Br(X ) = Br ′ (X ).
Under these assumptions, Br(X )/π ∗ Br(SpecK ) ∼ = H 1( Gal(K/K ), Pic(X K
) ) .
These assumptions are fulfilled, e.g., for X a smooth proper surface such that
X K
is a (not necessarily minimal) rational surface.
We may also suppose X (K) ≠ ∅ as, otherwise, the Brauer-Manin obstruction
would not be of much interest.
4.2. –––– Usually, α ∈ Br(X ) is not given itself, but only its image
α ∈ Br(X )/π ∗ Br(SpecK ) ∼ = H 1( Gal(K/K ), Pic(X K
) ) .
In order to compute ev(α, x) = ∑ ν ev ν (α, x ν ), it is therefore necessary to explicitly
lift α to Br(X ). Note that the ev ν , as opposed to ev itself, do not factor
via Br(X )/π ∗ Br(SpecK).
4.3. Lemma. –––– LetK beanumberfieldandπ: X → SpecK beageometrically
integral scheme which is proper over K.
Then, there is a natural isomorphism
ker(Br ′ (X ) → i′
Br ′ (X K
))/π ∗ ∼=
Br(SpecK ) −→ H 1( Gal(K/K ), Pic(X K
) )

Sec. 4] COMPUTING THE BRAUER-MANIN OBSTRUCTION 111
induced by the commutative diagram
Br(SpecK )
Br(SpecK )
π ∗
0 ker(Br ′ (X ) → i′
Br ′ (X K )) H 2( Gal(K/K ),Q(X K ) ∗) H 2( Gal(K/K ),Div(X K ) )
0 H 1( Gal(K/K ),Pic(X K ) ) H 2( Gal(K/K ),Q(X K ) ∗ /K ∗) H 2( Gal(K/K ),Div(X K ) ) .
Here, the bottom row is part of the long exact sequence of cohomology associated to
the exact sequence
0 −→ Q(X K
) ∗ /K ∗ −→ Div(X K
) −→ Pic(X K
) −→ 0 .
The middle column is part of the long exact cohomology sequence associated to the
exact sequence
0 −→ K ∗ −→ Q(X K
) ∗ −→ Q(X K
) ∗ /K ∗ −→ 0 .
Finally, the middle row is obtained when mapping the short exact sequence from
Proposition II.4.8 to
0 −→ H 2( Gal(Q(X K
)/K ), Q(X K
) ∗) −→ H 2( Gal(Q(X K
)/K ), Q(X K
) ∗) −→ 0
0
and taking kernels.
□
4.4. Remark. –––– The isomorphism given in Lemma 4.3 is the same as the
isomorphism provided by the Hochschild-Serre spectral sequence (Proposition
II.8.11). This is asserted in [Lic, Sec. 2].
4.5. –––– The diagram above leads to the following general strategy.
General strategy for computing ev(α, x).
a) Compute the image of α in H 2( Gal(K/K ), Q(X K
) ∗ /K ∗) .
b) Lift that to a cohomology class in H 2( Gal(K/K ), Q(X K
) ∗) .
c) For each ν,
i) restrict to H 2( Gal(K ν /K ν ), Q(X Kν
) ∗) .
ii) In a neighbourhood U of x ν such that Pic(U Kν
) = 0, use the exact sequence
H 2( Gal(K ν /K ν ), Γ(U Kν
, O ∗ U Kν
) ) −→ H 2( Gal(K ν /K ν ), Q(X Kν
) ∗) −→
−→ H 2( Gal(K ν /K ), Div(U Kν
) )

112 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
in order to lift to H 2( Gal(K ν /K ν ), Γ(U Kν
, O ∗ U Kν
) ) .
iii) Apply the evaluation map Γ(U Kν
, O ∗ U Kν
) → K ∗ ν at x ν to get a cohomology
class in H 2( Gal(K ν /K ν ), K ∗ ν)
.
iv) Take its invariant inÉ/.
d) Take the sum of all these invariants.
4.6. Remark. –––– In practice, there are several problems with that strategy.
First of all, our skills in dealing with general Galois cohomology
classes are limited. Anyway, for some suitable finite field extension L/K,
α ∈ H 1( Gal(K/K ), Pic(X K
) ) is the image under inflation of a class in
H 1( Gal(L/K ), Pic(X L ) ) .
4.7. Observations (Inflation from a finite quotient Gal(L/K )). —–
a) There is an exact sequence
0 −→ Q(X L ) ∗ /L ∗ −→ Div(X L ) −→ Pic(X L ) −→ 0
inducing a map H 1( Gal(L/K ), Pic(X L ) ) → H 2( Gal(L/K ), Q(X L ) ∗ /L ∗) .
b) ThehomomorphismH 2( Gal(L/K ), Q(X L ) ∗) → H 2( Gal(L/K ), Q(X L ) ∗ /L ∗)
is not surjective, in general. It is, however, when Gal(L/K ) is cyclic. Indeed, in
this case there is a non-canonical isomorphism
H 3( Gal(L/K ), L ∗) ∼ = H
1 ( Gal(L/K ), L ∗)
and the latter group vanishes by Hilbert’s Theorem 90.
c) The restriction in i) goes to H 2( Gal(L w /K ν ), Q(X Lw ) ∗) for w a prime above ν.
In step ii), one can work with U = Spec O XLw ,x ν
. There is the exact sequence
H 2( Gal(L w /K ν ), Γ(U Lw , O ∗ U Lw
) ) −→ H 2( Gal(L w /K ν ), Q(X Lw ) ∗) −→
−→ H 2( Gal(L w /K ), Div(U Lw ) )
such that one may lift to H 2( Gal(L w /K ν ), Γ(U Lw , O ∗ U Lw
) ) .
4.8. Remark. –––– To summarize, assume Gal(L/K ) is cyclic and we start with
a class in the image under inflation of H 1( Gal(L/K ), Pic(X L ) ) . Then, all the
cohomology classes obtained according to the general strategy are in the image
of H 2( Gal(L/K ), ·) or H 2( Gal(L w /K ν ), ·),
respectively.
4.9. –––– Thus, we assume that G := Gal(L/K ) is a cyclic group of order n.
In this case, H 2 (G,) ∼ =/nand the cup product with a generator
induces an isomorphism Ĥ q (G, A) → Ĥ q+2 (G, A) for all integers q and

Sec. 5] THE EXAMPLES OF MORDELL 113
all G-modules. We fix a generator of H 2 (G,) in order to determine these
isomorphisms uniquely.
Note that there is no distinguished generator of H 2 (G,) unless a generator
of G is fixed. Thus, the isomorphisms discussed above are not canonical.
Nevertheless, there is a commutative diagram, analogous to the one in
Lemma 4.3, with H 2 replaced by Ĥ 0 and H 1 replaced by Ĥ −1 .
4.10. –––– We therefore have the following plan.
Plan for computing ev(α, x).
One has
Ĥ −1 (G, Pic(X L )) ∼ = [Div 0 (X L ) G ∩ N Div(X L )]/N Div 0 (X L ) .
The map from step a) becomes the canonical map to Div 0 (X L ) G /N Div 0 (X L ).
Step b) is the lift to Q(X L ) G /NQ(X L ) = Q(X K )/NQ(X L ) under f ↦→ div f .
For each ν, steps c.i), ii) and iii) amount to the evaluation
Q(X K )/NQ(X L ) −→ K ∗ ν/NL ∗ w ,
f ↦→ [ f (x ν )]
at x ν . Note that this map is well-defined although a representative in Q(X K )
might have a pole at x ν .
Finally, in step c.iv), one has to apply the chosen isomorphism backwards,
K ∗ ν/NL ∗ w = Ĥ 0 (G, L ∗ w ) ∼ =
−→ H 2 (G, L ∗ w ) inv ν
−→É/.
ev(α, x) is the sum of all these local invariants.
5. The examples of Mordell
i. Formulation of the results. —
5.1. –––– In this section, we present a series of examples of cubic surfaces
overÉfor which the Hasse principle fails. Our series generalizes the examples
of Mordell [Mord]. It was observed by Yu. I. Manin [Man,кVI,º½¹º]
himself that the failure of the Hasse principle in Mordell’s examples may be
explained by the Brauer-Manin obstruction.

114 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
5.2. –––– Let p 0 ≡ 1 (mod 3) be a prime number and K/Ébe the unique
cubic field extension contained in the cyclotomic extensionÉ(ζ p0 )/É. We fix
the explicit generator θ ∈ K given by
More concretely, we write
where n is given by p 0 = 6n + 1.
θ := trÉ(ζ p0 )/K (ζ p0 − 1) .
θ = −2n + ∑
i∈(∗ p0 ) 3 ζ i p 0
5.3. Proposition. –––– Consider the cubic surface X ⊂ P 3É, given by
T 3 (a 1 T 0 + d 1 T 3 )(a 2 T 0 + d 2 T 3 ) =
3
∏
i=1
(
T0 + θ (i) T 1 + (θ (i) ) 2 T 2
)
.
Here, a 1 , a 2 , d 1 , d 2 ∈. The θ (i) are the images of θ under Gal(K/É).
i) Let A ≡ 1 (mod 3) be the unique integer such that 4p 0 = A 2 + 27B 2 .
Suppose that d 1 d 2 ≠ 0, a 1 ≠ 0, a 2 ≠ 0, a 1 /d 1 ≠ a 2 /d 2 , and that
T (a 1 + d 1 T )(a 2 + d 2 T ) + 4p 0−A 2
(p 2 0 −3p 0−A) 2 = 0
has no multiple zeroes. Then, X is smooth.
ii) Assume that p 0 ∤ d 1 d 2 , that gcd(a 1 , d 1 ) and gcd(a 2 , d 2 ) contain only prime factors
which decompose in K, and that T (a 1 + d 1 T )(a 2 + d 2 T ) − 1 = 0 has at least one
simple zero inp 0
. Then, X (É) ≠ ∅.
iii) Assume that X is smooth. Suppose further that p 0 ∤ d 1 d 2 and gcd(d 1 , d 2 ) = 1.
Then, there is a class α ∈ Br(X ) with the following property.
For an adelic point x = (x ν ) ν , the value of ev(α, x) depends only on the component
x νp0 . Write x νp0 =: (t 0 : t 1 : t 2 : t 3 ). Then, one has ev(α, x) = 0 if and
only if
a 1 t 0 + d 1 t 3
t 3
is a cube in∗ p0
.
5.4. Remarks. –––– i) K/Éis an abelian cubic field extension. It is totally
ramified at p 0 and unramified at all other primes. A prime p is completely
decomposed in K if and only if p is a cube modulo p 0 .
ii) We have ∏
3 ( )
T0 + θ (i) T 1 + (θ (i) ) 2 T 2 = NK/É(T 0 + θT 1 + θ 2 T 2 ) .
i=1
iii) The reduction of T (a 1 + d 1 T )(a 2 + d 2 T ) + 4p 0−A 2
modulo p
(p0 2−3p 0−A) 2 0 is exactly
T (a 1 + d 1 T )(a 2 + d 2 T ) − 1.

Sec. 5] THE EXAMPLES OF MORDELL 115
5.5. Remark. –––– Let (t 0 : t 1 : t 2 : t 3 ) ∈ X (Ép 0
). Then, for the reduction,
one has (t 0 : t 3 ) = (1 : s) for s a solution of T (a 1 + d 1 T )(a 2 + d 2 T ) − 1 = 0.
On the other hand, if s is a simple solution then, by Hensel’s lemma, there
exists (t 0 : t 1 : t 2 : t 3 ) ∈ X (Ép 0
) such that (t 0 : t 3 ) = (1 : s).
The possible values of s are in bijection with the up to three planes a p 0 -adic
point on X may reduce to.
We will show in Fact 5.19 that noÉp 0
-valued point on X reduces to the triple line
“T 0 = T 3 = 0”.
5.6. Observation. –––– As
a 1 t 0 + d 1 t 3
t 3
= a 1 + d 1 s
s
the value of ev(α, x) depends only on the plane to which its component x νp0 is
mapped under reduction.
5.7. Remark. –––– Since s is a solution of T (a 1 + d 1 T )(a 2 + d 2 T ) − 1 = 0, we
have s ≠ 0 and a 1 + d 1 s ≠ 0. Hence, a 1+d 1 s
s
is well defined and non-zero inp 0
.
ii. Smoothness. —
5.8. Lemma. –––– The minimal polynomial of θ is
T 3 + p 0 T 2 p 0 − 1
+ p 0 T + p3 0 − 3p0 2 − Ap 0
.
3 27
5.9. Remark. –––– This is an Eisenstein polynomial. We see once more that
K/Éis totally ramified at p 0 . Further, ν(θ) = 1 3 for ν the extension of ν p 0
to K.
5.10. Proof of the lemma. ––––∗ p0
is decomposed into three cosets C 1 , C 2 ,
and C 3 when factored by (∗ p0
) 3 . We have
θ (i) = −2n + ∑
j∈C i
ζ j p 0
.
As the sum over all p 0 -th roots of unity vanishes, this immediately implies
θ (1) + θ (2) + θ (3) = −p 0 .
Further, multiplying terms and adding-up shows that
θ (1) θ (2) + θ (2) θ (3) + θ (3) θ (1) = 12n 2 + 2(−2n) ∑ ζ j p 0
+ p 0 − 1
j∈∗ p0
3
∑
[ ] 2 p0 − 1
= 12 + 2 p 0 − 1
− p 0 − 1
6 3 3
= p 0 · p0 − 1
.
3
,
j∈∗ p0
ζ j p 0

116 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
In order to establish θ (1) θ (2) θ (3) = −p3 0 +3p2 0 +Ap 0
27
, multiplying expressions leads to
θ (1) θ (2) θ (3) = −8n 3 + 4n 2 ∑ ζ j p 0
− 2n p 0 − 1
j∈∗ p0
3
∑ ζ j p 0
+
j∈∗ p0
( )(
+ ∑ ζ j p 0
j∈C 1
( )( )( )
= −8n 3 + ∑ ζ j p 0 ∑ ζ j p 0 ∑ ζ j p 0
.
j∈C 1 j∈C 2 j∈C 3
)(
∑ ζ j p 0
j∈C 2
)
∑ ζ j p 0
j∈C 3
To calculate the remaining triple product is more troublesome as one needs to
know how often a product ζ j 1
p 0
ζ j 2
p 0
ζ j 3
p 0
for j i ∈ C i is equal to one.
Sublemma 5.11 shows that this happens exactly p−1 · p0+1+A times. As the
3 9
expression is invariant under Gal(É(ζ p0 )/É) and there are all in all (p−1)3 summands,
we find
27
(
)(
∑ ζ j p 0
j∈C 1
)(
∑ ζ j p 0
j∈C 2
)
∑ ζ j p 0
= p 0 − 1
(p 0 + 1 + A)
j∈C 3
27
+ 1
27 [(p 0 − 1) 2 − (p 0 + 1 + A)] ∑
j∈∗ p0
ζ j p 0
= 3p 0 − 1 + Ap 0
.
27
Further, −8n 3 = −8( p 0−1
) 3 = −p3 0 +3p2 0 −3p 0+1
. The assertion follows. □
6 27
5.11. Sublemma. –––– Let D be a non-cube in∗ p0
. Then,
#{(x, y) ∈2 p0
| Dx 3 + D 2 y 3 = 1} = p 0 + 1 + A .
Proof. The number of solutions inp 0
of such an equation may be counted
using Jacobi sums. As in [I/R, Chapter 8, §3], we see
#{(x, y) | Dx 3 + D 2 y 3 = 1} =
where χ is a fixed cubic character.
=
2 2
∑ ∑ ∑
i=0 j=0 a+b=1
2
∑
i=0
χ i (a/D)χ j (b/D 2 )
2
∑ χ i (1/D)χ j (1/D 2 )J (χ i , χ j ) .
j=0
The summands for i = j are the same as in the case D = 1. If i = 0 and j ≠ 0
or vice versa then the corresponding summand is 0. Finally, by [I/R, Chapter 8,
§3, Theorem 1.c)], we have J (χ, χ 2 ) = J (χ 2 , χ) = −1. Altogether,
#{(x, y) | Dx 3 + D 2 y 3 = 1} = #{(x, y) | x 3 + y 3 = 1} + 3 .

Sec. 5] THE EXAMPLES OF MORDELL 117
The claim now follows from a theorem of C. F. Gauß [I/R, Chapter 8, §3,
Theorem 2)].
□
5.12. Notation. –––– We will write
F (T 0 , T 1 , T 2 , T 3 ) := T 3 (a 1 T 0 +d 1 T 3 )(a 2 T 0 +d 2 T 3 )−
3
∏
i=1
(
T0 +θ (i) T 1 +(θ (i) ) 2 T 2
)
.
5.13. Proof of Proposition 5.3.i) –––– Since d 1 d 2 ≠ 0, the projection to the first
three coordinates induces a morphism of K-schemes q : X → P 2Éwhich is finite
of degree three.
The conditions ∂F
∂T 1
= 0 and ∂F
∂T 2
= 0 for a singular point x depend only on q(x).
Let us first analyze them.
Writing l i := T 0 + θ (i) T 1 + (θ (i) ) 2 T 2 , we get the system of equations
the solution of which is
θ (1) l 2 l 3 + θ (2) l 3 l 1 + θ (3) l 1 l 2 = 0 ,
θ (1)2 l 2 l 3 + θ (2)2 l 3 l 1 + θ (3)2 l 1 l 2 = 0 ,
(l 2 l 3 : l 3 l 1 : l 1 l 2 ) = (θ (2) θ (3) (θ (2) − θ (3) ) : θ (3) θ (1) (θ (3) − θ (1) ) : θ (1) θ (2) (θ (1) − θ (2) )) .
As ̟ : (T 0 : T 1 : T 2 ) → (l 2 l 3 : l 3 l 1 : l 1 l 2 ) is a quadratic transformation, there are
exactly four points in P 2 which fulfill that condition. These are the three points
of indeterminacy of ̟ which we denote by P 1 , P 2 , and P 3 and a non-trivial
solution P.
P 1 , P 2 , and P 3 are given by l i = l j = 0 for a pair of indices i ≠ j. This implies
that these points are not contained in the line “T 0 = 0”. The fiber of q over
any of these three points is therefore given by T 3 (a 1 + d 1 T 3 )(a 2 + d 2 T 3 ) = 0
which shows that q is unramified. Consequently, the points over P 1 , P 2 , and P 3
are smooth points of X.
It remains to consider the fiber of P. For that point, we find
( )
θ
(1)
(l 1 : l 2 : l 3 ) =
θ (2) − θ : θ (2)
(3) θ (3) − θ : θ (3)
.
(1) θ (1) − θ (2)
The linear system of equations
T 0 + θ (1) T 1 + (θ (1) ) 2 T 2 =
T 0 + θ (2) T 1 + (θ (2) ) 2 T 2 =
T 0 + θ (3) T 1 + (θ (3) ) 2 T 2 =
θ (1)
θ (2) − θ (3) ,
θ (2)
θ (3) − θ (1) ,
θ (3)
θ (1) − θ (2)

118 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
has the obvious solution
(T 0 : T 1 : T 2 )
= (−3θ (1) θ (2) θ (3) : 2[θ (1) θ (2) + θ (2) θ (3) + θ (3) θ (1) ] : [−(θ (1) + θ (2) + θ (3) )])
( p
3
= 0 − 3p0 2 − Ap 0
: 2 )
9 3 (p 0 − 1)p 0 : p 0
(
)
6(p 0 − 1)
= 1 :
p0 2 − 3p 0 − A : 9
p0 2 − 3p .
0 − A
which is unique since the coefficient matrix is Vandermonde.
Using Lemma 5.8again, a direct calculation which is conveniently done inmaple
shows that
(
1 + 6(p )
0 − 1) 9
p0 2 − 3p 0 − A θ(i) +
p0 2 − 3p 0 − A (θ(i) ) 2 = − 4p 0 − A 2
(p0 2 − 3p 0 − A) . 2
3
∏
i=1
Therefore, the fiber of q over P is given by
4p 0 − A
T 3 (a 1 + d 1 T 3 )(a 2 + d 2 T 3 ) +
2
(p0 2 − 3p 0 − A) = 0 . 2
By assumption, this equation has no multiple solutions. Therefore, q is unramified
over P. The points above P are hence smooth points of X. □
5.14. Remark. –––– The fact that X is smooth is not mentioned in the original
literature, neither in [Mord], nor in [Man].
iii. Existence of an adelic point. —
5.15. Proof of Proposition 5.3.ii) –––– We have to show that X (Éν) ≠ ∅ for
every valuation ofÉ.
X (Ê) ≠ ∅ is obvious. For a prime number p, in order to show X (Ép) ≠ ∅, we
use Hensel’s lemma. It is sufficient to verify that the reduction X p has a smooth
p-valued point.
Case 1: p = p 0 .
As ν p0 (θ) > 0, the reduction X p0 is given by T 3 (a 1 T 0 +d 1 T 3 )(a 2 T 0 +d 2 T 3 ) = T0 3.
This is the union of three planes meeting in the line given by T 0 = T 3 = 0. By assumption,
one of them has multiplicity one and is defined overp 0
. It contains
p0 2 smooth points.
Case 2: p ≠ p 0 , p ∤ d 1 d 2 .
It suffices to show that there is a smoothp-valued point on the intersection X ′
p
of X p with the hyperplane “T 0 = 0”. This curve is given by the equation
∏
d 1 d 2 T 3
3 = θ (1) θ (2) θ (3) 3
i=1
(T 1 + θ (i) T 2 ) .

Sec. 5] THE EXAMPLES OF MORDELL 119
If p ≠ 3 then this equation defines a smooth genus one curve. It has anp-valued
point by Hasse’s bound.
If p = 3 then the projection X ′
p → P 1 given by (T 1 : T 2 : T 3 ) ↦→ (T 1 : T 2 ) is oneto-one
onp-valued points. At least one of them is smooth since ∏ 3 i=1 (T +θ(i) )
is a separable polynomial.
Case 3: p ≠ p 0 , p|d 1 d 2 .
X ′ p := X p ∩ “T 0 = 0” is given by 0 = θ (1) θ (2) θ (3) ∏ 3 i=1 (T 1+θ (i) T 2 ). In particular,
x = (0 : 0 : 0 : 1) ∈ X p (p). We may assume that x is singular.
Then, X p is given as Q(T 0 , T 1 , T 2 )T 3 +K (T 0 , T 1 , T 2 ) = 0 for Q a quadratic form
and K a cubic form. If Q ≢ 0 then there is anp-rational line l through x such
that Q| l ≠ 0. Hence, l meets X p twice in x and once in anotherp-valued point
which is smooth.
Otherwise, (F mod p) does not depend on T 3 , i.e., the left hand side of the
equation of X vanishes modulo p. This means, one of the factors on the left
hand side vanishes modulo p. Say, we have a 1 ≡ d 1 ≡ 0 (mod p).
Then, by assumption, p decomposes completely in K. At such a prime, X p ′ is
the union of three lines which are all defined overp, different from each other,
and meeting in one point. We have plenty of smooth points.
□
iv. Construction of a Brauer class. —
5.16. –––– We write G := Gal(K/É).
According to the general strategy, described in the section above, an element
of H 1 (G, Pic(X K )) ⊆ H 1( Gal(É/É), Pic(XÉ) ) may be given by a rational
function f ∈ Q(X ) such that div( f ) ∈ N Div(X K ). Such a function is
f := a 1T 0 + d 1 T 3
T 3
.
Indeed, “a 1 T 0 + d 1 T 3 = 0” defines a triangle, the three lines of which are given
by that equation and T 0 + θ (i) T 1 + (θ (i) ) 2 T 2 = 0 for i = 1, 2, or 3, respectively.
Considered as a Weil divisor, this triangle is the norm of the divisor given by
one of the lines.
We write [ f ] for the Brauer class defined by f .
5.17. Remarks. –––– i) One might work as well with [ f ′ ] for f ′ := a 2T 0 +d 2 T 3
T 3
or
even with both, [ f ] and [ f ′ ], as Manin does in [Man,кVI,ÈÖÐÓÒº].
However,
[ ]
[ f ]·[ f ′ (a1 T 0 + d 1 T 3 )(a 2 T 0 + d 2 T 3 )
] =
=
Consequently, X (É)
T3
2
[ f ] = X (É)
[ f ′] .
[ ( )]
T0 + θT 1 + θ 2 T 2
N
= 0 .
T 3

=/3
120 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
ii) Under very mild hypotheses on X, one has Br(X )/π ∗ Br(SpecÉ) ∼
and [ f ] is a generator. We will discuss this point in Subsection vi.
5.18. Lemma. –––– Let ν be any valuation ofÉdifferent from ν p0 . Then,
for every adelic point x ∈ X (É).
Proof. First step: Elementary cases.
ev ν ([ f ], x) = 0
If ν = ν ∞ then ev ν ([ f ], x) = 0 since #G = 3 while only the values 1 2 and 0
are possible.
If ν = ν p and p is decomposed in K then every element ofÉ∗ p is a norm.
Therefore, ev ν ([ f ], x) = 0.
Second step: Preparations.
It remains to consider the case that p remains prime in K. We have to show that
a 1 t 0 +d 1 t 3
t 3
is a norm for each point (t 0 : t 1 : t 2 : t 3 ) ∈ X (Ép). An element w ∈É∗ p
is a norm if and only if 3|ν p (w).
It might happen that θ is not a unit in K ν . As K ν /Ép is unramified, there exists
t ∈É∗ p such that θ := tθ ∈ K ν is a unit. The surface X ′ given by
T 3 (a 1 T 0 + d 1 T 3 )(a 2 T 0 + d 2 T 3 ) =
3
∏
i=1
(
T0 + θ (i) T 1 + (θ (i) ) 2 T 2
)
is isomorphic to X. The map ι: (t 0 : t 1 : t 2 : t 3 ) ↦→ (t 0 :
isomorphism X → X ′ which leaves the rational function a 1T 0 +d 1 T 3
T 3
Hence, we may assume without restriction that θ ∈ K ν is a unit.
Third step: The case θ is a unit.
We may assume t 0 , t 1 , t 2 , t 3 ∈p are coprime.
t 1t
:
t 2
t 2
: t 3 ) is an
unchanged.
We could have ∏ 3 i=1(
t0 +θ (i) t 1 + (θ (i) ) 2 t 2
) = 0 only when t0 = t 1 = t 2 = 0 since
θ generates a cubic field extension ofÉp. Then, t 3 (a 1 t 0 + d 1 t 3 )(a 2 t 0 + d 2 t 3 ) = 0
implies t 3 = 0 which is a contradiction.
Therefore, both sides of the equation are different from zero. In particular, we
automatically have t 3 ≠ 0.
If ν ( t 3 (a 1 t 0 + d 1 t 3 )(a 2 t 0 + d 2 t 3 ) ) = 0 then a 1t 0 +d 1 t 3
t 3
is clearly a norm. Otherwise,
we have
ν ( ∏
3 (t 0 + θ (i) t 1 + (θ (i) ) 2 t 2 ) ) > 0 .
i=1
This implies that ν(t 0 ), ν(t 1 ), ν(t 2 ) > 0. Then, t 3 must be a unit.
From the equation of X, we deduce ν(d 1 d 2 ) > 0. If ν(d 2 ) > 0 then, according
to the assumption, d 1 is a unit. This shows ν(a 1 t 0 + d 1 t 3 ) = 0 from which the
assertion follows.

Sec. 5] THE EXAMPLES OF MORDELL 121
Thus, assume ν(d 1 ) > 0. Then, d 2 is a unit and, therefore, ν(a 2 t 0 + d 2 t 3 ) = 0.
Further, we note that 3|ν ( ∏ 3 i=1 (t 0 + θ (i) t 1 + (θ (i) ) 2 t 2 ) ) since the right hand side
is a norm. By consequence,
3|ν ( t 3 (a 1 t 0 + d 1 t 3 )(a 2 t 0 + d 2 t 3 ) ) .
Altogether, we see that 3|ν(a 1 t 0 + d 1 t 3 ) and 3|ν ( )
a 1 t 0 +d 1 t 3
t 3
. The claim follows.
□
5.19. Fact. –––– i) The reduction X p0 of X at p 0 is given by
T 3 (a 1 T 0 + d 1 T 3 )(a 2 T 0 + d 2 T 3 ) = T 3
0 .
ii) Over the algebraic closure, X p0 is the union of three planes meeting in a triple line
“T 0 = T 3 = 0”.
iii) NoÉp 0
-valued point on X reduces to the triple line.
Proof. i) and ii) are clear.
iii) Let (t 0 : t 1 : t 2 : t 3 ) ∈ X (Ép 0
). We may assume t 0 , t 1 , t 2 , t 3 ∈p 0
are coprime.
We write ν for the extension of ν p0 to K. Then ν(t 0 ) ≥ 1 and ν(t 3 ) ≥ 1 together
imply ν(t 3 (a 1 t 0 + d 1 t 3 )(a 2 t 0 + d 2 t 3 )) ≥ 3. On the other hand,
ν ( ∏
3 (t 0 + θ (i) t 1 + (θ (i) ) 2 t 2 ) )
i=1
is equal to 1 or 2, since t 1 or t 2 is a unit and ν(θ (i) ) = 1 3 .
□
5.20. Lemma. –––– Let (t 0 : t 1 : t 2 : t 3 ) ∈ X (Ép 0
).
Then, ev νp0 ([ f ], (t 0 : t 1 : t 2 : t 3 )) = 0 if and only if a 1t 0 +d 1 t 3
t 3
is a cube in∗ p0
.
Proof. (p 0 ) = p 3 is totally ramified. InÉp 0
, there is a uniformizer which is
a norm. Further, a p 0 -adic unit u is a norm if and only if u := (u mod p 0 ) is a
cube in∗ p0
.
We have that a 1t 0 +d 1 t 3
t 3
is automatically a p 0 -adic unit. Indeed, suppose that
t 0 , t 1 , t 2 , t 3 ∈p 0
are coprime. Modulo p 0 , the equation of X is
T 3 (a 1 T 0 + d 1 T 3 )(a 2 T 0 + d 2 T 3 ) = T 3
0 .
As no point may reduce to the singular line, we see that t 0 ≠ 0. This implies
t 3 ≠ 0 and a 1 t 0 + d 1 t 3 ≠ 0 which is the assertion.
□

122 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
v. Examples. —
5.21. Corollary. –––– Let p 0 ≡ 1 (mod 3) be a prime number and consider the
cubic surface X ⊂ P 3É, given by
T 3 (a 1 T 0 + d 1 T 3 )(a 2 T 0 + d 2 T 3 ) =
3
∏
i=1
(
T0 + θ (i) T 1 + (θ (i) ) 2 T 2
)
.
Here, a 1 , a 2 , d 1 , d 2 ∈and assume that a 1 ≠ 0, a 2 ≠ 0, a 1 /d 1 ≠ a 2 /d 2 , p 0 ∤d 1 d 2 ,
and that gcd(a 1 , d 1 ) and gcd(a 2 , d 2 ) contain only prime factors which decompose
in K.
i) Assume that
T (a 1 + d 1 T )(a 2 + d 2 T ) − 1 = 0
has three different zeroes t (1) , t (2) , t (3) ∈p 0
.
If a 1+d 1 t (1)
, a 1+d 1 t (2)
, and a 1+d 1 t (3)
are non-cubes in∗ t (1) t (2) t (3)
p0
then X (É) = ∅. On X, there
is a Brauer-Manin obstruction to the Hasse principle.
If exactly one of the three expressions is a cube in∗ p0
then, on X, there is a Brauer-
Manin obstruction to weak approximation.
ii) Assume that
T (a 1 + d 1 T )(a 2 + d 2 T ) − 1 = 0
has exactly one zero t ∈p 0
which is simple.
a 1 +d 1 t
t
If is not a cube in∗ p0
then X (É) = ∅. On X, there is a Brauer-Manin
obstruction to the Hasse principle.
5.22. Remark. –––– In the case of three different solutions, it is impossible that
exactly two of the expressions a 1+d 1 t (1)
, a 1+d 1 t (2)
, and a 1+d 1 t (3)
are cubes. Indeed, a
t (1) t (2) t (3)
direct calculation shows
a 1 + d 1 t (1) + d 1 t
·a1 (2) + d 1 t
·a1 (3)
= d 3
t (1) t (2) t (3) 1 .
This means, the Brauer-Manin obstruction might exclude no, one, or all three
of the planes, the reduction X p0 consists of, but not exactly two of them.
5.23. Example. –––– Let p 0 ≡ 1 (mod 3) be a prime number and assume that
d 1 and d 2 are non-cubes in∗ p0
such that d 1 d 2 is a cube, p 0 |a 1 and p 0 |a 2 . Further,
suppose a 1 ≠ 0, a 2 ≠ 0, and a 1 /d 1 ≠ a 2 /d 2 , as well as that gcd(a 1 , d 1 )
and gcd(a 2 , d 2 ) contain only prime factors which decompose in K.
Then, the cubic surface X given by
T 3 (a 1 T 0 + d 1 T 3 )(a 2 T 0 + d 2 T 3 ) =
is a counterexample to the Hasse principle.
3
∏
i=1
(
T0 + θ (i) T 1 + (θ (i) ) 2 T 2
)

Sec. 5] THE EXAMPLES OF MORDELL 123
Indeed, one has
a 1 t 0 + d 1 t 3
t 3
≡ d 1 (mod p 0 )
which is a non-cube. The assumption that d 1 d 2 is a cube is important to
guarantee X (Ép 0
) ≠ ∅.
More concretely, for p 0 = 19, a counterexample to the Hasse principle is
given by
T 3 (19T 0 + 5T 3 )(19T 0 + 4T 3 ) =
3
∏
i=1
(
T0 + θ (i) T 1 + (θ (i) ) 2 T 2
)
= T0 3 − 19T0 2 T 1 + 133T0 2 T 2 + 114T 0 T1
2
− 1 539T 0 T 1 T 2 + 5 054T 0 T2 2 − 209T1
3
+ 3 971T1 2 T 2 − 23 826T 1 T2 2 + 43 681T2 3 .
5.24. Example. –––– For p 0 = 19, consider the cubic surface X given by
3 ( )
T 3 (T 0 + T 3 )(12T 0 + T 3 ) = ∏ T0 + θ (i) T 1 + (θ (i) ) 2 T 2 .
i=1
Then, X (É) ≠ ∅ but X (É) = ∅. On X, there is a Brauer-Manin obstruction
to the Hasse principle.
Indeed, in19, the cubic equation
T (1 + T )(12 + T ) − 1 = 0
has the three solutions 12, 15, and 17. However, in19, 13/12 = 9, 16/15 = 15,
and 18/17 = 10 which are three non-cubes.
5.25. Example. –––– For p 0 = 19, consider the cubic surface X given by
T 3 (T 0 + T 3 )(6T 0 + T 3 ) =
3
∏
i=1
(
T0 + θ (i) T 1 + (θ (i) ) 2 T 2
)
.
Then, on X, there is a Brauer-Manin obstruction to weak approximation.
Indeed, in19, the cubic equation
T (1 + T )(6 + T ) − 1 = 0
has the three solutions 8, 9, and 14. However, in19, 10/9 = 18 is a cube while
9/8 = 13 and 15/14 = 16 are non-cubes.
The smallestÉ-rational point on X is (14 : 15 : 2 : (−7)). Note that indeed
T 3 /T 0 = −7/14 ≡ 9 (mod 19).

124 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
5.26. Example. –––– For p 0 = 19, consider the cubic surface X given by
3 ( )
T 3 (T 0 + T 3 )(2T 0 + T 3 ) = T0 + θ (i) T 1 + (θ (i) ) 2 T 2 .
Then, on X, there is a Brauer-Manin obstruction to the Hasse principle.
Indeed, in19, the cubic equation
∏
i=1
T (1 + T )(2 + T ) − 1 = 0
has T = 5 as its only solution. The other two solution are conjugate to each
other in19 2. However, in19, 6/5 = 5 is a non-cube.
5.27. Example (Swinnerton-Dyer [S-D62]). —– For p 0 = 7, consider the cubic
surface X given by
T 3 (T 0 + T 3 )(T 0 + 2T 3 ) =
3
∏
i=1
(
T0 + θ (i) T 1 + (θ (i) ) 2 T 2
)
= T0 3 − 7T0 2 T 1 + 21T0 2 T 2 + 14T 0 T1 2 − 77T 0 T 1 T 2
+ 98T 0 T2 2 − 7T1 3 + 49T1 2 T 2 − 98T 1 T2 2 + 49T2 3 .
Then, on X, there is a Brauer-Manin obstruction to the Hasse principle.
Indeed, in7, the cubic equation
T (1 + T )(1 + 2T ) − 1 = 0
has T = 5 as its only solution. However, in7, 6/5 = 4 is not a cube.
5.28. Remark. –––– From each of the examples given, by adding multiplies
of p 0 to the coefficients a 1 , d 1 , a 2 , and d 2 , a family of surfaces arises which are
of similar nature. In Example 5.23, some care has to be taken to keep a 1 and a 2
different from zero and a 1 /d 1 different from a 2 /d 2 . In all examples, gcd(a 1 , d 1 )
and gcd(a 2 , d 2 ) need some consideration.
5.29. Remark (Lattice basis reduction). —– The norm form in the p 0 = 19 examples
produces coefficients which are rather large. An equivalent form with
smaller coefficients may be obtained using lattice basis reduction. In its simplest
form, this means the following.
For the rank-2 lattice inÊ3 , generated by the vectors v 1 := (θ (1) , θ (2) , θ (3) ) and
v 2 := ( (θ (1) ) 2 , (θ (2) ) 2 , (θ (3) ) 2) , in fact {v 1 , v 2 + 7v 1 } is a reduced basis. Therefore,
the substitution T 1 ′ := T 1−7T 2 simplifies the norm form. Actually, we find
3
∏
i=1
(
T0 + θ (i) T 1 + (θ (i) ) 2 T 2
)
=T
3
0 −19T0 2 T 1+114T ′
0 T ′
1 2 +57T 0 T ′
1 T 2−133T 0 T2
2
− 209T ′
1 3 − 418T ′
1 2 T 2 + 1045T ′
1 T 2 2 − 209T2 3 .

Sec. 5] THE EXAMPLES OF MORDELL 125
vi. Calculation of the complete Br(X )/π ∗ Br(SpecÉ). —
5.30. Notation. –––– Let X be a cubic surface in P 3 given by
for six linear forms l 1 , l 2 , l 3 , λ 1 , λ 2 , λ 3 .
l 1 l 2 l 3 = λ 1 λ 2 λ 3
Then, we write L ij for the line on X given by l i = λ j = 0.
The subgroup of Pic(X ) generated by these nine lines will be denoted by P.
Further, we let S be the free abelian group over all lines L ij .
5.31. Facts. –––– i) P ⊂ Pic(X ) is a lattice of rank five and discriminant 3.
ii) The kernel S 0 of the canonical homomorphism S → P is generated by
D 1 := L 11 +L 12 +L 13 −L 21 −L 22 −L 23 , D 2 := L 11 +L 12 +L 13 −L 31 −L 32 −L 33 ,
D 3 := L 11 + L 13 − L 22 − L 32 , and D 4 := L 11 + L 12 − L 23 − L 33 .
Proof. We have D 1 = div(l 1 /l 2 ), D 2 = div(l 1 /l 3 ), D 3 = div(l 1 /λ 2 ),
and D 4 = div(l 1 /λ 3 ). Thus, D 1 , D 2 , D 3 , D 4 ∈ ker(S → P). It is easy to
check that D 1 , D 2 , D 3 , D 4 are linearly independent. This shows that rkP ≤ 5.
On the other hand, the intersection matrix of L 11 , L 22 , L 23 , L 32 , and L 33 is
⎛
⎞
−1 0 0 0 0
0 −1 1 1 0
⎜ 0 1 −1 0 1
⎟
⎝ 0 1 0 −1 1⎠ .
0 0 1 1 −1
The determinant of this matrix is equal to 3 which is a square-free integer.
Assertion i) is proven.
It remains to verify that D 1 , D 2 , D 3 , and D 4 generate ker(S → P). For this, we
have to show that the canonical injection
is actually bijective.
〈L 11 , L 22 , L 23 , L 32 , L 33 〉 −→ S/〈D 1 , D 2 , D 3 , D 4 〉
For surjectivity, observe that, modulo 〈D 1 , D 2 , D 3 , D 4 〉, the remaining generators
are given by L 12 ≡ −L 11 + L 23 + L 33 , L 13 ≡ −L 11 + L 22 + L 32 ,
L 21 ≡ L 11 + L 12 + L 13 − L 22 − L 23 ≡ −L 11 + L 32 + L 33 , and, finally,
L 31 ≡ L 11 + L 12 + L 13 − L 32 − L 33 ≡ −L 11 + L 22 + L 23 .
□
5.32. Proposition. –––– Let K/Ébe a Galois extension of degree three. Assume
that λ 1 , λ 2 , λ 3 are defined over K and form a Galois orbit and that l 1 , l 2 , l 3
are linear forms defined overÉ.
Then, Ĥ −1( Gal(K/É), P ) =/3and [l 1 /l 2 ] is a generator.

126 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
Proof. By Lemma II.8.19, we have a canonical isomorphism
Ĥ −1( Gal(K/É), P ) ∼ =
−→ (NS ∩ S 0 )/NS 0 .
Let us calculate NS ∩ S 0 .
D 1 = N (L 11 − L 21 ) and D 2 = N (L 11 − L 31 ) are clearly norms. If
aD 3 + bD 4 = (a + b)L 11 + bL 12 + aL 13 − aL 22 − bL 23 − aL 32 − bL 33
is a norm then the coefficients of L 31 , L 32 , and L 33
Hence, a = b = 0. Consequently,
must coincide.
NS ∩ S 0 = 〈D 1 , D 2 〉 .
Further, N (D 1 ) = 3D 1 , N (D 2 ) = 3D 2 , and N (D 3 ) = N (D 4 ) = D 1 + D 2 .
Hence,
NS 0 = 〈3D 1 , 3D 2 , D 1 + D 2 〉 .
The assertion follows.
□
5.33. –––– Return to the case of a cubic surface given an equation of type
3 ( )
T 3 (a 1 T 0 + d 1 T 3 )(a 2 T 0 + d 2 T 3 ) = T0 + θ (i) T 1 + (θ (i) ) 2 T 2 .
To deduce information on H 1( Gal(É/É), Pic(XÉ) ) from H 1 (G, P) ∼ =/3,
we need some understanding of the remaining 18 lines.
OverÉ(θ), there are defined at least nine lines. The list of the 350 conjugacy
classes of subgroups of W (E 6 ), established using GAP (cf. II.8.23 and the appendix),
contains only four classes which fix nine or more lines. The corresponding
extract looks like this.
1 #U = 1 [ ], #H^1 = 1, Rk(Pic) = 7, Orbits [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
2 #U = 2 [ 2 ], #H^1 = 1, Rk(Pic) = 6, Orbits [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 2, 2, 2, 2, 2, 2 ]
7 #U = 3 [ 3 ], #H^1 = 1, Rk(Pic) = 5, Orbits [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3,
3, 3 ]
24 #U = 6 [ 2 ], #H^1 = 1, Rk(Pic) = 5, Orbits [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3,
3, 3 ]
Thus, the remaining 18 lines are defined over an extension ofÉ(θ) which may
be of degree 6, 3, 2, or 1.
We see that rkPic(XÉ(θ)) ≥ 5. Equality holds if and only if the field of definition
of the 27 lines is of degree 3 or 6 overÉ(θ).
We also observe that, in any case,
∏
i=1
H 1( Gal(É/É(θ)), Pic(XÉ) ) = 0 .

Sec. 5] THE EXAMPLES OF MORDELL 127
5.34. Remark. –––– For every concrete choice of the coefficients, one may
determine the field of definition of the 27 lines by a Gröbner base calculation.
It turned out that it was of degree 6 overÉ(θ) in every example, we tested.
Thus, degree 6 seems to be the generic case.
If the field of definition of the 27 lines is a degree 6 or degree 3 extension ofÉ(θ)
then we may describe H 1( Gal(É/É), Pic(XÉ) ) , completely.
5.35. Proposition. –––– Let p 0 ≡ 1 (mod 3) be a prime number, θ (i) as above,
and X ⊂ P 3Ébe the cubic surface given by
T 3 (a 1 T 0 + d 1 T 3 )(a 2 T 0 + d 2 T 3 ) =
3
∏
i=1
(
T0 + θ (i) T 1 + (θ (i) ) 2 T 2
)
for a 1 , a 2 , d 1 , d 2 ∈. Assume that the field of definition of the 27 lines on X is of
degree 3 or 6 overÉ(θ).
Then, Br(X )/π ∗ Br(SpecÉ) ∼ =/3and [ a 1T 0 +d 1 T 3
T 3
] is a generator.
Proof. Since H 1( Gal(É/É(θ)), Pic(XÉ) ) = 0, the inflation map
H 1( Gal(É(θ)/É), Pic(XÉ) Gal(É/É(θ) ) ) −→ H 1( Gal(É/É), Pic(XÉ) )
is an isomorphism. Hence, in view of Proposition 5.32, it will suffice to
show that P = Pic(XÉ) Gal(É/É(θ)) .
“⊆” is obvious.
“⊇” Our assumption implies that rkPic(XÉ(θ)) = 5. Further, from Proposition
II.8.11, we see that Pic(XÉ(θ)) is always a subgroup of finite index
in Pic(XÉ) Gal(É/É(θ)) . Hence, rkPic(XÉ) Gal(É/É(θ)) = 5, too.
By Fact 5.31.i), we have rkP = 5. Thus, P ⊆ Pic(XÉ) Gal(É/É(θ)) is a sublattice
of finite index. Since DiscP = 3 is square-free, the claim follows. □
5.36. Example. –––– For p 0 = 19, consider the cubic surface X given by
T 3 (T 0 + T 3 )(7T 0 + T 3 ) =
3
∏
i=1
(
T0 + θ (i) T 1 + (θ (i) ) 2 T 2
)
.
Then, on X, there is no Brauer-Manin obstruction to weak approximation.
Indeed, a Gröbner base calculation shows that the 27 lines on X are defined over
a degree 6 extension ofÉ(θ). Thus, Br(X )/π ∗ Br(SpecÉ) ∼ =/3. A generator
is given by the class α := [ T 0+T 3
T 3
].
However, in19, the cubic equation
T (1 + T )(7 + T ) − 1 = 0

128 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
has the only solution T = 11. Hence, everyÉ19-valued point on X has a
reduction of the form (1 : y : z : 11). For the local evaluation, we find
= 12/11 = 8 ∈19 which is a cube.
T 0 +T 3
T 3
Br
Thus, ev(x, α) = 0 for every adelic point x ∈ X (É). By consequence,
X (É) = X (É). One might expect that X satisfies weak approximation.
5.37. Remarks (Degenerate cases). —– i) If the field of definition of the 27 lines
is quadratic overÉ(θ) then Br(X )/π ∗ Br(SpecÉ) = 0.
In fact, in this case, we have a cyclic group G of order 6 acting on the 27 lines.
There are seven conjugacy classes of cyclic groups of order 6 in W (E 6 ). We have
the following list.
26 #U = 6 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 6, 6, 6 ]
27 #U = 6 [ 2, 3 ], #H^1 = 4 [ 2, 2 ], Rk(Pic) = 1, Orbits [ 3, 6, 6, 6, 6 ]
28 #U = 6 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 6 ]
30 #U = 6 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 2, 6, 6, 6 ]
32 #U = 6 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 3, 3, 3, 6, 6 ]
36 #U = 6 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 2, 3, 3, 6, 6 ]
39 #U = 6 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 6, 6, 6, 6 ]
As the element of order two fixes 15 lines, we may not have more than two
orbits of size 6. This implies the claim.
Actually, we are in class No. 32.
ii) If the 27 lines are defined overÉ(θ) then, seemingly, there are two cases.
One might either have rk Pic(X ) = 3. Then, Br(X )/π ∗ Br(SpecÉ) = 0.
Or, otherwise, rk Pic(X ) = 1. In this case, Br(X )/π ∗ Br(SpecÉ) = (/3) 2 .
α := [ a 1T 0 +d 1 T 3
T 3
] is one of the generators.
We do not know of an example overÉin which any of these degenerate
cases occurs.
6. The “first case” of diagonal cubic surfaces
i. The statements. —
6.1. –––– In this section, we show how to compute the Brauer-Manin obstruction
for a large class of diagonal cubic surfaces. More precisely, we are concerned
with smooth diagonal cubic surfaces in P 3É, given by an equation of the form
a 0 T 3
0 + a 1 T 3
1 + a 2 T 3
2 + a 3 T 3
3 = 0
for a 0 , . . . , a 3 ∈\{0}, such that the following additional condition is satisfied.
(∗) There exists a prime number p 0 such that p 0 |a 3 but neither p 3 0|a 3 nor p 0 |a i
for i = 0, 1, 2.

Sec. 6] THE “FIRST CASE” OF DIAGONAL CUBIC SURFACES 129
6.2. –––– X has a model over Specgiven by the same equation. We will
denote that model by X .
On X, there is the smooth genus one curve E given by the equation T 3 = 0.
If p 0 ≠ 3 then assumption (∗) implies that the reduction X p0 is a cone over E p0 .
There is the mapping
red: X (Ép 0
) −→ E p0 (p 0
)
(x 0 : x 1 : x 2 : x 3 ) ↦→ ((x 0 mod p 0 ) : (x 1 mod p 0 ) : (x 2 mod p 0 )) .
6.3. Remark. –––– The Brauer-Manin obstruction on diagonal cubic surfaces
was studied by J.-L.Colliot-Thélène, D. Kanevsky, and J.-J.Sansuc in [CT/K/S].
For the case when (∗) is fulfilled, the main result asserts that X (É) Br is exactly
one third of the whole of X (É) in a sense made precise.
More concretely, there is the following theorem.
∅. Sup-
6.4. Theorem (Colliot-Thélène, Kanevsky, and Sansuc). —–
Let X ⊂ P 3Ébe →É/
a smooth diagonal cubic surface such that X (É) ≠
pose that (∗) is fulfilled for a certain prime number p 0 .
a) Then, Br(X )/π ∗ Br(SpecÉ) ∼ =/3.
b) The image of the evaluation map
ev: Br(X ) × X (É)
is 1 3/.
c) Choose an adelic point (x ν ) ν ∈ X ().
Let α ∈ Br(X ) be such that its image in Br(X )/π ∗ Br(SpecÉ) is non-trivial.
Assume that α is a normalized modulo π ∗ Br(SpecÉ) in such a way that
ev νp0 (α, x νp0 ) = 0.
i) Assume p 0
→É/
≠ 3.
Then, the local evaluation map
ev νp0 : Br(X ) × X (Ép 0
)
has the following property.
ev νp0 (α, ·) is the composition of red with a surjective group homomorphism
(E p0 (p 0
), x νp0 ) → 1 3/.
In particular, ev νp0 (α, x) depends only on the reduction x ∈ X (p 0
).
Further, we have
#{y ∈ X (p 0
) | ev νp0 (α, y)=0} = #{y ∈ X (p 0
) | ev νp0 (α, y)=1/3}
= #{y ∈ X (p 0
) | ev νp0 (α, y)=2/3} .

130 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
ii) Let p 0
→É/
= 3.
Then, the local evaluation map
ev ν3 : Br(X ) × X (Ép 0
)
has the following properties.
The image of ev ν3 (α, ·): X (É3) →É/is 1 3/.
There exists a positive integer n such that ev ν3 (α, x) depends only on the reduction
x (3n )
∈ X (/3 n).
For every y 0 ∈ X (3), we have
In particular,
#{y ∈ X (/3 n) | y = y 0 , ev ν3 (α, y)=0}
= #{y ∈ X (/3 n) | y = y 0 , ev ν3 (α, y)=1/3}
= #{y ∈ X (/3 n) | y = y 0 , ev ν3 (α, y)=2/3} .
#{y ∈ X (/3 n) | ev ν3 (α, y)=0} = #{y ∈ X (/3 n) | ev ν3 (α, y)=1/3}
= #{y ∈ X (/3 n) | ev ν3 (α, y)=2/3} .
6.5. Corollary. –––– Let X ⊂ P 3Ébe a smooth diagonal cubic surface such
that X (É) ≠ ∅. Suppose that (∗) is fulfilled for a certain prime number p 0 .
a) Then, on X, there is a Brauer-Manin obstruction to weak approximation.
b) There is, however, no Brauer-Manin obstruction to the Hasse principle.
6.6. Remark. –––– Colliot-Thélène, Kanevsky, and Sansuc verify the same behaviour
in their “first case” which is a bit more general than assumption (∗).
Another particular case in which the same result is true is when p 0 ∤ a 0 , a 1 , but
p 0 |a 2 , a 3 , p 2 0 ∤a 2, a 3 , and a 2 /a 3 is a p 0 -adic cube.
The three authors also show that, in the “Second case”, there exist diagonal cubic
surfaces such that X (É) ≠ ∅ and X (É) Br = X (É) or ∅. From Theorem 6.4,
it is clear that this may happen only under rather restrictive conditions. For example,
every prime number dividing one of the coefficients must necessarily
divide a second one.
6.7. Remark. –––– The statement that the three values are equally distributed
was not formulated in [CT/K/S].

Sec. 6] THE “FIRST CASE” OF DIAGONAL CUBIC SURFACES 131
6.8. Remark. –––– In order to prove Br(X )/π ∗ Br(SpecÉ) ∼ =/3, we make
heavy use of the list of all possible quotients Br(X )/π ∗ Br(SpecÉ) established
by help of the computer. Cf. II.8.23 and, for the list in full, the appendix.
ii. Calculating Br(X )/π ∗ Br(SpecÉ). —
6.9. Notation. –––– Following Manin [Man], we will denote the lines on X
as follows. We fix third roots of a 0 , . . . , a 3 , once and for all.
Then, L k (m, n) is the line given by
a 1 3 i T i + ζ m 3 a 1 3
j
T j = 0 ,
a 1 3
k T k + ζ n 3 a 1 3
3
T 3 = 0 .
Here, k ∈ {0, 1, 2}, i, j ∈ {0, 1, 2}\{k}, i < j, and m, n ∈/3.
6.10. –––– In this notation, the 45 triangles on X may easily be described.
i) The planes “a 1 3
k T k + ζ n 3 a 1 3
3
T 3 = 0”, for k ∈ {0, 1, 2}, n ∈/3, define 9 triangles
∆ 1 kn . They consist of L k(0, n), L k (1, n), and L k (2, n).
ii) Analogously, the planes “a 1 3 i T i + ζ m 3 a 1 3
j
T j = 0”, for i, j ∈ {0, 1, 2}, i < j,
and m ∈/3, define 9 triangles ∆ 2 km . They consist of L k(m, 0), L k (m, 1),
and L k (m, 2) for k ∈ {0, 1, 2} \ {i, j}.
iii) Finally, the planes given by a 1 3
0
T 0 + ζ a 3 a 1 3
1
T 1 + ζ b 3 a 1 3
2
T 2 + ζ c 3 a 1 3
3
T 3 = 0 for
a, b, c ∈ {0, 1, 2} define 27 triangles ∆ 3 abc . Those consist of L 0(b − a, c),
L 1 (b, c − a), and L 2 (a, c − b).
6.11. Notation. –––– We write L :=É( 3 √
a3 /a 0 , 3 √
a2 /a 0 , 3 √
a1 /a 0 , ζ 3 )/É) and
K :=É( 3 √
a2 /a 0 , 3 √
a1 /a 0 , ζ 3 ). Then, G := Gal(L/É) is the Galois group
acting on the 27 lines.
We have the subgroup G 1 := Gal(L/K ) which is isomorphic to/3by consequence
of assumption (∗). We choose a generator σ of G 1 .
Finally, we write τ for the involution ζ 3 ↦→ ζ3 2 fixing the three third roots.
Then, 〈σ, τ〉 ∼ = S 3 .
6.12. Proposition. –––– The restriction
is not the zero map.
H 1 (G, Pic(XÉ)) −→ H 1 (〈σ〉, Pic(XÉ))

132 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
Proof. The assumption (∗) guarantees that [É( 3 √
a3 /a 0 , ζ 3 ) :É] = 6. The adjunctions
of 3 √
a1 /a 0 and 3 √
a2 /a 0 might or might not lead to further cubic extensions.
Thus, there will be three cases to distinguish, #G may be either 6,
or 18, or 54.
We take for S the free group generated by the 27 lines and use the canonical
isomorphisms H 1 (G, Pic(XÉ)) ∼ = Hom((NG S ∩ S 0 )/N G S 0 ,É/) and
H 1 (〈σ〉, Pic(XÉ)) ∼ = Hom((N 〈σ〉 S ∩ S 0 )/N 〈σ〉 S 0 ,É/). The restriction is then
induced by the norm map N G/〈σ〉 : N 〈σ〉 S → N G S.
We will start with the divisor D := div ( (T 0 + (a 1 /a 0 ) 1 3 T1 )/(T 0 + (a 2 /a 0 ) 1 3 T2 ) )
which is the norm of L 2 (0, 0) − L 1 (0, 0). We have to show that N G/〈σ〉 D is not
the norm N G of a principal divisor.
First case: #G = 54.
This is the generic case. We find that
N G/〈σ〉 D = 6 div ( (a 0 T 3
0 + a 1 T 3
1 )/(a 0 T 3
0 + a 2 T 3
2 ) )
= 6 ∑
(m,n)∈(/3) 2
L 2 (m, n) − 6
∑
(m,n)∈(/3) 2
L 1 (m, n) .
On the other hand, the group S 0 of all principal divisors in S is generated by the
pairwise differences of all triangles [Man,кVI,ÄÑѺ].
We have
N G ∆ 1 kn = 18 ∑L k (i, j) ,
(i,j)∈(/3) 2
N G ∆km 2 = 18 ∑L k (i, j) ,
(i,j)∈(/3) 2
N G ∆ 3 abc = 6
∑
(i,j)∈(/3) 2
L 0 (i, j) + 6
∑
(i,j)∈(/3) 2
L 1 (i, j) + 6
and see that N G/〈σ〉 D is not generated by these elements.
Second case: #G = 18.
∑
(i,j)∈(/3) 2
L 2 (i, j)
Assume without restriction that a 0 = a 1 and a 2 /a 0 is not a cube inÉ∗ .
Then, we find
N G/〈σ〉 D = div ( (T0 3 + T1 3 )6 /(T0 3 + (a 2 /a 0 ) 3 T2 3 )2)
= 6[L 2 (0, 0) + L 2 (0, 1) + L 2 (0, 2)] − 2 ∑L 1 (i, j) .
(i,j)∈(/3) 2
On the other hand,
⎧
⎪⎨
N G L k (m, n) =
⎪⎩
2 ∑ L k (i, j) if k ≠ 2 ,
(i,j)∈(/3) 2
6 ∑
j∈/3L k (m, j) if k = 2 .

Sec. 6] THE “FIRST CASE” OF DIAGONAL CUBIC SURFACES 133
Hence,
N G ∆ 1 kn = 6 ∑L k (i, j),
⎧
⎪⎨
N G ∆ 2 km =
⎪⎩
N G ∆ 3 abc = 2
(i,j)∈(/3) 2
6 ∑ L k (i, j) if k ≠ 2 ,
(i,j)∈(/3) 2
18 ∑
j∈/3L k (m, j) if k = 2 ,
∑
(i,j)∈(/3) 2
L 0 (i, j) + 2
∑
(i,j)∈(/3) 2
L 1 (i, j) + 6 ∑ 2 (a, j) .
j∈/3L
Factoring modulo all summands of type L 2 and ignoring the round brackets,
these expressions are 54L 0 , 54L 1 , and 18L 0 + 18L 1 . They do not generate
(−18)L 1 .
Third case: #G = 6.
Here, we find that
On the other hand,
N G/〈σ〉 D = 2 div ( (T 0 + (a 1 /a 0 ) 1 3 T1 )/(T 0 + (a 2 /a 0 ) 1 3 T2 ) )
= 2 ∑
i∈/3L 2 (0, i) − 2 ∑
i∈/3L 1 (0, i).
N G ∆ 1 kn = 2 ∑
i∈/3L k (0, i) + 2 ∑
i∈/3L k (1, i) + 2 ∑
i∈/3L k (2, i)
N G ∆ 2 km = 6 ∑
i∈/3L k (m, i)
N G ∆ 3 abc = 2 ∑
i∈/3L 0 (b − a, i) + 2 ∑
i∈/3L 1 (b, i) + 2 ∑
i∈/3L 2 (a, i) .
Ignoring the round brackets, those are 18L k for k ∈ {0, 1, 2}and 6L 0 +6L 1 +6L 2
which do not generate 6L 2 − 6L 1 .
□
6.13. Corollary. –––– The restriction map
H 1 (G, Pic(XÉ)) −→ H 1 (〈σ, τ〉, Pic(XÉ))
is an isomorphism. The groups are isomorphic to/3.
Proof. H 1 (〈σ〉, Pic(XÉ)) is purely 3-torsion. Sir P. Swinnerton Dyer’s list
(cf. II.8.24) shows it is either/3or (/3) 2 . Further, we know that the
restriction H 1 (G, Pic(XÉ)) → H 1 (〈σ〉, Pic(XÉ)) is not the zero map. Thus, the
list implies that H 1 (G, Pic(XÉ)), too, is nothing but/3or (/3) 2 .

134 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
The restriction H 1 (G, Pic(XÉ)) → H 1 (〈σ〉, Pic(XÉ)) factors via
H 1 (G, Pic(XÉ)) → H 1 (〈σ, τ〉, Pic(XÉ)).
Thus, the latter is not the zero map, either. Again, a view on the list makes sure
that H 1 (〈σ, τ〉, Pic(XÉ)) may be only/3or (/3) 2 .
At this point, recall that (/3) 2 occurs only when the group acting nontrivially
on Pic(XÉ) is of order three. Since τ ∈ 〈σ, τ〉 ⊂ G is of order two and
acting non-trivially, we are not in this case.
The assertion follows.
6.14. Proposition. –––– The restriction map
H 1 (G, Pic(XÉ)) −→ H 1 (〈σ〉, Pic(XÉ)) τ
is an isomorphism. The groups are isomorphic to/3.
Proof. It suffices to show that the restriction map
H 1 (〈σ, τ〉, Pic(XÉ)) −→ H 1 (〈σ〉, Pic(XÉ)) τ
is an isomorphism. We know, the group on the left hand side is isomorphic
to/3. The group on the right hand side is clearly a 3-torsion group.
We consider the inflation-restriction spectral sequence
E p,q
2
:= H p (〈τ〉, H q (〈σ〉, Pic(XÉ))) =⇒ H n (〈σ, τ〉, Pic(XÉ))
and obtain the following exact sequence of terms of lower order,
0 −→ H 1 (〈τ〉, Pic(XÉ) σ ) −→ H 1 (〈σ, τ〉, Pic(XÉ)) −→
−→ H 1 (〈σ〉, Pic(XÉ)) τ −→ H 2 (〈τ〉, Pic(XÉ) σ ) .
Since the homomorphism considered is encircled by 2-torsion groups, the proof
is complete.
→É/
□
iii. Some observations. —
6.15. Lemma. –––– The image of the Manin map
ev: Br(X ) × X (É) is contained in 1 3/.
□
Proof. By Proposition 2.3.b.i) and iii), the Manin map is additive in the first
variable and factors via Br(X )/π ∗ Br(K ) ∼ =/3.
□

Sec. 6] THE “FIRST CASE” OF DIAGONAL CUBIC SURFACES 135
6.16. Lemma. –––– Let α ∈ Br(X ) be any Brauer class and (C, x 0 ) ⊂ XÉp 0
be a
smooth elliptic curve.
Then, one of the following two statements is true.
i) α| C ∈ Br(C )/π ∗ Br(SpecÉ) is zero. Then, ev νp0 (α, ·) is constant on C (Ép 0
).
ii) α| C ∈ Br(C )/π ∗ Br(SpecÉ) is non-zero.
Then, there is a surjective group homomorphism g : C (Ép 0
) → 1 3/such that
for every x ∈ C (Ép 0
).
ev νp0 (α, x) = g(x) + ev νp0 (α, x 0 )
Proof. i) is clear.
ii) follows directly from Theorem II.8.16.
□
6.17. Lemma. –––– Let p 0 ≠ 3 be a prime number, (C, O) an elliptic curve
overÉp 0
3/
, and C a minimal model of C overp 0
. Assume that there are no
Ép 0
-valued points on C with singular reduction.
Then, every continuous group homomorphism
g : C (Ép 0
) −→ 1
factors via the reduction map C (Ép 0
) → C (p 0
) modulo p 0 .
Proof. Continuity means that g factors via the reduction C (Ép 0
) → C (/p0)
n
for a certain n ∈Æ. The/p0-valued n points on C reducing to the neutral
element form a p 0 -group.
□
6.18. Proposition. –––– Suppose that p 0 ≠ 3. Then, for α ∈ Br(X ), the local
evaluation map ev νp0 (α, ·) factors via
red: X (Ép 0
) −→ E p (p 0
)
(x 0 : x 1 : x 2 : x 3 ) ↦→ ((x 0 mod p 0 ) : (x 1 mod p 0 ) : (x 2 mod p 0 )) .
Proof. Let (x 0 : x 1 : x 2 : x 3 ) ∈ X (Ép 0
). We may assume without restriction that
x 0 , x 1 , x 2 , x 3 ∈p 0
are coprime. Then, assumption (∗) implies that x 0 , x 1 , or x 2
is a unit. Assume, again without restriction, that x 2 is a unit. Then, x 0 and x 1
can not both be multiples of p 0 . Assume p 0 ∤x 1 .
Hensel’s lemma ensures that there exists a unique t ∈p 0
such that
t ≡ x 2 (mod p 0 ) and (x 0 : x 1 : t : 0) ∈ X (Ép 0
). We claim that
ev νp0 (α, (x 0 : x 1 : x 2 : x 3 )) = ev νp0 (α, (x 0 : x 1 : t : 0)) .
Proof of the claim: Both points are contained in the intersection of X with the
hyperplane “x 1 T 0 − x 0 T 1 = 0”. This is a genus one curve C. It is given by the

136 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
equation
[a 0 (x 0 /x 1 ) 3 + a 1 ]T 3
1 + a 2 T 3
2 + a 3 T 3
3 = 0.
The assumptions imply that the first coefficient is not divisible by p 0 .
As p 0 |a 3 , the curve C has bad reduction at p 0 . The equation above defines a
minimal model C ofC overp. Thereare noÉp 0
-valued points with singular reduction.
C p0
3/
consists of three lines meeting in the singular point (0 : 0 : 0 : 1).
It may happen that all three lines are defined overp 0
or that one line is defined
overp 0
and the other two overp 2 and conjugate to each other.
We choose a basepoint x ∈ C (Ép 0
). Then, by Lemma 6.15,
ev νp0 (α, ·)| C (Ép 0
) : C (Ép 0
) −→ 1
differs from a continuous homomorphism of groups just by a constant summand.
Lemma 6.17 shows that ev νp0 (α, ·)| C (Ép 0
) factors via the reduction
map C (Ép 0
) → C reg
p 0
(p 0
).
If only one line of C p0 is defined overp 0
then we have #C reg
p 0
(p 0
) = p 0 .
Every group homomorphism to 3/is 1 constant which implies the claim in
this case.
Otherwise, we have #C reg
p 0
(p 0
) = 3p 0 , each line having exactly p 0 smooth points.
Let x ′ be the third point of intersection of the line tangent to C in x with C.
Then, the group structure on C (Ép 0
) has the property that P+Q+R = 2·x+x ′
if and only if P, Q, and R are collinear. In particular, P + Q is the third point
on the line through R and the neutral element.
The reductions of three collinear points are either belonging to the same line
of C p0 or to three different lines. By consequence, the subgroup of C reg
p 0
(p 0
) of
order p 0 is provided by the line containing the neutral element. The three lines
form its cosets.
As the reductions of (x 0 : x 1 : x 2 : x 3 ) and (x 0 : x 1 : t : 0) are on the same line,
this implies the claim.
Now, we apply Theorem II.8.16 and Lemma 6.17 to the elliptic curve E given
by T 3 = 0. This shows
ev νp0
(
α, (x0 : x 1 : t : 0) ) = g ( (x 0 mod p 0 ) : (x 1 mod p 0 ) : (t mod p 0 ) )
for a map g : E p0 (p 0
) → 3/. 1 Since (t mod p 0 ) = (x 2 mod p 0 ), the proof
is complete.
□
6.19. Remark. –––– In the case p 0 ≠ 3, the only assertion still to be proven is
that ev νp0 (α, ·)| E(Ép 0
) is non-constant. For that, according to Theorem II.8.16,
it suffices to show that α| E ∈ Br(E)/π ∗ Br(SpecÉ) is non-zero. We will verify
this in the next subsection in Proposition 6.24.

Sec. 6] THE “FIRST CASE” OF DIAGONAL CUBIC SURFACES 137
The next lemma, although true in general, will be relevant for us only in the
case p 0 = 3.
6.20. Lemma. –––– Let p be an arbitrary prime number and α ∈ Br(X ) be any
Brauer class. Then, there exists a positive integer n such that ev νp (α, x) depends only
on the reduction x (pn )
∈ X (/p n).
Proof. This follows directly from the continuity of ev νp with respect to the
right argument.
□
iv. Evaluating on the elliptic curve. —
6.21. Notation. –––– For the fields K and L as in 6.11, we fix a valuation ν of K
lying above ν p0 and an extension w of ν to L.
6.22. Lemma. –––– Fix an isomorphism ι: H 2 (〈σ〉,) ∼ =/3. Under the
periodicity isomorphism
H 1 (〈σ〉, Pic(X Kν
))
∼=
−→ Ĥ −1 (〈σ〉, Pic(X Kν
))
= (N Div(X Lw ) ∩ Div 0 (X Kν ))/N Div 0 (X Lw )
induced by ι, a representative of a generator is given by
Proof. In fact,
f := (T 0 + (a 1 /a 0 ) 1 3 T1 )/(T 0 + (a 2 /a 0 ) 1 3 T2 ) .
div( f ) = L 2 (0, 0) + L 2 (0, 1) + L 2 (0, 2) − L 1 (0, 0) − L 1 (0, 1) − L 2 (0, 2)
= N (L 2 (0, 0) − L 1 (0, 0)) .
Further, the norms of the principal divisors are generated by the pairwise differences
of
N ∆ 1 kn = ∑
i∈/3L k (0, i) + ∑
i∈/3L k (1, i) + ∑
i∈/3L k (2, i) ,
N ∆ 2 km = 3 ∑
i∈/3L k (m, i) ,
N ∆ 3 abc = ∑ 0 (b − a, i) + ∑ 1 (b, i) + ∑ 2 (a, i) .
i∈/3L i∈/3L i∈/3L Ignoring the round brackets, these elements are 9L k for k ∈ {0, 1, 2}
and 3L 0 + 3L 1 + 3L 2 . They do not generate 3L 2 − 3L 1 .
□
6.23. Remark. –––– Theperiodicityisomorphism iscompatiblewiththe action
of the involution τ. As f is τ-invariant, it represents a non-zero cohomology
class in H 1 (〈σ〉, Pic(X Kν
)) τ .

138 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
6.24. Proposition. –––– Assume p 0 ≠ 3. Then, the pull-back
is not the zero map.
Br(X )/π ∗ Br(SpecÉ) −→ Br(EÉp 0
)/π ∗ Br(SpecÉp 0
)
Proof. It suffices to show that the restriction
Br(X )/π ∗ Br(SpecÉ) −→ Br(E Kν )/π ∗ Br(SpecK ν )
is not the zero map.
It factors via [Br(X Kν )/π ∗ Br(SpecK ν )] τ ∼ = H 1 (〈σ〉, Pic(X Kν
)) τ for which,
by Lemma 6.22, we have a generator f in explicit form.
It is, therefore, sufficient to show that there exists a point x ∈ E(K ν ) such
that f (x) ∈ K ∗ ν is not in the image of the norm map N : L ∗ w → K ∗ ν.
As L w /K ν is totally ramified, this follows directly from the lemma below.
□
6.25. Lemma. –––– Let p ≠ 3 be a prime number, K be a finite field extension
ofÉp, and L/K a totally ramified Galois extension of degree three. Further, let C
be the genus one curve over K given by x 3 + y 3 + z 3 = 0.
Then, the range of the rational function f := (x + y)/(x + z) on C (K ) contains
an element which is not a norm under N : L ∗ → K ∗ .
Proof. The residue field k := O K /m K is finite and #k ≡ 1 (mod 3). We have
to show that the range of the rational function f = (x + y)/(x + z) on C (k)
contains a non-cube in k ∗ .
For this, we write P := ((−1) : 1 : 0) and Q := ((−1) : 0 : 1).
Then, div f = 3(P) − 3(Q). Choosing an arbitrary point O ∈ C (k) as the
neutral element fixes a group law on C (k). We have the 3-division point P − Q
on C.
Consider the isogeny
i : C ′ ·3
:= C/〈P − Q〉 −→ C.
i corresponds to the field extension k(C ′ ) = k(C ) ( √ )
3 f /k(C ). We claim that
→
i ( C ′ (k) ) ∪ {P, Q} = {x ∈ C (k) | f (x) ∈ (k ∗ ) 3 } ∪ {P, Q} .
Indeed, togiveapointx ∈ C (k)isequivalenttogivingavaluationν x : k(C )
such that, for the corresponding residue field, one has O x /m x
∼ = k. x belongs to
the image of i(k) if and only if this valuation splits completely in k(C ′ ).
Thus, our task is to show that i(k): C ′ (k) → C (k) is not surjective. Note that
C (k) contains at least the nine points given by (1 : (−ζ3 i ) : 0) and permutations.

Sec. 6] THE “FIRST CASE” OF DIAGONAL CUBIC SURFACES 139
The exact sequence of Gal(k/k)-modules
induces a long exact sequence
0 −→ µ 3 −→ C ′ (k) −→ C (k) −→ 0
C ′ (k) −→ C (k) −→ k ∗ /(k ∗ ) 3 −→ 0 .
Observe, by Hasse’s bound, every genus one curve has a point over k. Thus,
there are no non-trivial torsors over C ′ and we have H 1( Gal(k/k), C ′ (k) ) = 0.
The assertion follows.
□
v. Completing the proof for p 0 = 3. —
6.26. Observation. –––– We have K ν =É3(ζ 3 ) or K ν =É3(ζ 3 , 3 √
2).
√ 3√
Proof. By construction, K ν =É3(ζ 3 , 3 a, b) for a, b units inÉ∗ 3 . Modulo third
powers, units inÉ∗ √ 3 fall into √ three classes, represented by 1, 2, and 4, respectively.
AsÉ3(ζ 3 , 3 2) =É3(ζ 3 , 3 4), the assertion follows.
□
6.27. Notation. –––– For F ∈3, we denote the genus one curve, given by the
equation T 3 = FT 0 on X, by E F . In particular, E 0 = E in our previous notation.
The projection of E F to the first three coordinates is “Ax 3 0 + a 1 x 3 1 + a 2 x 3 2 = 0”
for A := a 0 + F 3 a 3 . The assumptions 3 ∤a 0 and 3|a 3 make sure that A is a
3-adic unit.
We write E F for the model of E F over Spec3 given by T 3 = FT 0 on X .
6.28. Proposition. –––– For every F ∈3, the pull-back
is not the zero map.
Br(X )/π ∗ Br(SpecÉ) −→ Br(E FÉ3 )/π∗ Br(SpecÉ3)
Proof. It suffices to show that the restriction
is not the zero map.
Br(X )/π ∗ Br(SpecÉ) −→ Br(E F K ν
)/π ∗ Br(SpecK ν )
It factors via [Br(X Kν )/π ∗ Br(SpecK ν )] τ ∼ = H 1 (〈σ〉, Pic(X Kν
)) τ for which,
by Lemma 6.22, we have a generator [ f ] in explicit form.
It is, therefore, sufficient to show that there exists a point x ∈ E F (K ν ) such that
f (x) ∈ Kν ∗ is not in the image of the norm map N : L ∗ w → Kν. ∗ Since a 2 /a 1 is a
cube in Kν ∗ , this follows directly from the lemma below.
□

140 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
6.29. Lemma. –––– Let A ∈ O Kν be different from zero and C be the genus one
curve over K ν given by Ax 3 + y 3 + z 3 = 0.
Then, the range of the rational function f := (x + y)/(x + z) on C (K ν ) contains
an element which is not a norm under N : L ∗ w → K ∗ ν.
Proof. First step: Description of the norms.
Let π be a uniformizing element of K ν . Then, according to [Se62, Chap. V,
Proposition 5], there exists a positive integer t such that the homomorphism
N n : (1 + m n L w
)/(1 + m n+1
L w
) −→ ( 1 + (π) n) / ( 1 + (π) n+1)
is a bijection for n < t and three-to-one for n = t. In particular, there is a
unit α ∈ O ∗ K ν
such that every element
is not a norm.
For t, there is the formula
w ≡ 1 + απ t (mod π t+1 )
t = ν L(D Lw /K ν
)
2
where D Lw /K ν
denotes the different of L w /K ν .
Second step: Calculation of the different.
Case 1: K ν =É3(ζ 3 , 3 √
2).
Then, L w =É3(ζ 3 , 3 √
2,
3√
3). For the discriminants, one calculates
d Lw /É3 = (3)37 and d Kν /É3 = (3)7 . This results in ν L (D Lw /K ν
) = 16 and t = 7.
For comparison, ν K (3) = 6.
Case 2: K ν =É3(ζ 3 ).
− 1
Then, L w =É3(ζ 3 , 3 √
3),É3(ζ 3 , 3 √
6), orÉ3(ζ 3 , 3 √
12).
For each possibility the discriminant is the same, d Lw /É3 = (3)11 . Together with
d Kν /É3 = (3), this shows ν L(D Lw /K ν
) = 8 and t = 3.
For comparison, ν K (3) = 2.
Third step: Construction of the point.
We claim that, on C, there is a K-valued point (x : y : z) such that x = π t ,
y ≡ −1/α (mod π), and z ≡ 1/α (mod π). Then, the assertion follows since
is not a norm.
x + y
x + z ≡ πt − 1/α
π t + 1/α ≡ απt − 1
απ t + 1 ≡ −(1 + απt ) (mod π t+1 )

Sec. 6] THE “FIRST CASE” OF DIAGONAL CUBIC SURFACES 141
To show the existence of the point, we choose y ≡ −1/α (mod π), arbitrarily.
Then, the equation
g(Z) := Z 3 + (y 3 + Aπ 3t ) = 0
has a solution Z ≡ −y (mod π) by Hensel’s lemma since
ν K (g(−y)) = ν K (Aπ 3t ) ≥ 3t ,
ν K (g ′ (−y)) = ν K (3y 2 ) = ν K (3) ,
and 3t > 2ν K (3).
□
6.30. Proposition. –––– Assume that a 2 /a 1 ∈É∗ 3 is a non-cube and let
t = (t 0 : t 1 : t 2 : t 3 ) ∈ X (3) be a point with t 0 ≠ 0. Choose n ∈Æsuch
that ev νp (α, x) depends only on the reduction x (pn )
∈ X (/p n).
Then, for any F ∈3 such that (F mod 3) = t 3 /t 0 , one has
#{y ∈ E F (/3 n) | y = t, ev ν3 (α, y)=0}
= #{y ∈ E F (/3 n) | y = t, ev ν3 (α, y)=1/3}
= #{y ∈ E F (/3 n) | y = t, ev ν3 (α, y)=2/3} .
Proof. Proposition 6.28 shows that α| E ∈ Br(E FÉ3
FÉ3 )/π∗ Br(SpecÉ3) is different
from zero. Thus, α| E yields a surjective homomorphism of groups
FÉ3
E F (É3) → 3/. 1 From this, we immediately see
#{y ∈ E F (/3 n) | ev ν3 (α, y) = 0} = #{y ∈ E F (/3 n) | ev ν3 (α, y) = 1/3}
We have to show the same for points reducing to t.
= #{y ∈ E F (/3 n) | ev ν3 (α, y) = 2/3} .
A unit u ∈3 is a cube if and only if u ≡ ±1 (mod 9). Permuting coordinates
and changing a sign, if necessary, we may therefore assume that a 1 ≡ 1 (mod 9)
and a 2 ≡ 7 (mod 9).
E F is given by Ax 3 0 + a 1 x 3 1 + a 2 x 3 2 = 0 for A := a 0 + F 3 a 3 . In principle, there
are three cases.
First case: A ≡ ±7 (mod 9).
We may assume A ≡ 7 (mod 9). Then, modulo 3, allÉ3-valued points on E F
reduce to (1 : 0 : (−1)). Hence, the assertion is true in this case.
Second case: A ≡ ±4 (mod 9).
This is impossible as 4x 3 0 + 1x 3 1 + 7x 3 2 = 0 allows no solutions inÉ3 3 except
for (0, 0, 0).
Third case: A ≡ ±1 (mod 9).

142 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
Assume without restriction that A ≡ 1 (mod 9). AÉ3-valued point on E F
may reduce either to (1 : (−1) : 0) or to (1 : 1 : 1).
Take (1 : (−a) : 0) ∈ E F (É3), fora thethirdrootofA/a 1 , astheneutralelement.
Then, Lemma 6.31 shows that the reduction map E F (É3) → E F (3) gives rise
to a surjective group homomorphism E F (É3) →/2.
As 2 is prime relative to 3, the asserted equidistribution of ev ν3 (α, ·) holds in
each class, separately.
□
6.31. Lemma. –––– Consider the elliptic curve C overÉ3, given by
Take (1 : (−1) : 0) as its basepoint.
x 3 + y 3 + 7z 3 = 0 .
C has a minimal Weierstraß model such that C 0 (É3), the group of points with
non-singular reduction, coincides with the set of the points reducing (naively)
to (1 : (−1) : 0). Further, C (É3)/C 0 (É3) ∼ =/2.
Proof. The substitutions
lead to the Weierstraß equation
x ′ := −21z , y ′ := 63
2 (x − y) , z′ := x + y
z ′ y ′2 = x ′3 − 72 3 3
4 z′3 .
If the reduction of (x : y : z) is (1 : 1 : 1) then (x ′ : y ′ : z ′ ) reduces to (0 : 0 : 1).
This is the cusp.
On the other hand, consider the case that x = 1, z = 3k, and, therefore,
y ≡ −1−7·3 2 k 3 (mod 27). Dividing the substitution formulas by 63, we obtain
x ′ = −k, y ′ ≡ 1+ 7 2 32 k 3 (mod 27), i.e., y ′ ≡ 1 (mod 9), andz ′ = −k 3 (mod 3).
In particular, y ′ is always a unit. The reduction of (x ′ : y ′ : z ′ ) is never equal to
the cusp.
It remains to show that C (É3)/C 0 (É3) ∼ =/2and that the Weierstraß model
found is minimal. For this, we follow Tate’s algorithm [Ta75].
We have the affine equation
Y 2 = X 3 − 72 3 3
4 .
According to [Ta75, Summary], we are in case 6). The polynomial P is given
by P(T ) = T 3 − 72 . It has a triple zero modulo 3.
4
We observe that 72
≡ 1 (mod 9) and writeu for the 3-adic unit such 4 thatu3 = 72
. 4
The substitution X ′ := X − 3u leads to Y 2 = X ′3 + 9uX ′2 + 27u 2 X ′ .

Sec. 6] THE “FIRST CASE” OF DIAGONAL CUBIC SURFACES 143
J. Tate’s equation (8.1) becomes
Y 2 2 = 9X 3 2 + 9uX 2 2 + 3u 2 X 2 .
Here, Y =: 9Y 2 and X ′ =: 9X 2 . Equation (9.1) is
3Y 2 3 = 3X 3 2 + 3uX 2 2 + u 2 X 2
for Y 2 =: 3Y 3 . As u 2 is a unit, we see that the Weierstraß model is minimal and
of Kodaira type III ∗ . [Ta75, table on p. 46] indicates C (É3)/C 0 (É3) ∼ =/2.
□
6.32. Proposition. –––– Let X be any diagonal cubic surface fulfilling (∗).
Choose n ∈Æsuch that ev νp (α, x) depends only on the reduction
x (pn )
∈ X (/p n).
Then, for every t = (t 0 : t 1 : t 2 : t 3 ) ∈ X (3), we have
#{y ∈ X (/3 n) | y = t, ev ν3 (α, y)=0}
= #{y ∈ X (/3 n) | y = t, ev ν3 (α, y)=1/3}
= #{y ∈ X (/3 n) | y = t, ev ν3 (α, y)=2/3} .
Proof. The case a 0 ≡ a 1 ≡ a 2 (mod 9) is treated in Example 6.33.
We may therefore assume that two of the quotients a 0 /a 1 , a 1 /a 2 , and a 2 /a 0 ,
say a 1 /a 2 and a 2 /a 0 , are non-cubes inÉ∗ 3 . It is impossible that t 0 = t 1 = 0.
Again without restriction, assume t 0 ≠ 0.
We consider the projection
π : X −→ P 1 ,
(x 0 : x 1 : x 2 : x 3 ) ↦→ (x 3 : x 0 ) .
On aÉ3-valued point x reducing to t, the birational map π is defined.
We have π(x) = (F : 1) for some F ∈3 such that (F mod 3) = t 3 /t 0 .
By Proposition 6.30, the asserted equality is true in every fiber of π. It is
therefore true in general.
□
6.33. Example. –––– Let d ∈and consider the cubic surface overÉgiven by
T0 3 + T1 3 + T2 3 + 3dT3 3 = 0 .
=/3
By the results shown, we have
Br(X )/π ∗ Br(SpecÉ) = H 1( Gal(L/K ), Pic(X L ) ) ∼

144 AN APPLICATION: THE BRAUER-MANIN OBSTRUCTION [Chap. III
for L =É(ζ 3 , 3 √
3d) which is equal to the field of definition of the 27 lines
on X. As K =É(ζ 3 ) does not contain any cubic extensions, the Brauer-
Manin obstruction may be described completely explicitly, without relying on
Lichtenbaum’s duality.
In fact,
H 1( Gal(L/É), Pic(X L ) ) ∼ = H
1 ( 〈σ〉, Pic(X L ) ) τ
= [Br(XÉ(ζ 3 ))/π ∗ Br(SpecÉ(ζ 3 ))] τ .
For the latter, we have the explicit generator [ f ] for f := (T 0 +T 1 )/(T 0 +T 2 ).
This means, for x = (x 0 : x 1 : x 2 : x 3 ) ∈ X (É3(ζ 3 )) and ν ∈ Val(É(ζ 3 )) the
extension of ν 3 , we have
(
ev ν ([ f ], x) = i ν (t0 + t 1 )/(t 0 + t 2 ) ) .
Here, i ν is the homomorphism
É3(ζ 3 ) ∗ −→É3(ζ 3 ) ∗ /NL ∗ ∼=
w −→ Ĥ 0 (〈σ〉, L ∗ ∼=
w) −→ H 2 (〈σ〉, L ∗ w) inv ν
−→É/
for w the extension of ν to L.
However, we want to considerÉ3-valued points, notÉ3(ζ 3 )-valued ones.
Thus, let α ∈ Br(X ) be a Brauer class mapping to [ f ] under restriction.
Then, for x ∈ X (É3),
2 · ev ν3 (α, x) = ev ν ([ f ], x) .
Fortunately, multiplication by 2 is an automorphism of 1 3/.
On X, two kinds ofÉ3-valued points may be distinguished.
First kind: x 0 , x 1 , and x 2 are units.
Then, x 0 ≡ x 1 ≡ x 2 (mod 3) and d ∤ 3. We have
ev ν3 (α, x) = 0 ⇐⇒ x 1 ≡ x 2 (mod 9) .
Second kind: Among x 0 , x 1 , and x 2 , there is an element which is a multiple of 3.
Suppose 3|x 0 . Then, x 1 ≡ −x 2 (mod 3). We have
ev ν3 (α, x) = 0 ⇐⇒ 2x 0 + x 1 + x 2 (mod 9) .
6.34. Corollary. –––– If x 3 0 +x 3 1 +x 3 2+3x 3 3 = 0 for x 0 , . . . , x 3 ∈then we have
the following non-trivial congruences.
i) If x 0 , x 1 , and x 2 are units then
x 0 ≡ x 1 ≡ x 2 (mod 9) .

Sec. 6] THE “FIRST CASE” OF DIAGONAL CUBIC SURFACES 145
ii) If 3|x 0 then
x 0 ≡ 0 (mod 9) .
Proof. This is the particular case d = 1. Then, L =É(ζ 3 , 3 √
3) is unramified at
every prime different from 3. As ν p
(
(t0 + t 1 )/(t 0 + t 2 ) ) is always divisible by 3,
we have ev νp (α, x) = 0 for every p ≠ 3.
The congruences obtained from ev ν3 (α, x) = 0 imply the congruences asserted.
□
6.35. Remark. –––– Thus, we have shown that, for the equation
T 3
0 + T 3
1 + T 3
2 + 3T 3
3 = 0
overÉ, the restrictions coming from the Brauer-Manin obstruction are exactly
the congruences obtained by R. Heath-Brown [H-B92a] in 1993.
Note that Heath-Brown used rather different methods. His main tool is the law
of cubic reciprocity.

PART B
HEIGHTS

CHAPTER IV
THE CONCEPT OF A HEIGHT
Equations are just the boring part of mathematics. I attempt to see things
in terms of geometry.
STEPHEN HAWKING (A Biography (2005) by Kristine Larsen, p. 43)
1. The naive height on the projective space overÉ
1.1. –––– Heights have been studied by number theorists for a very long time.
A height is a function measuring the size or, more precisely, the arithmetic complexity
of certain objects. These objects are classically solutions of Diophantine
equations or rational points on an algebraic variety. A height then might answer
the question how many bits one would need in order to store the solution or
the point on a computer.
More recently, starting with G. Faltings’ ideas of heights on moduli spaces, it
became more common to consider heights for more complicated objects such
as cycles.
1.2. Definition. –––– For (x 0 : . . . : x n ) ∈ P n (É), put
Here,
H naive (x 0 : . . . : x n ) := max
i=0,...,n |x′ i| .
(x ′ 0 : . . . : x′ n ) = (x 0 : . . . : x n )
such that all x ′ i are integers and gcd(x′ 0 , . . . , x′ n ) = 1.
The function H naive : P nÉ(É) →Êis called the naive height.
1.3. Fact (Fundamental finiteness). —– For every C ∈Ê, there are only finitely
many points x ∈ P nÉ(É) such that
H naive (x) < C .
Proof. For eachcomponentof (x 0, ′ . . . , x n), ′ wehave −C < x i ′ < C. Thus, there
are only finitely many choices.
□

150 THE CONCEPT OF A HEIGHT [Chap. IV
1.4. –––– The naive height is probably the simplest function one might think
of which fulfills the fundamental finiteness property. For more general height
functions, the fundamental finiteness property will always be required.
1.5. Remark. –––– Let X ⊂ P nÉbe a subvariety. Then, everyÉ-rational point
on X is also aÉ-rational point on P n .
We call the restriction of H naive to X (É) the naive height on X.
Is is obvious that the naive height on X fulfills the fundamental finiteness property.
1.6. Notation. –––– i) For a prime number p, denote by ‖ . ‖ p the normalized
p-adic valuation.
I.e., for x ∈É\{0}, let x = ±2 v 2 ·3
v
v pk
3 · . . . · p
k
prime factors. Then, put
‖x‖ p := p −v p
.
Further, ‖0‖ p := 0.
be its decomposition into
ii) We use ‖ . ‖ ∞ as an alternative notation for the usual absolute value,
{ x if x ≥ 0 ,
‖x‖ ∞ :=
−x if x < 0 .
1.7. Fact. –––– For p a prime number or infinity, ‖ . ‖ p is indeed a valuation.
This means, for x, y ∈É,
i) ‖x‖ p ≥ 0,
ii) ‖x‖ p = 0 if and only if x = 0,
iii) ‖xy‖ p = ‖x‖ p · ‖y‖ p ,
iv) ‖x + y‖ p ≤ ‖x‖ p + ‖y‖ p .
For p ≠ ∞, one even has that ‖x + y‖ p ≤ max{‖x‖ p , ‖y‖ p }.
1.8. Fact (Product formula). —– For x ∈É\{0}, one has
∏ ‖x‖ p = 1 .
p prime or ∞
1.9. Lemma. –––– Let (x 0 : . . . : x n ) ∈ P nÉ(É). Then,
[
H naive (x 0 : . . . : x n ) =
∏
p prime or ∞
]
max ‖x i‖ p
i=0,...,n
Proof. The product formula implies that the right hand side remains unchanged
when (x 0 : . . . : x n ) is replaced by (λx 0 : . . . : λx n ) for λ ≠ 0. Thus, we may
suppose that all x i are integers and gcd(x 0 , . . . , x n ) = 1.
.

Sec. 2] GENERALIZATION TO NUMBER FIELDS 151
The assumptions that x 0 , . . . , x n ∈and gcd(x 0 , . . . , x n ) = 1 imply
max
i=0,...,n ‖x i‖ p = 1
for every prime number p. Hence, the formula on the right hand side may be
simplified to max i=0,...,n |x i |.
This is precisely the assertion.
1.10. Remark. –––– Despite being so primitive, the naive height is actually
sufficient for most applications. For example, in the numerical experiments
described in Part C, we will always work with the naive height.
□
2. Generalization to number fields
i. The definition. —
2.1. –––– Let K be a number field. I.e., K is a finite extension ofÉ. It is
well known from algebraic number theory [Cas67] that there is a set Val(K ) of
normalized valuations ‖ . ‖ ν on K satisfying the following conditions.
a) The functions ‖ . ‖ ν : K →Êare indeed valuations. I.e., for x, y ∈É,
i) ‖x‖ ν ≥ 0,
ii) ‖x‖ ν = 0 if and only if x = 0,
iii) ‖xy‖ ν = ‖x‖ ν · ‖y‖ ν ,
iv) ‖x + y‖ ν ≤ ‖x‖ ν + ‖y‖ ν .
b) There is the product formula
∏ ‖x‖ ν = 1
∈Æ
ν∈Val(K )
for every x ∈ K \{0}.
2.2. –––– Further, for L/K a degree d extension of number fields, the sets
Val(K ) and Val(L) are compatible in the following sense.
i) For every ‖ . ‖ µ ∈ Val(L), there are a valuation ‖ . ‖ ν ∈ Val(K ) and d µ
such that
‖ . ‖ µ | K = ‖ . ‖ d µ
ν .
In this case, it is said that ‖ . ‖ µ is lying above ‖ . ‖ ν .
ii) For every ‖ . ‖ ν ∈ Val(K ), there are only a finite number of valuations
‖ . ‖ µ1 , . . . , ‖ . ‖ µl ∈ Val(L) lying above ‖ . ‖ ν . One has
l
∑
i=1
d µi = d .

152 THE CONCEPT OF A HEIGHT [Chap. IV
This implies
for every x ∈ K.
‖x‖ d ν =
l
∏
i=1
‖x‖ µi
2.3. –––– A valuation is called archimedean if it lies above the valuation ‖ . ‖ ∞
ofÉ. Otherwise, it is called non-archimedean.
If ν is non-archimedean then one has the ultrametric triangle inequality
‖x + y‖ ν ≤ max{‖x‖ ν , ‖y‖ ν } .
2.4. Definition. –––– Let K be a number field of degree d. Then, for
(x 0 : . . . : x n ) ∈ P n (K ), one puts
[ ] 1
H naive (x 0 : . . . : x n ) := ∏ max ‖x d
i‖ ν .
i=0,...,n
ν∈Val(K )
This height function is the number field version of the naive height on P n . It is
usually called the absolute height.
2.5. Lemma. –––– Let K be a number field and (x 0 : . . . : x n ) ∈ P n (K ). Further,
let L ⊃ K be a finite extension.
Then, the absolute height H naive (x 0 : . . . : x n ) remains unchanged when
(x 0 : . . . : x n ) is considered as an L-rational point.
Proof. Put d ′ := [L : K ]. Then, by the properties of the valuations, we have
[
H naive (x 0 : . . . : x n ) = ∏
ν∈Val(K )
= ∏
ν∈Val(K )
= ∏
µ∈Val(L)
] 1
max ‖x d
i‖ ν
i=0,...,n
[
∏
µ∈Val(L)
µ above ν
[
] 1
max ‖x dd
i‖ ′
µ
i=0,...,n
] 1
max ‖x dd
i‖ ′
µ
i=0,...,n
which is exactly the formula for H naive (x 0 : . . . : x n ) considered as an
L-rational point.
□
2.6. Proposition (Northcott). —– Let B, D ∈Ê. Then,
is a finite set.
{x ∈ P n (É) | x ∈ P n (K ) for [K :É] < D and H naive (x) < B}
Proof. We may work with the number fields K of a fixed degree d.

Sec. 2] GENERALIZATION TO NUMBER FIELDS 153
For x, we choose homogeneous coordinates such that some coordinate equals 1.
Then, it is clear that, for every valuation ‖ . ‖ ν and every index i, we have
max{‖x 0 ‖ ν , . . . , ‖x n ‖ ν } ≥ max{1, ‖x i ‖ ν } .
Multiplying over all ν and taking the d-th root therefore shows
Hence, it suffices to verify that the set
H naive (x 0 : . . . : x n ) ≥ H naive (1 : x i ) .
{(1 : x) ∈ P 1 (É) | (1 : x) ∈ P 1 (K ) for [K :É] = d and H naive (1 : x) < B}
is finite.
For this, wewriteσ 1 (x), . . . , σ d (x) ∈Éfor theelementarysymmetricfunctions
in the conjugates of x. According to Vieta, x is a zero of the polynomial
F x (T ) := T d − σ 1 (x)T d−1 + σ 2 (x)T d−2 + . . . + (−1) d σ d (x) ∈É[T ] .
Lemma 2.7 shows that
(
H naive 1 : σr (x) ) ( d
)
( d
)
≤ · H naive (1 : x) rd ≤ · B rd .
r r
Thus, by Fact 1.3, we know that for each σ r (x) there are only finitely many possibilities.
Hence, there are only finitely many possibilities for the polynomial F x
and, therefore, only finitely many possibilities for x.
This completes the proof.
2.7. Lemma. –––– Let K be number field of degree d which is Galois overÉ.
For x ∈ K, denote by σ 1 (x), . . . , σ d (x) ∈Éthe elementary symmetric functions
in the conjugates of x. Then,
(
H naive 1 : σr (x) ) ( d
)
≤ · H naive (1 : x) rd .
r
Proof. We denote the conjugates of x by x 1 , . . . , x d . Let ν be any valuation
of K. Then,
∥ ∥∥∥∥ν
‖σ r (x)‖ ν =
∥
∑ x i1 · . . . · x ir
1≤i 1

154 THE CONCEPT OF A HEIGHT [Chap. IV
By consequence,
max{1, ‖σ r (x)‖ ν } ≤ C (d, r, ν) · max max{1, ‖x i ‖ ν } r
i
≤ C (d, r, ν) · ∏ max{1, ‖x i ‖ ν } r .
i
Multiplying over all valuations of K yields
(
H naive 1 : σr (x) )d ( d
) d
≤ ·
r
∏ H naive (1 : x i ) rd
i
( d
) d
= · Hnaive (1 : x) rd2
r
since conjugate points have the same height. This inequality is equivalent to
the assertion.
□
2.8. –––– For most theoretical investigations and, may be, some practical ones,
it is actually better to work with the logarithm of the absolute height.
2.9. Definition. –––– For (x 0 : . . . : x n ) ∈ P n (É), put
h naive (x 0 : . . . : x n ) := log H naive (x 0 : . . . : x n ) .
The height function h naive is called the logarithmic height.
ii. Application: The height defined by an ample invertible sheaf. —
2.10. Definition. –––– Let X be a projective variety over a number field K
and L be an ample invertible sheaf on X.
Then, a height function h L induced by L is given as follows. Let m ∈Æbe such
that L ⊗m is very ample. Put
h L (x) := 1 m h (
naive iL ⊗m(x) )
for x ∈ P a closed point. Here, i L ⊗m : P → P N denotes a closed embedding
defined by L ⊗m .
2.11. Lemma. –––– Let X be a projective variety over a number field K and L
be an ample invertible sheaf on X. If h (1)
L and h(2) L are two height functions induced
by L then there is a constant C such that
for every x ∈ X (É).
| h (1)
L (x) − h(2) L (x)| ≤ C

Sec. 3] GEOMETRIC INTERPRETATION 155
Proof. First, it is clear that the k-uple embedding fulfills
h naive
(
ρk (x) ) = k·h naive (x)
for every k ∈Æand x ∈ P N (É). Hence, if h L is defined using L ⊗m then the
same height function may be defined using L ⊗km .
We may therefore assume that h (1)
L
and h(2) L
are defined using the same tensor
L
differ by an automor-
⊗m
power of L . Then, the two embeddings i (1)
L
and i (2)
⊗m
phism ι: P N → P N .
We have to verify that
| h naive
(
ι(x)
) − hnaive (x)| ≤ C
for every x ∈ P N (É). ι is explicitly given by
(x 0 : . . . : x N ) ↦→ (x ′ 0 : . . . : x ′ N )
for some a = (a ij ) ij ∈ M N +1 (K ).
:= (a 00 x 0 + . . .+ a 0N x N ) : . . . : (a N 0 x 0 + . . .+ a NN x N )
For all but finitely many ν ∈ Val(K ), we have that ν is non-archimedean
and ‖a ij ‖ ν = 1 for all i and j. This yields
max
i=0,...,N ‖x′ i‖ w ≤ max ‖x i‖ w
i=0,...,N
for every number field L and every w ∈ Val(L) lying above such a valuation.
For every other valuation of K, we find a constant D ν such that
The desired inequality
follows immediately from this.
max
i=0,...,N ‖x′ i‖ w ≤ D d w/ν
ν
h naive
(
ι(x)
) − hnaive (x) ≤
· max
i=0,...,N ‖x i‖ w .
∑
ν∈Val(K )
log D ν
For the inequality the other way round, one may work with ι −1 instead of ι.
□
3. Geometric interpretation
3.1. –––– Heights are more closely related to modern algebraic geometry than
this might seem from the definitions given in the sections above. The geometric
interpretation of the concept of a height is the starting point of arithmetic
intersection theory, a fascinating theory which we may only touch upon here.

156 THE CONCEPT OF A HEIGHT [Chap. IV
The reader is advised to consult the articles [G/S 90, G/S 92] of H. Gillet
and C. Soulé, the textbook [S/A/B/K], and the references therein. More recent
developments are described by J. I. Burgos Gil, J. Kramer, and U. Kühn
in [B/K/K].
The particular case of the arithmetic intersection theory on a curve over a
number field had been developed earlier. The articles of S. Yu. Arakelov [Ara]
and G. Faltings [Fa84] present the point of view taken before around 1990 which
is a bit different from today’s.
3.2. –––– We return to the assumption that the ground field is K =É. This is
done mainly in order to ease notation. The theory would work equally well
over an arbitrary number field.
3.3. Definition. –––– An arithmetic variety is an integral scheme which is projective
and flat over Spec.
3.4. Definition. –––– A hermitian line bundle on an arithmetic variety X is a
pair (L , ‖ . ‖) consisting of an invertible sheaf L ∈ Pic(X ) and a continuous
hermitian metric on the line bundle Lassociated with L on the complex
space X ().
All hermitian line bundles on X form an abelian group which is denoted
by ̂Pic C 0 (X ).
For the group of all smooth (C ∞ ) hermitian line bundles, one writes ̂Pic(X ).
̂Pic(X ) is usually called the arithmetic Picard group.
3.5. Example. –––– The projective space P nover Specis an arithmetic variety.
On P n, there is the tautological invertible sheaf O(1). It is generated by
the global sections X 0 , . . . , X n ∈ Γ(P n, O(1)).
Let us show two explicit hermitian metrics on O(1)making O(1) to a hermitian
line bundle.
i) The Fubini-Study metric is given by
for l ∈ Γ(P n, O(1)).
‖l‖ FS :=
√
(X0
ii) The minimum metric is given by
‖l‖ min := min
l
1
) 2
+ . . . + ( X n
) 2
l
i=0,...,n
∣ l ∣∣∣
∣ .
X i

Sec. 3] GEOMETRIC INTERPRETATION 157
In the case of the Fubini-Study metric, the formula should be interpreted in
such a way that ‖l‖ FS (x) = 0 at all points x ∈ P n () where l vanishes. Similarly,
in the definition of ‖l‖ min (x), the minimum is actually to be taken over
all i such that x i ≠ 0.
The Fubini-Study metric is smooth (C ∞ ). The minimum metric is continuous
but not even C 1 .
3.6. Definition (Arakelov degree). —– For a hermitian line bundle (L , ‖ . ‖)
on Spec, define its Arakelov degree by
̂deg (L , ‖ . ‖) := log #(L /sL ) − log ‖s‖
for s ∈ Γ(Spec, L ) a non-zero section.
3.7. Remark. –––– L is associated with a free-module L of rank one. Lis
the-vector space L⊗. Thus, we work with a non-zero element s ∈ L and
with the norm ‖s ⊗ 1‖.
The definition is independent of the choice of s since
#(L /nsL ) = n·#(L /sL ) ,
therefore log #(L /nsL ) = log #(L /sL ) + log n.
log ‖ns‖ = log ‖s‖ + logn.
On the other hand,
3.8. Fact. –––– Let L be a hermitian line bundle on Spec. If there is a section
s ∈ Γ(Spec, L ) of norm less than one then ̂deg L > 0.
Proof. This follows immediately from the definition.
3.9. Fact. –––– For two hermitian line bundles L 1 , L 2 on Spec, one has
̂deg (L 1 ⊗ L 2 ) = ̂deg L 1 + ̂deg L 2 .
Proof. Apply the definition to arbitrary non-zero sections s 1 ∈ Γ(Spec, L 1 ),
s 2 ∈ Γ(Spec, L 2 ), and to their tensor product s 1 ⊗s 2 ∈ Γ(Spec, L 1 ⊗L 2 ). □
3.10. –––– AÉ-rational point on the projective space is actually a morphism
x : SpecÉ→P nof schemes. The valuative criterion for properness implies
that there is a unique extension to a morphism from Spec,
x
SpecÉ n P
∩
x
Spec.
□

158 THE CONCEPT OF A HEIGHT [Chap. IV
3.11. Observation (Arakelov). —– Let x ∈ P n (É) and x : Spec→P nits
extension to Spec. Then,
h naive (x) = ̂deg ( x ∗ (O(1), ‖ . ‖ min ) ) .
Proof. Write x in coordinates as (x 0 : . . . : x n ) for x 0 , . . . , x n ∈. Then, the
pull-back of X i ∈ Γ(P n, O(1)) is x i ∈ Γ(Spec, x ∗ (O(1))). Since O(1) is generated
by the global sections X i , we have
x ∗ (O(1)) = 〈x 0 , . . . , x n 〉 = gcd(x 0 , . . . , x n ) ·.
We choose an index i 0 such that x i0 ≠ 0. Using this section, we obtain
̂deg (x ∗ (O(1)), ‖ . ‖) = log # ( x ∗ (O(1))/x i0 x ∗ (O(1)) ) − log ‖x i0 ‖
which is exactly the logarithmic height of x.
= log # ( gcd(x 0 , . . . , x n )/x i0) − log min
∣ = log
x i0
∣gcd(x 0 , . . . , x n ) ∣ + log max
x i ∣∣∣
i=0,...,n
∣x i0
max |x i|
i=0,...,n
= log
gcd(x 0 , . . . , x n )
i=0,...,n
∣ x i0 ∣∣∣
∣ x i
□
3.12. –––– This motivates the following general definition.
Definition. Let X be an arithmetic variety and L be a hermitian line bundle
on X .
Then, the height with respect to L of theÉ-rational point x ∈ X (É) is given by
h L
(x) = ̂deg ( x ∗ L ) .
Here, x : Spec→X is the extension of x to Spec.
3.13. Examples. –––– i) Let X = P nand let L be the invertible sheaf O(1)
equipped with the minimum metric. Then, as shown in 3.11, the height with
respect to L is the naive height.
ii) Let X = P nand let L be the invertible sheaf O(1) equipped with the
Fubini-Study metric. Then, the height with respect to L is the l 2 -height which
is given by
√
h l 2(x 0 : . . . : x n ) := log x
0 ′2 + . . . + x′2 n
for
(x ′ 0 : . . . : x′ n ) = (x 0 : . . . : x n )

Sec. 3] GEOMETRIC INTERPRETATION 159
projective coordinates such that all x ′ i are integers and gcd(x ′ 0, . . . , x ′ n) = 1.
3.14. Lemma. –––– Let f : X → Y be a morphism of arithmetic varieties and
L be a hermitian line bundle on Y .
Then, for every x ∈ X (É),
h f ∗ L (x) = h L ( f (x)) .
Proof. Let x be the extension of x to Spec. Then, f ◦x is the extension of f (x)
to Spec. Thus,
h f ∗ L (x) = ̂deg ( x ∗ ( f ∗ L ) ) = ̂deg ( ( f ◦x) ∗ L ) = h L
( f (x)) .
□
3.15. Proposition. –––– Let X be an arithmetic variety.
a) Let ‖ . ‖ 1 and ‖ . ‖ 2 be hermitian metrics on one and the same invertible sheaf L .
Then, there is a constant C such that
for every x ∈ X (É).
|h (L ,‖ .‖1 )(x) − h (L ,‖ .‖2 )(x)| < C
b) Let L 1 and L 2 be two hermitian line bundles. Then, for every x ∈ X (É),
h (L1 ⊗L 2 ) (x) = h L 1
(x) + h L2
(x) .
c) LetL beahermitianlinebundlesuchthattheunderlyinginvertiblesheafisample.
Then, for every C ∈Ê, there are only finitely many points x ∈ X (É) such that
h L
(x) < C .
Proof. a) For x the extension of x to Spec, we have
h (L ,‖ .‖1 )(x) − h (L ,‖ .‖2 )(x) = ̂deg ( x ∗ L , x ∗ ‖ . ‖ 1
) − ̂deg
(
x ∗ L , x ∗ ‖ . ‖ 2
)
.
Working with a non-zero section s ∈ Γ(Spec, x ∗ L ), we see that the latter difference
is equal to
log #(x ∗ L /sx ∗ L ) − log ‖s‖ 1 (x) − ( log #(x ∗ L /sx ∗ L ) − log ‖s‖ 2 (x) )
= log ‖ . ‖ 2(x)
‖ . ‖ 1 (x) .
Since X () is compact, there is a positive constant D such that
1
D < ‖ . ‖ 2(x)
‖ . ‖ 1 (x) < D
for every x ∈ X (). This implies the claim.

160 THE CONCEPT OF A HEIGHT [Chap. IV
b) Clearly, for every x ∈ X (É), one has
h (L1 ⊗L 2 ) (x) = ̂deg ( x ∗ (L 1 ⊗L 2 ) ) = ̂deg ( (x ∗ L 1 )⊗(x ∗ L 2 ) )
= ̂deg (x ∗ L 1 ) + ̂deg (x ∗ L 2 ) = h L1
(x) + h L2
(x) .
c) There is some n ∈Æsuch that L ⊗n is very ample. Part b) shows that
it suffices to verify the assertion for L ⊗n . Thus, we may assume that L is
very ample.
Let i : X ֒→ P n be the closed embedding induced by L . Then, L = i ∗ O(1).
It follows from Tietze’s Theorem that there is a hermitian metric on O(1) such
that L = i ∗ O(1). Then, h L
(x) = h O(1)
(i(x)). It, therefore, suffices to show
fundamental finiteness for the height function h O(1)
on P n (É).
Part a) together with Observation 3.11 shows that h O(1)
differs from h naive by a
bounded summand. Fact 1.3 yields the assertion.
□
4. Basic arithmetic intersection theory
4.1. Remark. –––– It should be noticed that there is a strong formal analogy of
the concept of a height on an arithmetic variety to the concept of a degree in
algebraic geometry over a ground field. The only obvious difference is that the
role of the sections of an invertible sheaf is now played by small sections, say, of
norm less than one. Nevertheless, it seems that the height of a point is actually
some sort of arithmetic intersection number.
This is an idea which has been formalized first by S. Yu. Arakelov [Ara] for twodimensional
arithmetic varieties and later by H. Gillet and C. Soulé [G/S 90]
for arithmetic varieties of arbitrary dimension.
Let us briefly recall the most important definitions and some fundamental results
from arithmetic intersection theory.
4.2. Definition (Gillet-Soulé). —– Let X be a regular arithmetic variety
and i ∈Æ.
i) Then, an arithmetic cycle of codimension i on X is a pair (Z, g Z ) consisting a
codimensioni cycleZ ontheschemeX together withaGreen’scurrent g Z forZ.
The latter means that g Z ∈ D p−1,p−1 (X ()) is a real current on the complex
manifold X () invariant under the complex conjugation F ∞ : X () → X ()
such that
ω Z := 1
2πi ∂∂g Z + δ Z
is a smooth differential form on X (). Here, δ Z is the current given by
integration along Z() ⊆ X ().

Sec. 4] BASIC ARITHMETIC INTERSECTION THEORY 161
The abelian group of all codimension i arithmetic cycles on X is denoted
by Ẑi (X ).
ii) A codimension i arithmetic cycle on X is said to be linearly equivalent to
zero if it belongs to the subgroup of Ẑi (X ) generated by all arithmetic cycles
of types
(0, ∂u)
for u ∈ D p−2,p−1 (X ()),
(0, ∂v)
for v ∈ D p−1,p−2 (X ()), and
(div f , − log | f | 2 )
for f a non-zero rational function on an integral subscheme of X of codimension
i − 1.
The abelian group of all codimension i arithmetic cycles on X which are
linearly equivalent to zero is denoted by ̂R i (X ).
iii) The i-th arithmetic Chow group of X is defined by
ĈH i (X ) := Ẑi (X )/̂R i (X ) .
=
4.3. Remark. –––– − log | f | 2 is indeed a Green’s current for div f . This fact is
classically known as the Poincaré-Lelong equation [G/H, p. 388].
4.4. Example. –––– For every regular arithmetic variety X , we have
ĈH 0 (X ) = CH 0 (X )
since, on X (), there are no currents of type (−1, −1).
4.5. Example-Definition. –––– The arithmetic degree
is an isomorphism.
̂deg : ĈH1 (Spec) −→Ê
[ (∑
i
a i (p i ), r )] ↦→ 1 2 r + ∑a i log p i
i
Indeed, the homomorphism ̂deg is well-defined since (div n, − log |n| 2 ) is
mapped to zero. It is injective and surjective as one has CH 1 (Spec) = 0
for the usual Chow group and [D 0,0 (pt)] − =Ê.

162 THE CONCEPT OF A HEIGHT [Chap. IV
4.6. Example. –––– Obviously,
for i ≥ 2.
ĈH i (Spec) = CH i (Spec) = 0
4.7. Definition. –––– Let X be a regular arithmetic variety and L ∈ ̂Pic(X )
be a hermitian line bundle. Then, the first arithmetic Chernclass of L is given by
for a non-zero section s of L .
ĉ 1 (L ) := [(div s, − log ‖s‖ 2 )] ∈ ĈH1 (X )
4.8. Lemma. –––– For every regular arithmetic variety X , the first arithmetic
Chern class provides an isomorphism
ĉ 1 : ̂Pic(X ) −→ ĈH1 (X ) .
Proof. It follows form the Poincaré-Lelong equation that the homomorphism
ĉ 1 is well-defined.
To show injectivity, assume [(div s, − log ‖s‖ 2 )] = 0 ∈ ĈH1 (X ) for a non-zero
rational section s of a hermitian line bundle L . Since there are no (0, 0)-
currents of types ∂u or ∂v, this means (div s, − log ‖s‖ 2 ) = (div f , − log ‖ f ‖ 2 )
for a rational function f .
Hence, s/f is a global section of L without zeroes or poles and of constant
norm one. This implies that L represents the neutral element of ̂Pic(X ).
For surjectivity, let any arithmetic 1-cycle (Z, g Z ) be given. The invertible
sheaf O(Z) may be equipped with a smooth hermitian metric. This reduces the
question to the case Z = 0.
This means, we consider an arithmetic 1-cycle (0, g). I.e., g is a (0, 0)-current
such that ω := ∂∂g is a smooth (1, 1)-form. We claim this implies already that
g is smooth.
Indeed, the ∂∂-lemma [G/H, p. 149] yields a smooth function g ′ such
that ω = ∂∂g ′ . Hence, ∂∂(g ′ − g) = 0. This means that the current g ′ − g
vanishes on all test forms of type ∂∂η. By the ∂∂-lemma, these are all d-closed
forms of top degree. By consequence of Poincaré duality for de Rham cohomology,
g ′ − g is the current associated with a constant function.
For the arithmetic 1-cycle (0, g), we therefore have (0, g) = ĉ 1 (O X ) where O X
is the structure sheaf equipped with the smooth hermitian metric given
by ‖ f ‖ := e − g 2 | f |.
□
4.9. Remark. –––– For a hermitian line bundle L ∈ ̂Pic(Spec), its Arakelov
degree as defined in the previous section is the same as ̂deg ( ĉ 1 (L ) ) .

)⊗É
Sec. 4] BASIC ARITHMETIC INTERSECTION THEORY 163
4.10. –––– H. Gillet and C. Soulé [G/S 90, Theorem 4.2.3] define an intersection
product
·: ĈHi (X ) × ĈHj (X ) −→ ĈHi+j (X
with the following properties.
i) “·” is commutative and associative.
ii) If Y and Z are irreducible subvarieties meeting properly, i.e.,
codim(Y ∩ Z) ≥ codim Y + codim Z ,
then
[(Y, g Y )] · [(Z, g Z )] = [Y ·Z, g Y ∗ g Z ] .
Here, Y ·Z is the usual intersection cycle defined by the Tor-formula [Fu84,
Section 20.4]. The ∗-product of two Green’s currents is morally given by
the formula
g Y ∗ g Z := g Y ·δ Z + ω Y ·g Z .
This definition is obviously problematic as g Y ·δ Z is formally a product of
two currents. It may be justified in the expected manner when g Y is a Green’s
form of log type along Y [G/S 90, 2.1.3]. The latter is always fulfilled if Y is a
cycle of codimension one.
4.11. –––– For a morphism f : X → Y of regular arithmetic varieties,
H. Gillet and C. Soulé [G/S 90, 4.4.3] define a pull-back homomorphism
It has the properties below.
f ∗ : ĈHi (Y ) −→ ĈHi (X ) .
i) For a smooth hermitian line bundle L ∈ ̂Pic(Y ), one has
ii) For y ∈ ĈHi (Y ) and z ∈ ĈHj (Y ),
f ∗ ĉ 1 (L ) = ĉ 1 ( f ∗ L ) .
f ∗ (y·z) = f ∗ y · f ∗ z .
iii) For two morphisms f : X → Y and g : Y → Z of regular arithmetic
varieties, the equality
is true.
(g ◦ f ) ∗ = f ∗ ◦ g ∗ : ĈHi (Z ) −→ ĈHi (X )

164 THE CONCEPT OF A HEIGHT [Chap. IV
4.12. –––– For a morphism f : X → Y of regular arithmetic varieties such
that fÉ: XÉ→YÉis smooth, there is a push-forward homomorphism
with the following properties.
i) For [(Z, g Z )] ∈ ĈHi (X ), one has
f ∗ : ĈHi (X ) −→ ĈHi−dimX+dim Y (Y )
f ∗ [(Z, g Z )] = [( f ∗ Z, f ∗ g Z )] .
Here, f ∗ Z is the usual push-forward cycle [Fu84, Section 1.4] and f ∗ g Z is the
push-forward of g Z as a current.
ii) There is the projection formula
for x ∈ ĈHi (X ) and y ∈ ĈHj (Y ).
f ∗ (x · f ∗ y) = f ∗ x · y
iii) For two morphisms f : X → Y and g : Y → Z of regular arithmetic
varieties such that fÉand gÉare smooth, one has that
(g ◦ f ) ∗ = g ∗ ◦ f ∗ : ĈHi (X ) −→ ĈHi−dimX+dimZ (Z ) .
4.13. Remarks. –––– It requires rather hard work to establish all the results
described in 4.10 through 4.12. There are two different sorts of problems.
i) On the scheme-theoretic side, it is already a problem to define the intersection
of two cycles a ∈ CH i (X ) and b ∈ CH j (X ) in the ordinary Chow groups.
The point is that on a scheme of finite type over, there is no moving lemma
available except for divisors. Thus, in order to define the intersection product
of two cycles, a highly indirect approach must be chosen.
H. Gillet and C. Soulé use the fact that the Brown-Gersten-Quillen spectral
sequence degenerates after tensoring withÉ. The Chow groups of X appear
as part of the E 2 -terms while the spectral sequence converges versus the algebraic
K-groups K ∗ (X ). The product structure on K ∗ (X ) provided by the
tensor product of vector bundles then induces the desired intersection product
·: CH i (X ) × CH j (X ) −→ CH i+j (X )⊗É.
Many more details are given in [S/A/B/K, Chapter I].
Today, there is a second approach to this intersection product which is based
on the resolution of singularities by alterations found by A. J. de Jong [dJo].
This method is rather indirect, too. It yields the same intersection product as
the K-theory approach.

Sec. 5] AN INTERSECTION PRODUCT ON SINGULAR ARITHMETIC VARIETIES 165
One would prefer to have an intersection product
·: CH i (X ) × CH j (X ) −→ CH i+j (X ) .
In other words, one would like to get rid of the tensor product withÉ.
For schemes over, there is no such intersection product with reasonable
properties known up to now.
ii) On the other hand, it requires enormous technical efforts to justify the
definitions made on the complex-analytic side. H. Gillet and C. Soulé show that
for every arithmetic cycle there is an equivalent one which is of the form (Y, g Y )
for g Y a Green’s form of log type along Y. This is necessary to give the ∗-product
of Green’s currents a precise meaning.
Still, a lot of work has to be done in order to verify all the properties of
the intersection product which are asserted. Here, the hardest point is to
prove associativity.
4.14. –––– Over the years, a lot of progress has been made on this part of
the theory. Several people tried to define the arithmetic intersection product in
a more canonical manner. It turned out that the Gillet-Soulé intersection theory
may be understood as a particular case of a more general framework. From such
a more general point of view, the ∗-product of two Green’s currents appears as
being induced by the cup product on Deligne-Beilinson cohomology. Such an
approach therefore reduces commutativity and associativity of the intersection
product into formalities.
The clearest and most thorough framework for arithmetic intersection theory
on regular arithmetic varieties seen up to now has been provided by J. I. Burgos
Gil, J. Kramer, and U. Kühn [B/K/K].
4.15. Remark. –––– One of the main drawbacks of the arithmetic intersection
theory developed so far is that the push-forward is defined only for morphisms
which are smooth on the generic fiber. Replacing in Definition 4.2 the requirement
that ω Z has to be smooth by a slightly weaker condition, one obtains
a theory which is free of this drawback. Such theories are due to J. I. Burgos
Gil [Bu] and A. Moriwaki [Mori]. The reader may find a description
in [B/K/K, Theorem 6.35].
5. An intersection product on singular arithmetic varieties
5.1. –––– For arbitrary, i.e., possibly singular, arithmetic varieties, arithmetic
intersection theory is considerably less far developed. The following theorem is
a consequence from the theory presented so far which is valid in more generality.

166 THE CONCEPT OF A HEIGHT [Chap. IV
5.2. Theorem. –––– a) For every arithmetic variety π : X → Specsuch that its
generic fiber XÉis smooth of dimension d, there is a unique-multilinear map
pGS X : ̂Pic(X ) × . . . × ̂Pic(X ) −→Ê
} {{ }
(d+1) times
satisfying the following conditions.
i) If X is regular then
p X GS (L 1, . . . , L d+1 ) = ̂deg ( π ∗
(ĉ1 (L 1 ) · . . . · ĉ 1 (L d+1 ) ))
for every L 1 , . . . , L d+1 ∈ ̂Pic(X ).
ii) If f : X → Y is a generically finite morphism of arithmetic varieties such that
XÉand YÉare smooth then
p X GS ( f ∗ L 1 , . . . , f ∗ L d+1 ) = deg f · p Y GS (L 1, . . . , L d+1 )
for every L 1 , . . . , L d+1 ∈ ̂Pic(Y ).
b) pGS X is symmetric.
Proof. a) Let π : X → Specbe any arithmetic variety such that its generic
fiber XÉis smooth of dimension d.
Uniqueness: Let r : X ′ → X be a generically finite morphism of arithmetic
varieties such that X ′ is regular. The existence of such a morphism is provided
by the main result of [dJo]. Then, conditions i) and ii) imply that, necessarily,
p X GS (L 1, . . . , L d+1 ) := 1
degr ̂deg ( (π◦r) ∗
(ĉ1 (r ∗ L 1 ) · . . . · ĉ 1 (r ∗ L d+1 ) )) .
Existence: We take formula (∗) as the definition for p GS .
One has to show that the definition does not depend on the choice of r.
For this, suppose there are given two generically finite morphisms r ′ : X ′ → X
and r ′′ : X ′′ → X such that X ′ and X ′′ are regular.
The fiber product X ′ × X X ′′ is generically finite over X . Again by virtue
of [dJo], there is a generically finite morphism X ′′′ → X ′ × X X ′′ such
that X ′′′ is regular. Thus, we are reduced to the case that r ′′ = r ′ ◦ g
for g : X ′′ → X ′ a morphism of regular schemes.
Then, r ′′∗ L i = g ∗ r ′∗ L i for every i. We may now apply the results from
the previous section as g is a morphism of regular arithmetic varieties.
Thus, ĉ 1 (r ′′∗ L i ) = g ∗( ĉ 1 (r ′∗ L i ) ) and
ĉ 1 (r ′′∗ L 1 ) · . . . · ĉ 1 (r ′′∗ L d+1 ) = g ∗( ĉ 1 (r ′∗ L 1 ) · . . . · ĉ 1 (r ′∗ L d+1 ) ) .
(∗)

Sec. 5] AN INTERSECTION PRODUCT ON SINGULAR ARITHMETIC VARIETIES 167
The projection formula implies
(π◦r ′′ ) ∗
(
g
∗ ( ĉ 1 (r ′∗ L 1 ) · . . . · ĉ 1 (r ′∗ L d+1 ) ))
= (π◦r ′ ◦g) ∗
(
g
∗ ( ĉ 1 (r ′∗ L 1 ) · . . . · ĉ 1 (r ′∗ L d+1 ) ))
= (π◦r ′ ) ∗ g ∗
(
g
∗ ( ĉ 1 (r ′∗ L 1 ) · . . . · ĉ 1 (r ′∗ L d+1 ) ))
= (π◦r ′ ) ∗
[
(g∗ 1) · (ĉ
1 (r ′∗ L 1 ) · . . . · ĉ 1 (r ′∗ L d+1 ) )] .
This yields the claim as g ∗ 1 = deg g.
p X GS is-multilinear by definition.
It fulfills condition i) as one may work with r = id in the case that X is regular.
ii) Choose a generically finite morphism r : X ′ → X such that X ′ is regular.
The definitions of p X GS ( f ∗ L 1 , . . . , f ∗ L d+1 ) and p Y GS (L 1, . . . , L d+1 ) using r
and f ◦r, respectively, differ only by the degree divided out.
b) This follows from formula (∗) together with the commutativity and associativity
of the intersection product of H. Gillet and C. Soulé [cf. 4.10.i)]. □
5.3. Definition. –––– Let X be an arithmetic variety and L ∈ ̂Pic(X ) be a
hermitian line bundle.
Then, for every cycle Z ∈ Z(XÉ) of dimension i, one defines its height by
h BGS
L
(Z) := pZ GS(L | Z
, . . . , L | } {{ Z
) . }
(i+1) times
Here, Z is the Zariski closure of Z in X equipped with its induced reduced structure.
5.4. Remark. –––– This definition for the height of a cycle of arbitrary
dimension is, in slightly different form, due to J.-B. Bost, H. Gillet,
and C. Soulé [B/G/S, Section 3.1.1]. It fulfills a positivity condition and a
very strong fundamental finiteness statement.
5.5. Theorem. –––– Let i ∈Æ, X be an arithmetic variety, and L ∈ ̂Pic(X ) be
a hermitian line bundle.
a) If the curvature form c 1 (L ) is positive and some positive power L ⊗n is generated
by global sections of norm less than one then
for every effective cycle Z on XÉ.
h BGS
L (Z) > 0

168 THE CONCEPT OF A HEIGHT [Chap. IV
b) If L is ample on X then for every B ∈Êthere are only finitely many effective
cycles Z ∈ Z i (XÉ) such that deg L
(Z) < B and
h BGS
L (Z) < B .
Proof. These are [B/G/S, Proposition 3.2.4 and Theorem 3.2.5].
□
5.6. Remark. –––– There are rather few computational results for the Bost-
Gillet-Soulé height h BGS
L for cycles of positive dimension.
One of them is the height of projective space itself. In [B/G/S, Section 3.3], it
is shown that
Z
h (O(1),‖ .‖FS )(P n ) = h (O(1),‖ .‖FS )(P n−1 ) − log ‖x 0 ‖ FS ωFS n .
As the latter integral may easily be computed to 1 2
h (O(1),‖ .‖FS )(P n ) = 1 2
n
∑
k=1
k
∑
m=1
P n ()
n
1
∑
m
m=1
1
m = n + 1
2
n
∑
m=1
, this leads to
1
m − n 2 .
5.7. Definition. –––– Let X be an arithmetic variety, X its generic fiber,
and L ∈ ̂Pic(X ) be a hermitian line bundle.
Then, for a closed point x ∈ X, one defines its absolute height by
1 ( )
h L
(x) :=
[L :É] hBGS L (x) .
Here, L is the field of definition of x. (x) denotes the 0-dimensional cycle on X
associated with x.
5.8. Remarks. –––– i) Theorem 5.2.a.ii) implies that this definition is independent
of the choice of L.
ii) Since x ∈ Z 1 (X ) is a one-dimensional cycle, this definition actually does not
involve the arithmetic intersection product. Only the arithmetic degree is used.
For this reason, there is an explicit formula for the dependence of the height on
the choice of the metric. Indeed,
h (L ,‖ .‖ ′ )(x) = h (L ,‖ .‖) (x) − 1
[L :É] ∑
σ : L→log ‖ . ‖′( σ(x) )
‖ . ‖ ( σ(x) ) .
In particular, there is a constant C such that
| h (L ,‖.‖ ′ )(x) − h (L ,‖.‖) (x)| < C
for every x ∈ X. This naturally generalizes Lemma 2.11 and Proposition 3.15.

Sec. 6] THE ARITHMETIC HILBERT-SAMUEL FORMULA 169
6. The arithmetic Hilbert-Samuel formula
6.1. Definition. –––– Let X be a Noetherian scheme and h: X →Êbe
any function. For i ∈Æ, we define the i-th successive minimum of h by
e i (h) :=
sup
inf
codim X Y=i x∈X \Y
h(x) .
Here, Y runs through all (possibly reducible) closed subsets of X of codimension
i. x runs through the closed points.
6.2. Remark. –––– A priori, one has that e i (h) ∈ [−∞, ∞]. Further, directly
from the definition, we see
e 1 (h) ≥ e 2 (h) ≥ . . . ≥ e dim X+1 (h) = inf
x∈X h(x) .
6.3. Lemma. –––– Let X be an arithmetic variety the generic fiber of which is X.
Let L ∈ ̂Pic(X ) and suppose that L is ample. Then,
for every i ∈Æ.
e i (h L
) > −∞
Proof. First, we note that it is sufficient to show e dim X+1 (h L
) > −∞.
For this, we may suppose that L is very ample. Then, L is generated by
global sections s 1 , . . . , s k ∈ Γ(X , L ). Further, on L , there exists a hermitian
metric ‖ . ‖ ′ such that c 1 (L , ‖ . ‖ ′ ) is positive. Scale ‖ . ‖ ′ by a positive constant
factor such that ‖s 1 ‖ ′ , . . . , ‖s k ‖ ′ < 1.
Then, by Theorem 5.5.a), h (L ,‖ .‖ ′ )(x) > 0 for every closed point x ∈ X. Remark
5.8.ii) shows that there is a constant C such that
h L
(x) > C
for every x ∈ X. Consequently, e dim X+1 (h L
) > −∞.
□
6.4. Notation. –––– Let X be an arithmetic variety, X its generic fiber,
(L , ‖ . ‖) ∈ ̂Pic(X ), and n ∈Æ.
Further, let s ∈ Γ(X, LÉ) ⊗n be any global section. We define its norm by
‖s‖ := sup ‖s‖(x) .
x∈X ()
This induces a norm on theÊ-vector space Γ(X, LÉ)⊗ÉÊ.
⊗n
On Γ(X, LÉ)⊗ÉÊ, ⊗n we consider the normalized Lebesgue measure µ n such
that the unit ball
B n := {s ∈ Γ(X, LÉ)⊗ÉÊ|‖s‖ ⊗n
≤ 1}

É)⊗ÉÊ
170 THE CONCEPT OF A HEIGHT [Chap. IV
is of measure one.
Further, we observe that
Γ(X , L ⊗n ) ⊂ Γ(X, L ⊗n
is a lattice of maximal rank.
6.5. Theorem (Arithmetic Hilbert-Samuel formula). —–
Let X be an arithmetic variety, X its generic fiber, and L ∈ ̂Pic(X ).
Putd := dim X. Assume thatX is smooth and the curvature formc 1 (L ) is positive.
Then,
− log covol µn Γ(X , L ⊗n ) ∼ pX GS (L , . . . , L ) n d+1 .
(d + 1)!
Proof. When X is regular, this result may be deduced from the arithmetic
Riemann-Roch Theorem proven by H. Gillet and C. Soulé [G/S 92].
See also [Fa92]. This, in turn, is a very deep theorem. Its proof combines the
work of several people including, besides H. Gillet and C. Soulé, D. Quillen,
J.-M. Bismut, and G. Lebeau.
A direct proof for the arithmetic Hilbert-Samuel formula which does not rely
on the arithmetic Riemann-Roch Theorem has been provided by A. Abbes
and T. Bouche [A/B]. This proof works for singular arithmetic varieties, too.
□
6.6. Corollary (Theorem of successive minima, cf. [Zh95b, (5.2)]). —–
Let X be an arithmetic variety, X its generic fiber, and L ∈ ̂Pic(X ).
Putd := dim X. Assume that X is smooth and the curvature formc 1 (L ) is positive.
Then,
e 1 (h L
) ≥ pX GS (L , . . . , L )
(d + 1) deg L
X .
Proof. Write (L , ‖ . ‖) for L .
When one multiplies ‖ . ‖ by a constant factor e −C , the summand C is added
to the height function h L
. Therefore, the term e 1 (h L
) on the left hand side is
changed by C, too. On the other hand, from Lemma 6.9, we see that the right
hand side is changed by the same summand.
Consequently, it will suffice to show that e 1 (h L
) ≥ 0 under the assumption
p X GS (L , . . . , L ) > 0 .
For this, we first observe that the usual Hilbert-Samuel formula yields a constant
D such that
dim Γ(X, L ⊗n É)⊗ÉÊ

Sec. 6] THE ARITHMETIC HILBERT-SAMUEL FORMULA 171
Hence, by the arithmetic Hilbert-Samuel formula,
⊗n dim Γ(X,L
log [2
É)⊗ÉÊ· covol µn Γ(X , L ⊗n )]
= (log 2) dim Γ(X, LÉ)⊗ÉÊ+log ⊗n covol µn Γ(X , L ⊗n )
< D(log 2)n d − 1 2
< 0
p X GS (L , . . . , L )
(d + 1)!
n d+1
for n ≫ 0. Exponentiating, we see
⊗n dim Γ(X,L
2
É)⊗ÉÊ· covol µn Γ(X , L ⊗n ) < 1 = vol B n .
This shows that we may apply Minkowski’s lattice point theorem [B/S,
кII, §4]. The unit ball B n contains a lattice point which is different from
the origin.
In other words, there is a non-zero section s ∈ Γ(X , L ⊗n ) such that ‖s‖ < 1.
We now claim that h L
(x) ≥ 0 for every x ∈ X \div s| X . This clearly suffices for
the asserted inequality.
To verify the claim, let L be the field of definition of the point x and x its
Zariski closure in X . Then,
h L
(x) = 1 ( ) 1
[L :É] hBGS L (x) =
[L :É] px GS (L | x) .
The subscheme x might be singular but there is a morphism i : Spec O L → x of
schemes which is generically one-to-one. Theorem 5.2 shows
p x GS(L | x ) = p SpecO L
GS
(i ∗ L ) = ̂deg π ∗ ĉ 1 (i ∗ L )
for π : Spec O L → Specthe structural morphism.
By our assumption, i # s ∈ Γ(Spec O L , i ∗ L ) is a non-zero section. Thus,
ĉ 1 (i ∗ L ) = [(div i # s, − log ‖i # s‖ 2 )] .
Here, div i # s ∈ Z 0 (Spec O L ) is an effective cycle. − log ‖i # s‖ 2 is a function taking
only positive values on the 0-dimensional manifold Hom fields (L,). Hence,
π ∗ ĉ 1 (i ∗ L ) = [(π ∗ div i # s, π ∗ (− log ‖i # s‖ 2 ))] =: [(z, r)]
for an effective cycle z ∈ Z 0 (Spec) and r > 0.
It follows directly from Example-Definition 4.5 that ̂deg [(z, r)] > 0.
□

172 THE CONCEPT OF A HEIGHT [Chap. IV
6.7. Remark. –––– Using the same techniques, S. Zhang [Zh95b, (5.2)] also
shows the inequality
p X GS (L , . . . , L )
deg L
X
6.8. Remark. –––– By definition,
≥ e 1 (h L
) + . . . + e d+1 (h L
) .
p X GS (L , . . . , L ) = hBGS (X ) .
The theorem of successive minima therefore provides a comparison of the height
of the variety X itself with the heights of the points on it.
In order to simplify formulas, S. Zhang [Zh95b] normalizes the height of a
closed subvariety Z ⊆ X to be
h L
(Z) :=
h BGS
L (Z)
(d + 1) deg L
X .
This normalization has to be used with some care. The problem is that the
absolute height of a closed point is not the same as the normalized height of the
corresponding 0-cycle.
6.9. Lemma. –––– Let X be an arithmetic variety, X its generic fiber, and
(L , ‖ . ‖) ∈ ̂Pic(X ). Write d := dim X. Then, for every C > 0,
( ) ( )
(L , C ·‖ . ‖), . . . , (L , C ·‖ . ‖) = p
X
GS (L , ‖ . ‖), . . . , (L , ‖ . ‖)
p X GS
− (d + 1)(log C ) deg L
X .
Proof. By [dJo], we may choose a regular arithmetic variety X ′ and a generically
finite morphism f : X ′ → X which surjects to X . Theorem 5.2.ii) implies
that the assertion for X and L is equivalent to the assertion for X ′ and f ∗ L .
We are therefore reduced to the case that X is regular.
Then, by Theorem 5.2.i),
( )
(L , C ·‖ . ‖), . . . , (L , C ·‖ . ‖)
p X GS
= ̂deg π ∗
(ĉ1 (L , C ·‖ . ‖) · . . . · ĉ 1 (L , C ·‖ . ‖) )
= ̂deg π ∗
(
(ĉ1 (L , ‖ . ‖)+[(0, −2 logC)]) · . . . · (ĉ 1 (L , ‖ . ‖)+[(0, −2 logC)]) ) .
Here, π : X → Specdenotes the structural morphism.
Next, we apply the binomial theorem. The description of the intersection
product given in 4.10 immediately shows
[(0, −2 logC)]·[(Z, g Z )] = [(0, −2(logC )δ Z )]

Sec. 7] THE ADELIC PICARD GROUP 173
for (Z, g Z ) an arbitrary arithmetic cycle. In particular,
[(0, −2 logC )]·[(0, −2 logC)] = 0 .
By consequence, we find
( )
(L , C ·‖ . ‖), . . . , (L , C ·‖ . ‖)
p X GS
( )
= pGS
X (L , ‖ . ‖), . . . , (L , ‖ . ‖)
+ (d + 1) ̂deg π ∗
(
[(0, −2 logC )] · ĉ1 (L , ‖ . ‖) · . . . · ĉ 1 (L , ‖ . ‖) ) .
Here, ĉ 1 (L , ‖ . ‖) · . . . · ĉ 1 (L , ‖ . ‖) = [(Z, g Z )] for Z a cycle of dimension one.
The restriction of Z to the generic fiber is a 0-cycle of degree deg L
X. Therefore,
π ∗
(
[(0, −2 logC )] · [(Z, gZ )] ) = π ∗ [(0, −2(logC )δ Z )]
= [(0, −2(logC ) deg L
X )] .
The claim now follows from Example-Definition 4.5.
□
7. The adelic Picard group
7.1. –––– The arithmetic intersection theory discussed so far seems to be
very general. It is, however, still not general enough. An obvious problem
is that the naive height on P n is not covered. The requirement that
(L , ‖ . ‖) = L ∈ ̂Pic(X ) includes that ‖ . ‖ has to be smooth. For the naive
height, we want to include the minimum metric. This means, we need a generalization
to continuous hermitian metrics.
This may be done in a rather ad hoc manner by taking into account certain
limits of smooth hermitian metrics. A way to formalize this as well as the
dependence of the height on the choice of a model is provided by S. Zhang’s
adelic Picard group [Zh95a].
i. The local case. Metrics induced by a model. —
7.2. –––– Let K be an algebraically closed valuation field. The cases we have
in mind are K =Ép for a prime number p and K =É∞ =.
We will denote the valuation of x ∈ K by |x|. In the case K =Ép, we assume
| . | is normalized by |p| = 1 . We also write ν(x) := − log |x|.
p
7.3. Definition. –––– Let X be a K-scheme. Then, by a metric on an invertible
sheaf L ∈ Pic(X ), we mean a system of K-norms on the K-vector spaces L (x)
for x ∈ X (K ).

174 THE CONCEPT OF A HEIGHT [Chap. IV
7.4. Remark. –––– If K =then a metric on L is the same as a (possibly
discontinuous) hermitian metric.
7.5. Definition. –––– Assume K to be non-archimedean and let O K be the ring
of integers in K. Further, let X be a K-scheme and L ∈ Pic(X ).
Then, by a model of (X, L ) we mean a triple (X , ˜L , n) consisting of a natural
number n, a flat projective scheme π : X → O K such that X K
∼ = X, and an
invertible sheaf ˜L ∈ Pic(X ) fulfilling ˜L | X
∼ = L ⊗n .
7.6. Example. –––– Assume K to be non-archimedean, let O K be the ring of
integers in K, and let X be a K-scheme equipped with an invertible sheaf L .
Then, a model (X , ˜L , n) of (X, L ) induces a metric ‖ . ‖ on L as follows.
x ∈ X (K ) has a unique extension x : Spec O K → X . Then, x ∗˜L is a projective
O K -module of rank 1. Each l ∈ L (x) induces l ⊗n ∈ L ⊗n (x) and, therefore, a
rational section of x ∗˜L . Put
‖l‖(x) :=
[
inf
{
|a| | a ∈ K, l ∈ a · x ∗˜L }] 1
n
. (†)
7.7. Definition. –––– The metric ‖ . ‖ given by (†) is called the metric on L
induced by the model (X , ˜L , n).
7.8. Remark. –––– Note here that O K is, in general, a non-discrete valuation
ring. In particular, O K will usually be non-Noetherian.
Nevertheless, projectivity includes being of finite type [EGA, Chapitre II, Définition
(5.5.2)]. This means, for the description of X , only a finite number of
elements from O K are needed.
In the particular case K =Ép, the group ν(K ) is isomorphic to (É, +).
Thus, for any finite set {a 1 , . . . , a s } ⊂ , there exists a discrete valuation ring
OÉp
O ⊆ containing a OÉp
1, . . . , a s .
By consequence, X is the base change of some scheme which is projective over
a discrete valuation ring.
7.9. Definition. –––– Let K be an algebraically closed valuation field. Assume
K to be non-archimedean.
Then, a metric ‖ . ‖ on L ∈ Pic(X ) is called continuous, respectively bounded,
if ‖ . ‖ = f · ‖ . ‖ ′ for ‖ . ‖ ′ a metric induced by some model and f a function
on X (K ) which is continuous or bounded, respectively.
7.10. Remark. –––– If K =then we adopt the concepts of bounded, continuous,
and smooth metrics in their the usual meaning from complex geometry.
Note that smooth metrics are continuous and that continuous metrics are automatically
bounded in the case K =.

Sec. 7] THE ADELIC PICARD GROUP 175
ii. The global case. Adelicly metrized invertible sheaves. —
7.11. Definition. –––– Let X be a projective variety overÉand m ∈Æ.
Then, by a model of X over Spec[ 1 ], we mean a scheme X which is projective
m
and flat over Spec[ 1 ] such that the generic fiber of X is isomorphic to X.
m
7.12. Definition. –––– Let X be a projective variety overÉand L ∈ Pic(X )
be an invertible sheaf.
a) Then, an adelic metric on L is a system
‖ . ‖ = {‖ . ‖ ν } ν∈Val(É)
of continuous and bounded metrics on ∈ ) such that
LÉν
Pic(XÉν
i) for each ν ∈ Val(É), the metric ‖ . ‖ ν is Gal(Éν/Éν)-invariant,
ii) for some m ∈Æ, there exist a model X of X over Spec[ 1 ], an invertible
m
sheaf ˜L ∈ Pic(X ), and a natural number n such that
˜L | X
∼ = L
⊗n
and, for all prime numbers p ∤m, the metric ‖ . ‖ νp is induced by (X p , ˜L | Xp , n).
b) Aninvertiblesheafequipped withan adelic metriciscalled an adeliclymetrized
invertible sheaf.
All adelicly metrized invertible sheaves on X form an abelian group which will
be denoted by Pic ad (X ).
7.13. Notation. –––– Let X be a model of X over Spec. Then, taking the
induced metric yields two natural homomorphisms
i X : ̂Pic(X ) → Pic ad (X ) ,
a X : ker(Pic(X ) → Pic(X )) ⊗É→Pic ad (X ) .
Further, one has the forgetful homomorphism
v : Pic ad (X ) → Pic(X ) .
7.14. Notation. –––– The models of X together with all birational morphisms
between them form an inverse system of schemes. This is a filtered inverse
system since, for two models X and X ′ , the closure of the diagonal
∆ ⊂ X × SpecÉX ⊂ X × SpecX ′ projects to both of them.
Thus, the arithmetic Picard groups ̂Pic(X ) for all models X of X form a
filtered direct system. The injections ̂Pic(X ) ֒→ Pic ad (X ) fit together to yield
an injection
ι X : lim −→ ̂Pic(X ) ֒→ Pic ad (X ) .

176 THE CONCEPT OF A HEIGHT [Chap. IV
Similarly, the usual Picard groups Pic(X ) form a filtered direct system, too.
We get a homomorphism
α X : lim −→
ker(Pic(X ) → Pic(X )) ⊗É−→ Pic ad (X ) .
7.15. Definition (Metric on v −1 (L ) ⊆ Pic ad (X )). —–
Let X be a projective variety overÉ. On X, let (L , ‖ . ‖) and (L , ‖ . ‖ ′ ) be two
adelicly metrized invertible sheaves with the same underlying sheaf.
Then, the distance between (L , ‖ . ‖) and (L , ‖ . ‖ ′ ) is given by
δ ( (L , ‖ . ‖), (L , ‖ . ‖ ′ ) ) ( )
:= ∑ δ ν ‖ . ‖ν , ‖ . ‖ ′ ν
ν∈Val(É)
for
( δ ν ‖ . ‖ν , ‖ . ‖ ν) ′ := sup
∣ log ‖ . ‖′ ν(x)
‖ . ‖ ν (x) ∣ .
x∈X (Éν )
7.16. Lemma. –––– δ is a metric on the set v −1 (L ) of all metrizations of L .
Proof. We have to show that the sum is always finite.
For this, we note first that the metrics ‖ . ‖ ν and ‖ . ‖ ′ ν are bounded by definition.
Therefore, each summand is finite.
We may thus ignore a finite set S of primes and assume that ‖ . ‖ and ‖ . ‖ ′
are given by triples (X , ˜L , n) and (X ′ , ˜L ′ , n ′ ), respectively, in the sense of
Definition 7.12.a.ii).
=
The isomorphism XÉ∼
−→ X ∼ =
−→ X ′Émay be extended to an open neighbourhood
of the generic fiber. Therefore, enlarging S if necessary, we have
an isomorphism X ′ ∼ =
−→ X of schemes over Spec\S. Further, the
triple (X , ˜L , n) may be replaced by (X , ˜L ⊗n′ , nn ′ ) without any change of
the induced metric. Thus, without restriction, n = n ′ .
To summarize, we are reduced to the case that
XÉ‖ . ‖ and ‖ . ‖ ′ are given
by (X , ˜L , n) and (X , ˜L ′ , n). We have an isomorphism
= ˜L | XÉ∼
−→ L ⊗n ∼ =
−→ ˜L ′ |
which may be extended to an open neighbourhood of the generic fiber. Therefore,
in the definition of δ ( (L , ‖ . ‖), (L,‖.‖ ′ ) ) , all the summands vanish,
except finitely many.
Positivity, symmetry, and the triangle inequality are clear.
7.17. Remark. –––– It is convenient to consider two adelicly metrized invertible
sheaves with different underlying sheaves as of distance infinity. Then, the
distance δ is no longer a metric but only a separated écart in the sense of N. Bourbaki
[Bou-T, §1].
□

Sec. 7] THE ADELIC PICARD GROUP 177
7.18. Lemma. –––– Let f : X → Y be a morphism of projective varieties overÉ.
i) Then, the homomorphism f ∗ : Pic ad (Y ) → Pic ad (X ) is continuous with respect
to the metric topology.
ii) Even more,
δ( f ∗ L 1 , f ∗ L 2 ) ≤ δ(L 1 , L 2 )
for arbitrary adelicly metrized invertible sheaves L 1 , L 2 ∈ Pic ad (Y ).
Proof. ii) is obvious. i) follows immediately from ii).
□
iii. The adelic Picard group. —
7.19. Definition. –––– Let X be a regular, projective variety overÉ.
a) We call an adelicly metrized invertible sheaf (L , ‖ . ‖) on X semipositive if
there exist a model X of X over Specand a smooth hermitian line bundle
(L , ‖ . ‖ ∞ ) ∈ ̂Pic(X ) fulfilling
i) (L , ‖ . ‖) = i X (L , ‖ . ‖ ∞ ),
ii) L | Xp
is nef for every prime number p, and
iii) the curvature form c 1 (L, ‖ . ‖ ∞ ) is non-negative.
b) By the semipositive cone C ≥0 ⊂ X
Picad (X ), we mean the set of all adelicly
metrized invertible sheaves which may be written as positiveÉ-linear combinations
of semipositive elements.
7.20. Definition. –––– Let X be a regular, projective variety overÉ.
We define the adelic Picard group ˜Pic(X ) ⊆ Pic ad (X ) to be the subgroup generated
by C ≥0
X . Here, overlining indicates the topological closure in Picad (X ) with
respect to the metric topology.
If (L , ‖ . ‖) ∈ ˜Pic(X ) then ‖ . ‖ is called an integrable metric on L . (L , ‖ . ‖) is
then said to be an integrably metrized invertible sheaf.
7.21. Remark. –––– Since C ≥0
X
is a surjection.
diff : C ≥0
X
is closed under addition, the difference map
× C ≥0
X −→ ˜Pic(X )
7.22. Definition. –––– Let X be a regular, projective variety overÉ.
We define the integrable topology on ˜Pic(X ) to be the finest topology such that
the difference map diff is continuous.

178 THE CONCEPT OF A HEIGHT [Chap. IV
7.23. Remarks. –––– i) The integrable topology on ˜Pic(X ) is finer than the
metric topology.
ii) We will consider ˜Pic(X ) as being equipped with the integrable topology.
Not with the metric topology which is the same as the topology as a subspace
of Pic ad (X ).
iii) A map i : ˜Pic(X ) → M to any topological space is continuous if and only if
i ◦ diff : C ≥0
X
× C ≥0
X
−→ M
is continuous with respect to the metric topology.
7.24. Lemma. –––– Let X be a regular, projective variety overÉ. Then, the
difference map
diff : C ≥0 × X
C ≥0 −→ ˜Pic(X X
)
is open.
Proof. Let U ⊆ C ≥0 × X
C ≥0
X
be any open set. We have to show that
diff −1 (diff(U )) is open. But this is clear since
diff −1 (diff(U )) = [ [ ] [ ]
U + (x, x) ∩ C
≥0 ≥0
X
×CX
x∈Pic ad (X )
and addition is continuous with respect to the metric topology.
7.25. Lemma. –––– Let X be a regular, projective variety overÉ. Then, the
topological closures of C ≥0
X
with respect to the metric topology and the integrable
topology coincide.
Proof. Let C ≥0
≥0
X
be the closure of CX
with respect to the metric topology and
C be the closure with respect to the integrable topology. Since the integrable
topology is finer, we have C ⊆ C ≥0
X .
On the other hand, let l ∈ C ≥0
X . Then, there is a sequence (l i) i in C ≥0
X
which
converges to l. As the canonical map diff(·, 0): C ≥0 → ˜Pic(X X
) is continuous,
we see that l i → l with respect to the integrable topology. Hence, l ∈ C. □
7.26. Definition. –––– Let X be a smooth, projective variety overÉ
and L ∈ Pic(X ).
a) Let p be a prime number.
Then, a metric ‖ . ‖ on will be called semipositive if there is a sequence
LÉp
(‖ . ‖ i ) i≥0 of metrics on L satisfying the following conditions.
i) (‖ . ‖ i ) i≥0 converges uniformly to ‖ . ‖.
ii) Each ‖ . ‖ i is induced by a triple (X × Specp, Spec(p)
pr ∗ 1˜L , n) where X is a
scheme over(p) :=p ∩É, ˜L ∈ Pic(X ), and ˜L | Xp
is nef.
□

Sec. 7] THE ADELIC PICARD GROUP 179
b) We will call a metric ‖ . ‖ on Lpositive if ‖ . ‖ is smooth and the curvature
form c 1 (L , ‖ . ‖) is positive.
‖ . ‖ is said to be almost semiample if there is a sequence of positive metrics
(‖ . ‖ i ) i≥0 on Lthat converges uniformly to ‖ . ‖.
7.27. Lemma. –––– Let X be a regular, projective variety overÉ. Then, one has
C ≥0
X = {(L , {‖ . ‖ ν} ν∈Val(É)) ∈ Pic ad (X ) |
‖ . ‖ ν semipositive for ν non-archimedean,
Proof. We consider C ≥0
X
“⊆” is clear from Definitions 7.19 and 7.26.
‖ . ‖ ν almost semiample for ν archimedean}.
as the closure with respect to the metric topology.
“⊇” Let (L , {‖ . ‖ ν } ν ) ∈ Pic ad (X ) be such that ‖ . ‖ νp is semipositive for every
prime number p and ‖ . ‖ ν∞ is almost semiample. The assertion says that, for
every ε > 0, there exists (L , {‖ . ‖ ′ ν} ν ) ∈ C ≥0
X
such that
δ ( (L , {‖ . ‖ ν } ν ), (L , {‖ . ‖ ′ ν} ν
L
) ) < ε .
The archimedean valuation. The hermitian metric ‖ . ‖ ν∞ is almost semiample.
This means that, for every ε > 0, there is a hermitian metric ‖ . ‖ ′ ν ∞
on
which is smooth and positively curved such that
δ ν∞
( ‖ . ‖ν∞ , ‖ . ‖ ′ ν ∞
) < ε .
Good primes. By Definition 7.12, we have a finite set S of bad prime numbers,
a model X of X over Spec\S, an invertible sheaf ˜L ∈ Pic(X ), and a natural
number n such that
i) ˜L | X
∼ = L ⊗n and
ii) ‖ . ‖ νp is induced by (X ν , ˜L | Xν , n) for all prime numbers p ∉ S.
Bad primes. Let p ∈ S. According to Definition 7.26.a), for every ε > 0, there
exists an induced metric ‖ . ‖ ′ ν p
such that
δ νp
( ‖ . ‖νp , ‖ . ‖ ′ ν p
) < ε .
More precisely, ‖ . ‖ ′ ν p
is induced by a triple
(X p × Spec(p) Specp, pr ∗ 1˜L (p) , n p )
such that X p is a model of over(p), ˜L (p) ∈ Pic(M XÉ(p) p ), and ˜L (p) | Xp
is nef.

180 THE CONCEPT OF A HEIGHT [Chap. IV
Gluing. We claim that X and all the X p for p ∈ S may be glued together along
the generic fiber X to a scheme X ′ over Spec. Further, for
N := n ∏ n p ,
p∈S
the tensor powers ˜L ⊗N/n and [˜L (p) ] ⊗N/n p
fit together to give an invertible
sheaf ˜L ′ on X .
Then, {‖ . ‖ ′ ν} ν is defined by
(L , {‖ . ‖ ′ ν} ν ) := 1 N i X (˜L ′ , ‖ . ‖ ′ ν ∞
) .
This guarantees that ‖ . ‖ ′ ν p
and ‖ . ‖ νp coincide for all p ∉ S. In particular,
˜L ′ | Xp
is automatically nef for these primes. For the bad primes p ∈ S, the
same property follows immediately from the construction.
Gluing may, however, not be done directly as X ⊂ X is not an open set.
Instead, we may extend each X p to a scheme X (p) over Spec[ 1 ] for some
m
m ∈Æwhich is not a multiple of p. We may assume that m is divisible by all
bad primes different from p.
The isomorphism between the generic fibers of X and X (p) extends to an open
neighbourhood of X. More precisely, there exists some f p ∈not divisible by
any of the bad primes such that the induced birational map
is an isomorphism.
X (p) × SpecSpec[ 1
p f p
] −→ X × SpecSpec[ 1 f p
]
Putting f := ∏ p∈S f p and U (p) := X (p) × SpecSpec[ 1 ], we have an isomorphism
f
ϕ p : U (p) × SpecSpec[ 1 ] −→ X × p SpecSpec[ 1 ] . f
The definition
ϕ p1 ,p 2
:= ϕ −1
p 2
◦ϕ p1 : U (p1 ) × SpecSpec[ 1 p 1
] −→ U (p2 ) × SpecSpec[ 1 p 2
]
makes the gluing data complete.
The cocycle relations are fulfilled since the corresponding relations are satisfied
on the generic fiber. Observe, U (p) for p ∈ S and X are integral schemes.
By consequence, the U (p) and X may be glued together to give a-scheme X ′ .
To glue the invertible sheaves together, we need isomorphisms
ι p : ϕ ∗ p˜L ⊗N/n | U −→ [˜L (p) ] ⊗N/n p
| U(p) × SpecSpec[ 1 p ]
extending the isomorphism over X.

Sec. 7] THE ADELIC PICARD GROUP 181
Generally, there is an open neighbourhood O ⊆ U (p) × SpecSpec[ 1 p ] of X
such that ι p | O exists and is an isomorphism. Adding to f a suitable set of prime
factors guarantees that O = U (p) × SpecSpec[ 1 p ].
This completes the proof.
7.28. Remark (concerning the notion of a semipositive metric). —–
Condition ii) in Definition 7.26 seems to be somewhat unnatural. One might
want to allow arbitrary models overp or, even more generally, over the integer
ring O L of a finite extension L ofÉp.
It turns out from a standard descent argument that finite extensions actually
make no difference. Indeed, a metric is supposed to be Gal(Ép/Ép)-invariant.
Thus, it may be shown by descent that for every model over O L there is an
equivalent model overp. Similarly, for every model over the integer ring of a
finite extension ofÉ(p), there is an equivalent model over(p).
The descent needed here may actually be seen as a particular case of faithful
flat descent [K/O74] or as a slight generalization of Proposition I.3.5
to O K -schemes instead of K-schemes.
It seems, however, that allowing the completion leads to a more general notion
of semipositivity. We do not know whether Lemma 7.27 is still true for
that notion.
7.29. Lemma. –––– Let X be a regular, projective variety overÉ.
a) Then, the natural homomorphisms
and
factorize via ˜Pic(X ).
ι X : lim −→ ̂Pic(X ) → Pic ad (X )
α X : lim −→
ker(Pic(X ) → Pic(X )) ⊗É→Pic ad (X )
b) The subgroup 〈 im ι X , im α X 〉 is dense in ˜Pic(X ).
Proof. a) Let X be any model of X over Spec. We have to show that the
homomorphisms i X and a X factorize via ˜Pic(X ).
Since X is a projective scheme, every (L , ‖ . ‖) ∈ ̂Pic(X ) may be written
as (L 1 , ‖ . ‖ 1 )⊗ OX (L 2 , ‖ . ‖ 2 ) ∨ where both operands are ample invertible sheaves
with positive metrics. Further, we note that ampleness of L i implies that L i | Xp
is ample on Xp
for every p (cf. [EGA, Chapitre II, Corollaire (4.5.10.ii)]).
Hence,
i X (L , ‖ . ‖) = i X (L 1 , ‖ . ‖ 1 ) − i X (L 2 , ‖ . ‖ 2 ) ∈ diff(C ≥0
X , C ≥0
X ) .
□

182 THE CONCEPT OF A HEIGHT [Chap. IV
On the other hand, let k ∈Æand L ∈ ker(Pic(X ) → Pic(X )) be given.
We may write L = L 1 ⊗ OX L2 ∨ for two ample invertible sheaves such
that L 1 | X
∼ = L2 | X is k-divisible in Pic(X ). Then,
a X (L ⊗ 1/k) = 1 k ·i X (L 1 , ‖ . ‖) − 1 k ·i X (L 2 , ‖ . ‖) ∈ diff(C ≥0
X , C ≥0
X )
independently of the metric ‖ . ‖ chosen. Note that 1 k ·i X (L i , ‖ . ‖) ∈ Pic ad (X )
since L 1 | X is k-divisible.
b) Since the difference map diff : C ≥0
X
× C ≥0
X
surjective, it suffices to prove that 〈 im ι X , im α X 〉∩C ≥0
X
show that even 〈 im ι X , im α X 〉 ⊇ C ≥0
X .
For this, let (L , ‖ . ‖) ∈ C ≥0
X
. By definition, we have
→ ˜Pic(X ) is continuous and
≥0
is dense in C . We will
(L , ‖ . ‖) = a 1 i X (L 1 , ‖ . ‖ 1 ) + . . . + a n i X (L n , ‖ . ‖ n )
for a certain model X of X, (L i , ‖ . ‖ i ) ∈ ̂Pic(X ), and rational numbers a i ≥ 0.
We choose a common denominator k of all the a i . Then,
(L , ‖ . ‖) ⊗k = i X (L , ‖ . ‖ ′ )
for some (L , ‖ . ‖ ′ ) ∈ ̂Pic(X ). In particular, L ⊗k ∼ = L | X .
We may write L = R ⊗k ⊗ T for R ∈ Pic(X ) an arbitrary extension of L
and T ∈ ker(Pic(X ) → Pic(X )). Then,
(L , ‖ . ‖) = i X (R, ‖ . ‖ ′ 1 k ) + aX (T ⊗ 1/k) .
X
The assertion is proven.
□
iv. Pull-back and intersection product. —
7.30. Lemma. –––– Let f : X → Y be a morphism of projective varieties overÉ.
a) Then, there is a natural homomorphism
f ∗ : ˜Pic(Y ) → ˜Pic(X )
(L , ‖ . ‖) ↦→ ( f ∗ L , f ∗ ‖ . ‖) .
f ∗ admits the property that f ∗( )
CY
≥0 ⊆ C
≥0
X .
b) f ∗ is continuous.
Proof. a) It is clear that if ‖ . ‖ is an adelic metric then f ∗ ‖ . ‖ is an adelic metric.
According to Definition 7.20, for the existence of f ∗ : ˜Pic(Y ) → ˜Pic(X ), it will
be sufficient to show
f ∗( )
CY
≥0 ⊆ C
≥0
X .

Sec. 7] THE ADELIC PICARD GROUP 183
Let us first verify that f ∗ (CY ≥0 ) ⊆ C ≥0
X .
≥0
For this, let (L , ‖ . ‖) ∈ CY .
I.e., (L , ‖ . ‖) = i Y (L , ‖ . ‖ ∞ ) for a smooth hermitian line bundle (L , ‖ . ‖ ∞ )
on a model Y of Y such that L | Yp
is nef for every prime number p and the
curvature form c 1 (L, ‖ . ‖ ∞ ) is non-negative.
Fix an arbitrary model X ′ of X. We define X to be the closure of the
graph of f in X ′ ×Y . Then, the projection to the second factor yields a
morphism f : X → Y which extends f . This yields
f ∗ (L , ‖ . ‖) = i X ( f ∗ L , f ∗ ‖ . ‖ ∞ ) .
The pull-back of a nef invertible sheaf is always nef and the pull-back of
a smooth non-negatively curved hermitian line bundle is smooth and nonnegatively
curved. Hence, we indeed have f ∗ (CY ≥0 ) ⊆ C ≥0
X .
Further, by Lemma 7.18, the homomorphism f ∗ : Pic ad (Y ) → Pic ad (X ) is continuous
with respect to the metric topology. Lemma 7.25 yields the assertion.
b) f ∗ : CY
≥0 → C ≥0
X
is continuous with respect to the metric topology. Thus, we
have a continuous map
C ≥0
Y
× C ≥0
Y
f ∗ ×f
−→ ∗
C ≥0 × X
C ≥0 diff
X
−→ Pic ad (X ) .
The definition of the integrable topology yields the claim.
□
7.31. Theorem. –––– Let X be a regular, projective variety overÉ.
i) Then, there is exactly one continuous-multilinear map, the adelic intersection
product
pZ X : ˜Pic(X ) × . . . × ˜Pic(X )
Ê
−→Ê,
} {{ }
dim X+1 times
such that for every model X of X over Specthe diagram
˜Pic(X ) × . . . × ˜Pic(X
pZ
X
)
commutes.
ii) p X Z
is symmetric.
i dimX+1
X
̂Pic(X ) × . . . × ̂Pic(X )
Proof. i) By Theorem 5.2.ii), the intersection products pGS X are compatible with
pull-backs under birational maps. We obtain a-multilinear map p such that
p X GS

Ê
184 THE CONCEPT OF A HEIGHT [Chap. IV
the diagrams
p
im ι X × . . . × im ι X
i dimX+1
X
p X
̂Pic(X ) × . . . × ̂Pic(X
GS
)
commute.
Every element of im α X has an integral multiple in im ι X . Therefore, p extends
uniquely to a-multilinear map
p : 〈 im ι X , im α X 〉 × . . . × 〈 im ι X , im α X 〉 −→Ê.
By Lemma 7.29.b), 〈 im ι X , im α X 〉 is a dense subset of ˜Pic(X ). This implies that
pZ X is unique.
To show the existence, we need that p is continuous. This may be tested locally.
Let
We write
l = (l 0 , . . . , l dimX ) ∈ 〈 im ι X , im α X 〉 × . . . × 〈 im ι X , im α X 〉
⊆ ˜Pic(X ) × . . . × ˜Pic(X ) .
(l 0 , . . . , l dimX ) = (l (1)
0 , . . . , l(1) dim X ) − (l(2) 0 , . . . , l(2) dimX )
for l (i)
0 , . . . , l(i) ∈ dimX C ≥0
(i)
X
and define U
j
in C ≥0
X
for ε := 1. By Lemma 7.24,
to be the ε-neighbourhood of l (i)
j
U l := diff(U (1)
0 , U (2)
(1)
0
) × . . . × diff(U
dim X , U (2)
is an open neighbourhood of (l 0 , . . . , l dimX ).
dimX )
It is therefore sufficient to verify that p is continuous on U l . This is a direct
consequence of Lemma 7.32. Indeed, for every (L , ‖ . ‖) ∈ U (i)
j
the restriction
:= L | X is always the same. Put
L (i)
j
M 0 :=
2 2
∑ ∑
i 1 =1 i 2 =1
. . .
2
∑
i dimX =1
deg ( c 1 (L (i 1)
1
) · . . . · c 1 (L (i dimX )
dim X )) .
Then, p is Lipschitz continuous on U l with respect to the first variable. M 0 is a
Lipschitz constant.
Lipschitz continuity in the other variables may be shown analogously.
ii) This follows immediately from the uniqueness together with the symmetry
of the intersection products pGS X proven in Theorem 5.2.b).
□

Sec. 7] THE ADELIC PICARD GROUP 185
7.32. Lemma. –––– Let π : X → Specbe an arithmetic variety. Suppose its
generic fiber X is regular and put d := dim X.
Let L 1 , . . . , L d ∈ ̂Pic(X ). Suppose that L 1 | , . . . , L Xp
d| Xp
are nef for every
prime number p. Further, consider (L , ‖ . ‖) ∈ ̂Pic(X ) such that L | X
∼ = OX .
Then,
|p X GS(
(L , ‖ . ‖), L1 , . . . , L d
) | ≤
≤ δ ( i X (L , ‖ . ‖), 0 ) · deg ( c 1 (L 1 | X ) · . . . · c 1 (L d | X ) )
for 0 ∈ Pic ad (X ) the neutral element.
Proof. The assertion is compatible with pull-back under a generically finite, surjective
morphism. Thus, we may assume without restriction that X is regular.
The asserted inequality then reads
∣
∣̂deg π ∗
(ĉ1 (L , ‖ . ‖) · ĉ 1 (L 1 ) · . . . · ĉ 1 (L d ) )∣ ∣ ≤
To show this, we write first
≤ δ ( i X (L , ‖ . ‖), 0 ) · deg ( c 1 (L 1 | X ) · . . . · c 1 (L d | X ) ) .
ĉ 1 (L , ‖ . ‖) =: [(D, g)] .
Here, D = div s for s the rational section of L such that s| X = 1 ∈ Γ(X, O X ).
In particular, D is a divisor supported in the special fibers. Further,
g := − log ‖s‖ 2 .
It is sufficient to treat the cases (0, g) and (D, 0).
First Case: D = 0.
We have ĉ 1 (L 1 ) · . . . · ĉ 1 (L d ) = [(Z, g Z )] for Z a cycle of dimension
one. The restriction of Z to the generic fiber is a 0-cycle of degree
deg ( c 1 (L 1 | X ) · . . . · c 1 (L d | X ) ) . Therefore, by the description of the arithmetic
intersection product given in 4.10,
(
π ∗ [(0, g)] · ĉ1 (L 1 ) · . . . · ĉ 1 (L d ) ) (
= π ∗ [(0, g)] · [(Z, gZ )] )
= π ∗ [(0, gδ Z )] .
The absolute value of the arithmetic degree is bounded by
1
2 sup |g(x)|·degZÉ≤ sup |log ‖1‖(x)| · deg ( c 1 (L 1 | X ) · . . . · c 1 (L d | X ) ) .
x∈X ()
x∈X ()
Second Case: g = 0.
We may assume that D is contained in one special fiber, say, in Xp .

186 THE CONCEPT OF A HEIGHT [Chap. IV
Assume first that D = (V ) for an irreducible component V of of maximal
dimension. Then, by the definition of the push-forward and the arithmetic
Xp
degree, we have
̂deg π ∗
(
[(D, 0)] · ĉ1 (L 1 ) · . . . · ĉ 1 (L d ) ) = (log p)·deg ( c 1 (L 1 | V )·. . . ·c 1 (L d | V ) ) .
We observe, if D is effective then the expression on the right hand side is nonnegative.
Indeed, the intersection number of nef divisors is always non-negative
by a theorem of S. Kleiman [Laz, Example 1.4.16].
By consequence, if
for r ∈Êthen
∣ ̂deg
(
π ∗ [(D, 0)] · ĉ1 (L 1 ) · . . . · ĉ 1 (L d ) )∣ ∣ ≤
) ≤ D ≤ ) (‡)
−r(Xp r(Xp
≤ r(log p) · deg ( c 1 (L 1 | Xp ) · . . . · c 1(L d | Xp ))
= r(log p) · deg ( c 1 (L 1 | X ) · . . . · c 1 (L d | X ) ) (§)
since the fibers are linearly equivalent to each other.
We choose the minimal number r such that property (‡) is fulfilled.
This means, there is some ξ ∈ Xp
such that O X ,ξ (D) = O X ,ξ (±rXp ).
In particular, r = s/t is a rational number. We assume the plus sign without
restriction. Then, O X ,ξ (D) ⊗t = O X ,ξ (Xp )⊗s .
The definition of D shows that L = O X (D). By consequence,
L ⊗t
ξ
= O X,ξ (D) ⊗t = O X ,ξ (Xp )⊗s .
Definition 7.7implies that ‖1‖ p (x) = p −r for x ∈ X specializing to ξ. Hence, by
Lemma-Definition 7.15,
δ νp
(
iX (L , ‖ . ‖), 0 ) ≥ |log ‖1‖ p (x)| = r log p .
Formula (§) yields the assertion.
□
7.33. Lemma. –––– Let f : X → Y be a generically finite morphism of projective
varieties overÉ. Then, for every L 1 , . . . , L dim Y+1 ∈ ˜Pic(Y ),
p X Z ( f ∗ L 1 , . . . , f ∗ L dim Y+1 ) = p Y Z (L 1, . . . , L dim Y+1 ) .
Proof. This follows directly from Theorem 5.2.a.ii) together with
Lemma 7.29.b).
□

Sec. 7] THE ADELIC PICARD GROUP 187
7.34. Remark. –––– One might want to enlarge ˜Pic(X ). This is easily possible
by omitting condition ii) in Definition 7.12. This would not destroy any of the
properties proven except for the explicit description of the closure C ≥0
X
given
in Lemma 7.27.
7.35. Remark. –––– A priori, S. Zhang’s adelic Picard group [Zh95a] is somewhat
smaller.
S. Zhang works with a more restrictive notion of convergence. Instead of
the closure with respect to the metric topology, he works with all limits of
sequences ( {‖ . ‖ ν,i } ν
)i
of adelic metrics which are convergent in the following
sense.
(‖ . ‖ ν,i ) i is a constant sequence for all but finitely many valuations ν. It converges
uniformly for the other ν.
The proof of Lemma 7.27 indicates, however, that this should make no difference.
v. Examples. —
7.36. Example. –––– Let K be a number field. Then,
I.e., there is an exact sequence
˜Pic(SpecK ) = ̂Pic(Spec O K ) .
0 −→ Cl K −→ ˜Pic(SpecK ) ˜deg
−→Ê−→ 0 .
Here, the class group Cl K is a discrete subgroup of ˜Pic(SpecK ).
homeomorphism on every cofactor.
˜deg is a
Proof. The models of SpecK are the spectra of the orders of K. All induced
metrics come from the maximal order.
□
7.37. Example. –––– Let V be a semistable projective curve overÉ.
Then, there is an exact sequence
where= L
ν∈Val(É)
0 →→ ˜Pic(V ) → Pic(V ) → 0
C 0 (V (Éν)).
Here, C 0 (V (É∞)) = C(V ()) F ∞
is the space of all continuous F ∞ -invariant
functions on V ().
For p a prime number, C 0 (V (Ép)) is given as follows.
We choose a minimal semistable model V of V over SpecÉp. Let D 1 , . . . , D r
be the irreducible divisors in the special fiber. Further, for each k ∈Æ, we

→Ê
188 THE CONCEPT OF A HEIGHT [Chap. IV
define a countable disjoint covering {U i
(k) } i∈Æof V (Ép) by the requirement
that x, y ∈ V (Ép) are in the same set if they coincide modulo p k .
Then, C 0 (V (Ép)) consists of all Gal(Ép/Ép)-invariant functions V (Ép)
of the type
r
∞
∑ α i ‖1‖ (V ,O(Di )) + ∑ ∑ β ikl χ U
(k) .
i
i=1
l=1 ik
Here, ‖ . ‖ (V ,O(Di )) denotes the induced metric, induced by the model (V , O(D i )).
χ U
(k)
i
is the characteristic function of U (k)
i
. α i and β ikl are real numbers.
We require that the inner sum is finite for each l ∈Æand that the outer series
converges uniformly.
vi. Adelic heights. —
7.38. Definition. –––– Let X be a regular, projective variety overÉand
L ∈ ˜Pic(X ) be an integrably metrized invertible sheaf.
Then, the absolute height with respect to L of an L-valued point x ∈ X (L)
for L a number field is given by
1
h L
(x) :=
[L :É] ˜deg L | x .
7.39. Remark. –––– Lemma 7.33 shows that this definition is independent of
the choice of L.
7.40. Corollary (Theorem of successive minima). —–
Let X be a regular, projective variety overÉand L ∈ ˜Pic(X ). Assume
a) L ∈ C ≥0
X
or
b) L ∈ ˜Pic(X ) is such that ‖ . ‖ p is semipositive for all p ≠ ∞ and ‖ . ‖ ∞ is
almost semiample.
Then,
e 1 (h L
) ≥ pX Z (L , . . . , L )
(dim X + 1) deg L
X ≥ e 1(h L
) + . . . + e dim X+1 (h L
)
.
dim X + 1
Proof. Cf. [Zh95a, Theorem 1.10].
a) From Theorem 7.31, we see that Corollary 6.6, together with Remark 6.7,
immediately implies the assertion for L ∈ C ≥0
X . Further, pX Z and ˜deg are continuous.
This shows that the inequalities are true for L ∈ C ≥0
X , too.
b) Lemma 7.27 says that L = (L , ‖ . ‖) ∈ C ≥0
X
is equivalent to the statement
that (L , ‖ . ‖) ∈ ˜Pic(X ) such that ‖ . ‖ p is semipositive for all p ≠ ∞ and ‖ . ‖ ∞
is almost semiample.
□

CHAPTER V
ON THE DISTRIBUTION OF SMALL POINTS
ON ABELIAN AND TORIC VARIETIES ∗
1. Introduction
i. Classical results. —
1.1. –––– In 1972, J.-P. Serre proved the following remarkable result.
Theorem (Serre). Let K be an algebraic number field and E be an elliptic curve
over K without complex multiplication over K. Then, for almost every prime
number l, the Galois group Gal(K/K ) acts transitively on the l-torsion points of E.
1.2. Notation. –––– For x ∈ E(K ), we denote by δ x the Dirac measure associated
to its Galois orbit. I.e., if x ∈ E(F ) for some number field F ⊇ K then
for each ϕ ∈ C 0 (E()).
δ x (ϕ) :=
1
♯ Gal(F/K )
∑
σ : F֒→ϕ(σ(x))
1.3. –––– In this language, Serre’s theorem immediately implies the following
corollary.
Corollary. Let K be an algebraic number field and E be an elliptic curve over K
without complex multiplication over K. Further, let (x i ) i∈Æbe a sequence of torsion
points in E which does not contain any constant subsequence.
Then, the associated sequence (δ xi ) i∈Æof measures on E() converges weakly to the
Haar measure of volume one.
(∗) This chapter is a revised version of the article: On the distribution of small points on abelian
and toric varieties, Preprint.

190 DISTRIBUTION OF SMALL POINTS [Chap. V
1.4. –––– In 1997, L. Szpiro, E. Ullmo and S. Zhang [S/U/Z] showed a generalization
of this corollary from torsion points (which are of Néron-Tate height
zero) to points of a small positive height. The three authors heavily make use
of the powerful methods of Arakelov geometry as presented in Chapter IV.
Their result is as follows.
i∈Æ
1.5. Theorem (Szpiro, Ullmo, Zhang). —– Let A be an abelian variety over
some number field K and (x i ) i∈Æbe a sequence of closed points in A. Assume (x i )
converges to the generic point of A in the sense of the Zariski topology. Suppose further
that lim h NT (x i ) = 0.
i→∞
Then, the associated sequence (δ xi ) i∈Æof measures on A() converges weakly to the
Haar measure of volume one.
1.6. –––– Surprisingly, there is a completely parallel theorem for the naive
height on projective space which was proven by Yuri Bilu [Bilu97]. In fact,
for P 1 , he proved this result already in 1988 [Bilu88]. It would have been easy
to deduce the general result from this.
1.7. Theorem (Bilu). —– Let (x i ) i∈Æbe a sequence of closed points in P nÉ.
Assume that (x i ) i∈Æconverges to the generic point of P n in the sense of the
Zariski topology. Assume further that lim
i→∞
h naive (x i ) = 0.
Then, the associated sequence (δ xi ) i∈Æof measures on P n () converges weakly to the
Haar measure of volume one on (S 1 ) n ⊂n ⊂ P n ().
ii. Canonical heights. —
1.8. General situation. –––– Let P be a projective variety overÉequipped
with an ample invertible sheaf L ∈ Pic(P). Further, we assume there is given
a self map f : P → P such that there is some isomorphism Φ: L ⊗d ∼ =
−→ f ∗ L .
1.9. Definition. –––– The canonical height function h f ,L on P is given by
1
h f ,L (x) := lim
n→∞ d h ( n L f (n) (x) )
for x ∈ P a closed point. Here, f (n) means the n-fold iteration of f . h L denotes
the height defined by the invertible sheaf L introduced in Definition IV.2.10.
1.10. Remark. –––– The limit process defining h f ,L is a generalization of the
classical one for the Néron-Tate height [C/S, Chapter VI, §4].
We will see in Example 1.16.ii) that the naive height is a canonical height, too.

Sec. 1] INTRODUCTION 191
1.11. Remark. –––– h L is defined only up to a bounded summand. Nevertheless,
h f ,L is independent of the choice of a particular representative. Indeed, if
|h (1)
L (x) − h(2) L (x)| ≤ C for every x ∈ P then
1 (
∣d n h(1) L f (n) (x) ) − 1 (
d n h(2) L f (n) (x) )∣ ∣∣ ≤ C d . n
iii. An equidistribution result. —
1.12. –––– The goal of the present chapter is to prove a common generalization
of Theorems 1.5 and 1.7. We will give an Arakelovian approach to a situation
which covers the case of the Néron-Tate height as well as the naive height.
1.13. Situation. –––– Let P be a regular projective variety overÉcontaining
a group scheme G ⊆ P as an open dense subset. Further, let a morphism
f : P → P be given and assume that
i) the unit e is a repelling fixed point. I.e., all eigenvalues of the tangent
map T e f : T e G → T e G are of absolute value strictly bigger than 1.
ii) f | G is a group homomorphism f | G : G → G.
iii) There is some compact, Zariski dense subgroup K ⊆ G() which is both
backward and forward invariant under f.
Finally, let L ∈ Pic(P) be ample and Φ: L ⊗d ∼ =
−→ f ∗ L be an isomorphism.
This guarantees that we have the canonical height h f ,L .
1.14. Theorem (Equidistribution on P()). —– Let P, f , L , and Φ are as described
in 1.13.
Let (x i ) i∈Æbe a sequence of closed points in P. Assume that (x i ) i∈Æconverges
to the generic point of P in the sense of the Zariski topology. Further, suppose
that h f ,L (x i ) → 0.
Then, the associated sequence (δ xi ) i of measures on P() converges weakly to the
measure τ on P() being the zero measure on P() \ K and the Haar measure of
volume one on K.
1.15. Remark. –––– For a general self map f , iterating will lead to fractals.
Unfortunately, not much is known about the corresponding fractal heights.
1.16. Examples. –––– The condition that e is a repelling fixed point is fulfilled,
in particular, if G is commutative and f : g ↦→ g l is the homomorphism raising
to the l-th power for l ≥ 2. This includes the two standard cases. Namely,
i) let P = G = A be an abelian variety and L ∈ Pic(A) a symmetric, ample
invertible sheaf. One has
f = [l]: A → A

192 DISTRIBUTION OF SMALL POINTS [Chap. V
and puts K = A(). Then, independently of the choice of l, h f ,L is the
Néron-Tate height corresponding to L .
ii) Let P = P n , G =n+1 ∼
m /m =n m and L = O(1). We have
f = [l]: (x 0 , . . . , x n ) ↦→ (x l 0 , . . . , xl n )
and put K := U (1) n . Here, h f ,L is the naive height.
iii) One may combine abelian varieties and projective spaces and consider a
split semi-abelian variety A × P n . Here, G = A ×n m , K = A × U (1),
and L = S ⊠O(p) for S a symmetric, ample invertible sheaf on A. One may
put f := [l] × [l 2 ].
A. Chambert-Loir’s equidistribution result [C-L] for quasi-split semi-abelian
varieties can easily be deduced from this.
iv) Consider a free abelian group N of finite rank and a finite rational polyhedral
decomposition {σ α } α of N ⊗Ê. These data define a proper toric variety X.
The multiplication map N → N , n ↦→ ln induces a self-morphism
f = [l]: X → X.
Further, consider a function g : N ⊗Ê→Êwhich is strictly convex in the
sense of [K/K/M/S, Chapter I, Theorem 13]. Then, L := F g is a T -invariant
ample invertible sheaf on X. One has [l] ∗ F g
∼ = Flg = F ⊗l
g .
In this situation, P = X, G = T ∼ =n m is the Zariski dense torus in X, and
K ⊂ T is its maximal compact subgroup.
1.17. Remark. –––– The latter example clearly contains the case of the naive
height on projective space. Notice that it also contains the blow-up Bl (0:0:0) (P 2 )
of the projective plane in the origin and the canonical height defined by the
ample divisor nH − mE for n, m ∈Æ, n > m.
1.18. Remark. –––– For abelian varieties, our proof coincides with that of
Szpiro, Ullmo, and Zhang. In the case of projective space, we provide a more
geometric proof for Bilu’s theorem.
2. The dynamics of f
i. Arakelovian interpretation of the canonical height. —
2.1. –––– Let P, f , L , and Φ are as described in 1.8.
To understand the canonical height better, one has to use the language of
Arakelov geometry introduced in the previous chapter.

Sec. 2] THE DYNAMICS OF f 193
First, recall that, in Definition 1.9, we may replace h L by any other height
function which differs only by a bounded summand. For example, we may
work with h ′ L (x) := 1 m h l 2 (
iL ⊗m(x) ) .
Further, there is a projective model P of P over Specto which L ⊗m extends.
Indeed, L ⊗m defines a closed embedding i L ⊗m : P → P N and one may
put P := i L ⊗m(P).
Then, L := O(1)| P is an extension of L ⊗m to P. We equip L with the
restriction of the Fubini-Study metric. This guarantees that, for every closed
point x ∈ P,
h L ,0 (x) := h L (x) = 1 m h (L ,‖ .‖)(x) .
Here, h (L ,‖ .‖) is the absolute height defined by the smooth hermitian line bundle
(L , ‖ . ‖), cf. Definition IV.5.7.
The recursion is given by
h L ,i+1 (x) := 1 d h L ,i(
f (x)
)
.
Unfortunately, the isomorphism Φ: L ⊗d ∼ =
−→ f ∗ L does not extend to P. If it
would then we could put ‖ . ‖ 0 := ‖ . ‖ and, recursively,
This would then lead to
‖ . ‖ i := Φ −1 f ∗ ‖ . ‖ 1/d
i−1 .
h L ,i (x) = 1 m h (L ,‖ .‖ i )(x) .
2.2. –––– To make the plan above precise, one has to work in the more flexible
context of adelic Picard groups introduced in Section IV.7.
There is the adelic metric ‖ . ‖ (0)
∼ on L ⊗m given by
(L ⊗m , ‖ . ‖ (0)
∼ ) := i P(L , ‖ . ‖ 0 ) .
Since L is very ample and ‖ . ‖ 0 is a restriction of the Fubini-Study metric, we
see that (L ⊗m , ‖ . ‖ ∼ (0))
∈ C ≥0
P .
Further, we define, recursively,
‖ . ‖ ∼ (i) := (Φ⊗m ) −1 f ∗[ ]
‖ . ‖ ∼
(i−1) 1
d
.
Lemma IV.7.30.a) shows (L ⊗m , ‖ . ‖ ∼ (i) ) ∈ C ≥0
P
for every i ∈Æ.
We also observe that the isomorphism Φ does extend to an open neighbourhood
of the generic fiber of P. Therefore, the sequence (‖ . ‖ ∼ (i) ) i is actually constant
at all but finitely many valuations.

194 DISTRIBUTION OF SMALL POINTS [Chap. V
Furthermore, Lemma IV.7.18 implies that
δ ( (L ⊗m , ‖ . ‖ (i)
∼ ), (L ⊗m , ‖ . ‖ (i+1)
∼ )) ≤ 1 d i ·δ( (L ⊗m , ‖ . ‖ (0)
∼ ), (L ⊗m , ‖ . ‖ (1)
∼ )) .
This means, for every ν ∈ Val(É), the sequence (‖ . ‖ (i)
ν ) i of metrics is uniformly
convergent.
The limit ‖ . ‖ ∼ clearly fulfills all the requirements of Definition IV.7.12.
I.e., ‖ . ‖ ∼ is an adelic metric on L ⊗m . Even more,
L := (L ⊗m , ‖ . ‖ ∼ ) ∈ C ≥0
P
⊆ ˜Pic(P) .
By Lemma IV.7.27, ‖ . ‖ νp is semipositive for every prime number p ≠ ∞ and
‖ . ‖ ν∞ is almost semiample.
Finally, we have
h f ,L = 1 m h L = 1 m h (L ⊗m ,‖ .‖ ∼ ) .
2.3. Lemma. –––– Let P, f , L , and Φ are as described in 1.8. Then, f is
automatically finite of degree d dimP .
Proof. P is a projective variety overÉand f : P → P is a morphism such
that L ⊗d ∼ = f ∗ L for some ample L ∈ Pic(P).
In particular, f ∗ L is ample which implies f is quasi-finite. Since f is a projective
morphism, this is sufficient for finiteness. Further, we have deg L
P ≠ 0
and deg f ∗ L P = deg L
P = d dimP deg ⊗d L
P.
□
2.4. Lemma. –––– Let P, f , L , and Φ are as described in 1.8. Then,
i) p P Z(
(L ⊗m , ‖ . ‖ ∼ ), . . . , (L ⊗m , ‖ . ‖ ∼ ) ) = 0,
ii) e 1 (h L
) = . . . = e dim P+1 (h L
) = 0.
Proof. i) For every i ∈Æ, write
(
P i := pZ
P (L ⊗m , ‖ . ‖ ∼ (i) ), . . . , (L ⊗m , ‖ . ‖ ∼ (i) )) .
Then, since f is of degree d dimP , Lemma 7.33 shows
(
d dimP P i−1 := pZ
P ( f ∗ L ⊗m , f ∗ ‖ . ‖ ∼
(i−1) ), . . . , ( f ∗ L ⊗m , f ∗ ‖ . ‖ ∼ (i−1) ) ) = 0 .
Applying the isomorphism Φ ⊗m and dividing each of the (dim P +1) operators
by d yields
P i−1
d
= ( pP Z (L ⊗m , ‖ . ‖ ∼ (i) ), . . . , (L ⊗m , ‖ . ‖ ∼ (i) )) = P i .
(P i ) i is, therefore, a zero sequence.
ii) Lemma IV.6.3 implies that there is a constant C such that
h L ,0 (x) > C

Sec. 2] THE DYNAMICS OF f 195
for everyx ∈ P. Consequently, h L ,i (x) > C/d i and h L
(x) ≥ 0 for everyx ∈ P.
This means
e dim P+1 (h L
) ≥ 0 .
Corollary IV.7.40 together with the inequality
e 1 (h L
) ≥ . . . ≥ e dim P+1 (h L
)
implies the claim.
□
2.5. Remark. –––– As the construction of ‖ . ‖ ∼ depends on the iterated pullbacks
under f , one has to understand the dynamics of the system ( f i ) i≥1 . It will
turn out to be sufficient for our purposes to pay attention to the infinite place.
I.e., to the dynamics of that system on P().
ii. Analyzing the dynamics of f . —
2.6. –––– Let us analyze the special situation described in 1.13.
i) G ⊆ P is an open dense subset being a group scheme such that f has a
restriction m := f | G : G → G which is a homomorphism of group schemes.
Therefore, kerm is a group scheme of finite order d dimP .
As f is surjective, im m is a priori Zariski dense. Since m is a homomorphism,
it is surjective, too.
There is a compact subgroup K ⊆ G() which is both forward and backward
invariant under m. In particular, all the finite groups ker(m◦ . . . ◦m) are
contained in K.
ii) All eigenvalues of the tangent map T e m have absolute value strictly bigger
than 1. Therefore, on G(), there is a left invariant Riemannian metric µ
such that
q := ‖(T e m) −1 ‖ max < 1 .
Indeed, based on the Jordan normal form, one finds easily a hermitian scalar
product on T e G() satisfying the analogous inequality. We take its real part.
By transport of structure, we find a left invariant Riemannian metric µ on G().
We will denote by δ the metric on G() given by the lengths of the
µ-shortest paths.
2.7. Convention. –––– In order to simplify notation, we will often write f
and m instead of fand mwhen there is no danger of confusion.

196 DISTRIBUTION OF SMALL POINTS [Chap. V
2.8. Lemma. –––– a) m induces a mapping
which is a diffeomorphism.
m : G()/K → G()/K
b) m is expanding for the Riemannian metric µ on G()/K induced by µ.
More precisely,
for any z 1 , z 2 ⊆ G()/K.
δ ( m(z 1 ), m(z 2 ) ) ≥ 1 q δ(z 1, z 2 )
Proof. a) m is well-defined since K is forward invariant under m.
Backward invariance of K implies that m is even an injection. Further,
as m : G() → G() is a surjection, the same is then true for m.
Finally, the tangent map
T e m : T e G()/K → T e G()/K
is expanding, too. In particular, T e m is invertible. This yields that m is a
local diffeomorphism.
b) The preimage m −1 (w) of a path w in G()/K is a path, too. All tangent
maps of m are expanding by a factor ≥ 1 q . Therefore,
l ( m −1 (w) ) ≤ q · l(w) .
This implies the claim.
□
2.9. Corollary. –––– Let G() = G()∪{∞} be the one-point compactification
of G().
Then, for each x ∈ G()\K, the sequence (x i ) i such that x 0 = x and x i+1 = m(x i )
converges to ∞. In other words, K is the Julia set of the dynamical system (m i ) i≥1
on G().
Proof. We have to show that for each compact set L ⊆ G() there are only
finitely many i ∈Æsuch that x i ∈ L. Replacing L by KL, if necessary, we may
assume that L is left K-invariant.
This means, we are given a compact set L ⊆ G()/K and have to verify that
x i ∈ L for only finitely many values of i.
As m : G()/K → G()/K is expanding and x 0 ≠ [K ], we have
δ(x i , [K ]) → ∞ .
However, L is a compact set. In particular, L is bounded.
□

Sec. 2] THE DYNAMICS OF f 197
2.10. Corollary. –––– The union over all the groups ker(m ◦ . . . ◦ m) is dense
in K.
Proof. Otherwise, the topological closure of
K ′ := [ ker(m ◦ . . . ◦ m)
} {{ }
i
i times
would be a compact Lie group properly contained in K. It is obvious that K ′ is
forward and backward invariant under f. Therefore, the map induced by f
on the compact homogeneous space K/K ′ would be injective and expanding.
□
2.11. Notation. –––– For N ∈Ê, we will write U N (K ) for the set
{x ∈ G() | δ(x, K ) ≤ N } .
2.12. Proposition –––– Let µ i be the measure induced by the smooth differential
form
c 1 (L , ‖ . ‖ ∞) (i) ∧ . . . ∧ c 1 (L , ‖ . ‖ ∞)
(i)
of type (dim P, dimP) on P(). Then, the sequence (µ i ) i converges weakly to
the measure µ which is the zero measure on P()\K and the Haar measure of
volume deg L
P on K.
Proof. First step. Some observations.
Each µ i is of volume deg L
P. Further, the sequence (µ i ) i obeys the recursive low
Second step. µ i | P()\K → 0.
µ i+1 = 1
d dim P m∗ µ i .
For this, let g ∈ C 0 (P() \K) be a continuous function with compact support.
We consider the sequence (g i ) i such that g 0 := g and g i+1 := 1 f
d dimP ∗ g i . Here, f ∗ ,
the push-forward of a function, is given by summation over the fibers of f .
By definition, µ i (g) = µ 0 (g i ). Thus, let us show µ 0 (g i ) → 0 for i → ∞.
For this, we note that |g| is bounded by some constant C. This implies |g i | ≤ C
for every i. Further, there is some ε > 0 such that supp(g) ⊆ P()\U ε (K ).
Consequently,
Z
|µ i (g)| = |µ 0 (g i )| ≤ C χ (P()\U ε/q i ) dµ 0 .
Here, χ A denotes the characteristic function of a measurable set A. The integrand
converges monotonically to χ P()\G() which is of integral zero. Therefore,
the Theorem of Beppo Levi yields µ i (g) → 0.
P()

198 DISTRIBUTION OF SMALL POINTS [Chap. V
Third step. The assertion in general.
Let g ∈ C 0 (G()) be a continuous function with compact support. Again, we
put g 0 := g and g i+1 := 1 m
d dimP ∗ g i such that we have µ i (g) = µ 0 (g i ). As the sequence
(g i ) i is uniformly bounded, it suffices to show that it converges pointwise
to the constant
I := 1 Z
g dρ .
volK
We claim that (g i ) i converges even uniformly in every compact set A ⊆ G().
For this, we may assume without loss of generality that A = U N (K ) for
some N ∈Ê. Further, note that
g i (x) =
K
1
♯ kerm i
∑
k∈kerm i g(ky)
for any y such that f (y) = x. Here, we clearly have (m i ) −1 (A) ⊆ U Nq i(K )
and Nq i → 0 for i → ∞.
In addition, g is uniformly continuous on A and m is expanding. Thus, for each
ε > 0, there is some i ∈Æsuch that, for two arbitrary points x, x ′ ∈ A, one
can find y, y ′ such that f (y) = x, f (y ′ ) = x ′ , and δ(y, y ′ ) < ε. Consequently,
max g i | A − min g i | A
tends to zero for i → ∞.
Finally, for every i ∈Æ, I is the mean value of g i on K.
□
iii. Some further observations. —
2.13. Caution. –––– It is not true in general that the curvature current, given by
c 1 (L, ‖ . ‖ ∞ ) := 1 ∂∂ log ‖s‖2
2πi
for a non-zero rational section of L, vanishes on P() \K. Nevertheless, one
has at least the following.
2.14. Lemma. –––– The curvature current c 1 (L, ‖ . ‖ ∞ ) has the properties below.
a) The restriction c 1 (L, ‖ . ‖ ∞ )| G() is left and right K-invariant.
b) It satisfies the equation f ∗ c 1 (L, ‖ . ‖ ∞ ) = d · c 1 (L, ‖ . ‖ ∞ ).
Proof. a) By construction, c 1 (L, ‖ . ‖ ∞,i ) is invariant under ker(m i).
Thus, c 1 (L, ‖ . ‖ ∞ ), which is the weak limit of that sequence of currents,
is invariant under S i ker(m i). Invariance under any k ∈ K follows since
k i → k implies that the test forms k i ω converge to kω in the Schwartz space.
b) is clear. □

Sec. 3] PERTURBING ALMOST SEMIAMPLE METRICS 199
3. Perturbing almost semiample metrics
3.1. –––– One of the most natural ways to produce a new adelic metric from
an old one is to replace ‖ . ‖ ∞ by ‖ . ‖ ∞ · exp(−g) for some continuous function
g ∈ C (P()). As we want to use Corollary IV.7.40 as a fundamental tool,
it is necessary to understand for which g this metric is almost semiample.
3.2. Notation. –––– In this section we will continue to use the notations of 1.13.
Further, we denote by S the set of all continuous functions g ∈ P() such that
‖ . ‖ ∞ · exp(−g) is almost semiample.
3.3. Lemma. –––– a) One has C ∈ S for every constant C.
b) Let g 1 , g 2 ∈ S and 0 < a < 1. Then, ag 1 + (1 − a)g 2 ∈ S.
c) If g ∈ S then 1 d f ∗ g ∈ S.
d) Let g ∈ S be a function such that supp g ⊆ G(). Then, for each k ∈ K, one
has g · k ∈ S for
{ g(kx) if x ∈ G() ,
(g · k)(x) :=
0 otherwise .
e) S is closed under uniform convergence.
Proof. e) is trivial.
a) ‖ . ‖ ∞ is almost semiample. Thus, there is a sequence (‖ . ‖ i ∞ ) i of smooth
hermitian metrics with strictly positive curvature on Lwhich is uniformly
convergent to ‖ . ‖ ∞ . The metrics ‖ . ‖ ∞,i · exp(−C ) have the same curvatures
and converge uniformly to ‖ . ‖ ∞ · exp(−C ).
b) Let (‖ . ‖ i ′) i and (‖ . ‖ i ′′)
i be sequences uniformly convergent versus
‖ . ‖ ∞ · exp(−g 1 ) and ‖ . ‖ ∞ · exp(−g 2 ), respectively. Put
‖ . ‖ i := ‖ . ‖ ′a
i
′′ (1−a)
· ‖ . ‖ i .
These metrics have positive curvature and, uniformly,
‖ . ‖ i → ‖ . ‖ ∞ · exp ( − ag 1 − (1 − a)g 2
)
.
c) Put ‖ . ‖ ∞ ′ := ‖ . ‖ ∞ · exp(−g). Then,
(
Φ −1 f ∗ ‖ . ‖ ′ d
1
∞ = Φ −1 f ∗ d
‖ . ‖ 1 ∞ · exp − 1 )
d f ∗ g
(
= ‖ . ‖ ∞· exp − 1 )
d f ∗ g .
Therefore, if (‖ . ‖ ∞,i ′ ) i is a sequence of metrics with positive curvature
which converges uniformly to ‖ . ‖ ∞ ′ then (Φ −1 f ∗ ‖ . ‖ ′ 1 d
∞,i) i converges uniformly
to ‖ . ‖ ∞ · exp(− 1 f ∗ g).
d
d) We denote by
e k : G() → G()

200 DISTRIBUTION OF SMALL POINTS [Chap. V
multiplication by k from the left.
By e), we may assume that k ∈ ker(m j ) for some j ∈Æ. To show
‖ . ‖ ∞ · exp(−g · k)
is almost semiample, it suffices to consider the hermitian metric
‖ . ‖ d j
∞ · exp(−d j g · k)
on L.
⊗d j
Let us consider the restriction to G(), first. Iterated application of Φ induces
an isomorphism
e ∗ k L | ⊗d j ∼ G() = e ∗ k ( f j ) ∗ L| ∼ G() = ( f j ) ∗ L| ∼ ⊗d j
G() = L| G() . (∗)
Here, the isomorphism in the middle is canonical as f j ◦e k = f j .
By construction 2.2, ‖ . ‖ ∞ is the uniform limit of a sequence (‖ . ‖ ∞ (i))
i such that,
for every i ≥ j,
e ∗ k
[ ] ‖ . ‖
(i) d j
∞
under the identification (∗). By consequence,
= [ ]
‖ . ‖ ∞
(i) d j
e ∗ k‖ . ‖ d j
∞ = ‖ . ‖ d j
∞ ,
too, which shows
[ ‖ . ‖
d j
∞ · exp(−d j g) ] = ‖ . ‖ d j
∞ · exp(−d j g · k) .
e ∗ k
As almost semiampleness is preserved under holomorphic pull-back, we see that
‖ . ‖ d j
∞ · exp(−d j g · k) is almost semiample on G().
On the other hand, there is some neighbourhood U of P() \G() such
that g| U = 0. This implies −d j g · k = 0 and, therefore,
‖ . ‖ d j
∞ · exp(−d j g · k) = ‖ . ‖ d j
∞
on U. In particular, ‖ . ‖ d j
∞ · exp(−d j g · k) is almost semiample on U.
The assertion follows from Proposition 3.4.
□
3.4. Proposition –––– Let M be a projective complex manifold, G ∈ Pic(M ) be
ample, and {U 1 , . . . , U n } an open covering. Let ‖ . ‖ 1 , . . . , ‖ . ‖ n be continuous
hermitian metrics on G | U1 , . . . , G | Un such that, for some δ > 0, the
sets D 1,δ , . . . , D n,δ , given by
D i,δ := {x ∈ U i | ‖ . ‖ i,(x) ≤ (1 + δ) · min{‖ . ‖ 1,(x) , . . . , ‖ . ‖ n,(x) }} ⊆ U i

Sec. 3] PERTURBING ALMOST SEMIAMPLE METRICS 201
are compact. Assume, each ‖ . ‖ i is almost semiample.
Then, the continuous hermitian metric ‖ . ‖ := min i ‖ . ‖ i is almost semiample.
Proof. We may assume that G is very ample.
For every i, let (‖ . ‖ ij ) j be a sequence of smooth and positively curved metrics
on G | Ui which converges uniformly to ‖ . ‖ i . Put
‖ . ‖ − j
:= min{‖ . ‖ 1j , . . . , ‖ . ‖ nj } .
For j ≫ 0, ‖ . ‖ − j
is a continuous hermitian metric on the whole of G . The sequence
(‖ . ‖ − j ) j converges uniformly to ‖ . ‖.
We choose an embedding i : M ֒→ P N such that i ∗ O(1) ∼ = G . Further, we
extend the smooth metrics ‖ . ‖ ij to smooth metrics ‖ . ‖ ′ ij
on O(1) which are
defined on open subsets W i ⊇ U i of P N .
Under the tautological action PGL n () × P N → P N , each γ ∈ PGL n ()
defines an automorphism e γ : P N → P N such that there is a natural identification
eγ ∗ O(1) ∼ = O(1). We find an open neighbourhood O of e ∈ PGL n () such
that, for every γ ∈ O and every i ∈ {1, . . . , n}, the pull-back eγ ∗ ‖ . ‖
ij ′ is
well-defined on D i,δ and
eγ‖ ∗ . ‖ ′ ij | Ui ∩eγ −1 (W i )
has strictly positive curvature in every point of D i,δ .
We claim that, for each γ ∈ O, the perturbation
‖ . ‖ −,γ
j
:= e ∗ γ(
min{‖ . ‖
′
1j , . . . , ‖ . ‖ ′ nj} ) | M
of ‖ . ‖ − j
has a positive curvature current on each holomorphic curve inside M.
Indeed, this is a local statement. Let x ∈ M. Fix a holomorphic section
0 ≠ s ∈ Γ(U,G) defined in some neighbourhood U of x. Then,
c 1 (G , ‖ . ‖ −,γ
j
)| U = −dd c log ( ‖s‖ −,γ ) 2
j
= dd c( max{− logeγ‖s‖ ∗ ′ 1j 2 , . . . , − loge∗ γ‖s‖ nj} ) ′ 2 .
Observe that the maximum of a finite system of plurisubharmonic functions is
plurisubharmonic, again.
Now use Sobolev’s averaging procedure. For every non-negative smooth function
ϕ ≠ 0 on PGL n () such that supp ϕ ⊆ O, one gets an approximation
/
Z
Z
‖ . ‖ −,ϕ
j
:= ϕ(γ) · ‖ . ‖ −,γ
j
dρ γ ϕ(γ)dρ γ
PGL n ()
PGL n ()
of ‖ . ‖ − j . Here, ρ is the left Haar measure on PGL n().

202 DISTRIBUTION OF SMALL POINTS [Chap. V
Consider a sequence (ϕ k ) k of functions as above which converges weakly to the
delta distribution δ e . Every ‖ . ‖ −,ϕ k
j
is smooth and positively curved. The sequence
(‖ . ‖ −,ϕ k) k k
converges uniformly to ‖ . ‖.
3.5. Corollary. –––– If g 1 , g 2 ∈ S then g := max{g 1 , g 2 } ∈ S.
3.6. Corollary. –––– Let l ∈Æ, 0 < a < 1, and 0 ≠ s ∈ Γ(P(), L). ⊗l Then,
Proof. Consider the function g on
g := max{a log ‖s‖ l , 0} ∈ S .
U := {x ∈ P() | log ‖s‖(x) >−1}
given by g(x) := a log ‖s‖ l (x). We have to show that the hermitian metric
‖ . ‖ ′ ∞ := ‖ . ‖l ∞ · exp(−g)
on L| ⊗l U is the limit of a uniformly convergent sequence of smooth metrics
with positive curvature.
We put ‖ . ‖ ′ ∞,i := ‖ . ‖l ∞,i/ ( ‖s‖ l ∞,i) a. It is clear that, uniformly,
Further,
‖ . ‖ ′ ∞,i → ‖ . ‖ ′ ∞ .
c 1 (L| U , ‖ . ‖ ′ ∞,i ) = ddc (− log ‖s‖ ′2 ∞,i )
= dd c( − log(‖s‖ l ∞,i )2(1−a))
= (1 − a)l · c 1 (L| U , ‖ . ‖ ∞,i ) > 0 . □
□
3.7. Lemma. –––– Let g ∈ S suchthatsupp g ⊆ G(). Consider the K-invariant
function g given by
{
max g(kx) if x ∈ G() ,
g(x) := k∈K
g(x) otherwise .
Then, g ∈ S, too.
Proof. g is the limit of the uniformly convergent sequence (g i ) i , defined by
g i (x) := max g(kx) .
k∈ker(m i )
□

Sec. 3] PERTURBING ALMOST SEMIAMPLE METRICS 203
3.8. Proposition –––– Let ε > 0 and U ⊆ G() an open set containing K.
Then, there is some non-negative K-invariant function 0 ≠ g ∈Ê+·S such that
i) g(e) = 1,
ii) max x∈P() g(x) ≤ 1 + ε, and
iii) supp g ⊆ U.
Proof. Put D := P \G. For some j ≫ 0, the coherent sheaf L ⊗j ⊗I D has a
section which does not vanish in e ∈ G. I.e., there is a section s ∈ Γ(P, L ⊗j )
vanishing in D but not in e.
Using Corollary 3.6, we see
1
}
˜g C := max{
2j log ‖Cs‖j , 0
{ 1
= max
2j log ‖s‖j + logC }
, 0 ∈ S
2j
for every C > 0. It is clear that, for each C, supp ˜g C ⊆ G() is a compact set.
We put
and
A := max
x∈K
1
2j log ‖s‖j (x)
1
B := max
x∈P() 2j log ‖s‖j (x) .
Then, we choose C such that log C + A > 0 and
logC + B
log C + A ≤ 1 + ε .
Further, let g 0 be the K-invariant function associated to ˜g C . I.e.,
{
max ˜g C (kx) if x ∈ G() ,
g 0 := k∈K
g(x) otherwise .
By construction, g 0 (e) > 0 and
max g 0 (x)/g 0 (e) ≤ 1 + ε .
x∈P()
Further, supp g 0 ⊆ K ·supp ˜g C is compact. Lemma 3.7 shows that g 0 ∈ S.
Finally, we define a sequence (g i ) i of functions on P() by putting, recursively,
g i+1 := 1 d f ∗ g i . Then, one clearly has g i (e) = g 0 (e) > 0 and
max g i(x)/g i (e) = max g 0(x)/g 0 (e) ≤ 1 + ε
x∈P() x∈P()
for every i ∈Æ. Lemma 3.3.c) implies that g i ∈ S.
As supp g 0 is compact, there is some N ∈Êsuch that supp g 0 ⊆ U N (K ).
This yields, by Lemma 2.8.b), that supp g i ⊆ U Nq i(K ).
Therefore, the function g, given by g(x) := g i (x)/g i (e) for some i ≫ 0, has all
the properties desired.
□

204 DISTRIBUTION OF SMALL POINTS [Chap. V
4. Equidistribution
4.1. Definition. –––– Let X be a projective variety overÉ. Then, a sequence
(x i ) i of closed points on X is called generic if no infinite subsequence is
contained in a proper closed subvariety of X.
4.2. Remark. –––– In other words, (x i ) i is generic if it converges to the
generic point with respect to the Zariski topology.
4.3. Definition. –––– Let X be a projective variety overÉand L ∈ ˜Pic(X ).
Suppose that e dim X+1 (h L
) = 0.
Then, a sequence (x i ) i of closed points on X is called small if h L
(x i ) → 0.
4.4. Lemma. –––– Let X be a projective variety overÉand
L = (L , ‖ . ‖) ∈ ˜Pic(X )
be such that ‖ . ‖ p is semipositive for every prime number p and ‖ . ‖ ∞ is almost semiample.
Assume e dim X+1 (L ) = 0.
Then, pZ X (L , . . . , L ) = 0 is equivalent to the existence of a sequence of closed
points on X which is generic and small.
Proof. “=⇒” There are not more than countably many closed subvarieties
X 1 , X 2 , . . . ⊂ X. Further, Corollary IV.7.40 implies e 1 (h L
) = 0. Therefore,
we may choose a sequence (x i ) i of closed points on X such that
x i ∈ X \X 1 \X 2 \ . . . \X i
and h L
(x i ) < 1 i . (x i) i is generic and small.
“⇐=” Let (x i ) i be a sequence of closed points on X which is generic and small.
Let Y ⊂ X be a closed subset of codimension one. Then, only finitely many
of the x i are contained in Y. Hence, inf x∈X \Y h L
(x) = 0. As this is true for
every Y, we see e 1 (h L
) = 0. Corollary IV.7.40 yields pZ X (L , . . . , L ) = 0.
□
4.5. Lemma. –––– Let P, f , L = (L ⊗m , ‖ . ‖ ∼ ), and Φ be as described
in 1.13. Denote by S the set of all continuous functions g ∈ C (P()) such that
‖ . ‖ ∞· exp(−g) is almost semiample.
Let ϕ ∈ C (P()). Then, for every ε > 0, there exist a function ϕ 1 ∈ C (P())
supported in P()\K and a function ϕ 2 ∈Ê+·S such that
‖ϕ − ϕ 1 − ϕ 2 ‖ max < ε .

Sec. 4] EQUIDISTRIBUTION 205
Proof. The set
T := {h ∈ C (K ) | h = ϕ| K for some ϕ ∈Ê+·S}
fulfills all the assumptions of Sublemma 4.6, except for closedness under uniform
convergence.
Indeed, Lemma 3.3.a) and b) imply that assumptions i) and ii) are satisfied.
Corollary 3.6 is general enough to guarantee assumption iii). Lemma 3.3.d)
yields assumption iv). Finally, Corollary 3.5 implies that assumption v) is fulfilled.
Therefore, there is some ϕ 2 ∈Ê+·S such that |ϕ(x)−ϕ 2 (x)| < ε for every x ∈ K.
2
A decomposition of the unit adapted to a suitable open covering {P()\K, U }
of P() yields some ϕ 1 ∈ C (P()) such that supp ϕ 1 ⊆ P()\K and
∣ ( ϕ(x) − ϕ 2 (x) ) − ϕ 1 (x) ∣ < ε
for every x ∈ P().
□
4.6. Sublemma. –––– Let L be a compact group and T ⊆ C (L) a set of continuous
real-valued functions such that the following conditions are fulfilled.
i) T contains all the constant functions.
ii) T is closed under addition and multiplication by positive constants.
iii) For each x ∈ L such that x ≠ e, there is some g ∈ T fulfilling
g(e) > g(x) .
iv) For each x ∈ L and g ∈ T , the shift g · x is also in T .
v) If g ∈ T then g + ∈ T for
g + (x) := max {g(x), 0} .
vi) T is closed under uniform convergence.
Then, T = C (L) contains all continuous functions.
Proof. We fix a Haar measure ρ on L. Then, the conditions ii,) iv) and vi)
together imply the following.
(†) If h ∈ T and g is a non-negative, measurable, and bounded function then,
for the convolution, we have g∗h ∈ T .

206 DISTRIBUTION OF SMALL POINTS [Chap. V
Hence, it suffices to show that, for each open set U ⊆ L containing e, there
is some non-negative function g U ∈ T such that g U ≠ 0 and supp g U ⊆ U.
For this, by i), ii), and v), we only need a function g ∈ T having its maximum
entirely in U.
Assume, for some open U 0 , there would be no such function. Then, for
every g ∈ T having its maximum in e, there is a non-empty set
A g := {y ∈ L\U 0 | g(y) = g(e)} ⊆ L\U 0
where this maximum is taken, too. If
\
g∈T
A g = ∅
then, by compactness, already a finite intersection A g1 ∩ . . . ∩ A gn would
be empty. Then, g 1 + . . . + g n ∈ S had its maximum in U 0 , only.
Consequently,
is a non-empty set. Put
A U0 := \
g∈T
A := {e} ∪ [
A g
U ∋e,U open
By construction, this set has the property below.
A U .
(‡) Every function g ∈ T assuming its maximum in e is necessarily constant
on A. Further, A ⊆ L is the largest set containing e with this property.
The property (‡) implies that A is a closed set.
For y ∈ A and g ∈ T , the shift g·y −1 has its maxima in yA. Therefore, A = yA
for every y ∈ A. This means that A ⊆ L is a subgroup.
Fix a Haar measure ρ A on A and denote by i : A → L the natural inclusion.
Property (‡) together with i), ii) and v) implies that there is a sequence of
functions of T such that the associated distributions converge weakly to i ∗ ρ A .
Then, property (†) guarantees together with vi) that every A-left invariant
continuous function is an element of T .
Now, we use condition iii). Choose some x 0 ∈ A different from e. There exists
some g ∈ T such that g(e) > g(x 0 ). Shifting by an element of A, if necessary,
we may assume that g(e) = max x∈A g(x).
Define another continuous function M on L by
M (x) := − max
a∈A g(ax) .

Sec. 4] EQUIDISTRIBUTION 207
Obviously, M is A-invariant. Thus, M ∈ T . Consequently, ˜g := g + M ∈ T ,
too.
However, ˜g(x) ≤ 0 for every x ∈ L, ˜g(e) = 0, and ˜g(x 0 ) < 0 together form a
contradiction to property (‡).
□
4.7. Proposition. –––– Let P, f , L , and Φ be as described in 1.13.
Let (x i ) i be a sequence of closed points on P which is generic and small. Assume, some
non-negative ϕ ∈ C (P()) fulfills either supp ϕ ⊆ P()\K or ϕ ∈Ê+·S.
Then,
lim inf
i→∞
Z
P()
ϕ dδ xi ≥
Z
P()
ϕ dτ .
Here, τ is the zero measure on P() \K and the Haar measure of volume one on K.
Proof. For ϕ such that supp ϕ ⊆ P() \K, the right hand side is zero. Thus, in
this case, the assertion is clear.
Let ϕ ∈Ê+·S. Then, for any positive λ ∈Ê, let ‖ . ‖ ∼ λ be the adelic metric
on L ⊗m given by ‖ . ‖ λ p := ‖ . ‖ p for the non-archimedean valuations and
by ‖ . ‖ ∞ λ := ‖ . ‖ ∞ · exp(−λϕ) for the archimedean valuation.
For λ → 0, the adelic metric ‖ . ‖ λ fulfills the assumptions of Corollary IV.7.40.
Further, one clearly has
Z
h (L ⊗m ,‖ .‖ λ )(x i ) = h L
(x i ) + λ ϕ dδ xi .
As (x i ) i is generic and h L
(x i ) → 0, we obtain, according to the very definition
of e 1 ,
Z
λ·liminf ϕ dδ xi ≥ e 1 (h (L ⊗m ,‖.‖ λ )) . (§)
On the other hand,
=
1
(dim X + 1)c 1 (L ⊗m
i→∞
X ()
É)
dim X pX Z
1
(dim X + 1)m dimX deg L
X pX Z (L , . . . , L )
X ()
(
(L ⊗m , ‖ . ‖ λ ∼ ), . . . , (L ⊗m , ‖ . ‖ λ ∼ ))
1
+
m dim X deg L
X ·
(
(OX , ‖ . ‖ ∞· exp(−λϕ)), (L ⊗m , ‖ . ‖ ∼ ), . . . , (L ⊗m , ‖ . ‖ ∼ ) )
+ O(λ 2 ) .
p X Z

208 DISTRIBUTION OF SMALL POINTS [Chap. V
By virtue of Lemma 4.4, we have p X Z (L , . . . , L ) = 0. Further,
1 (
m dimX deg L
X ·pX Z (OX , ‖ . ‖ ∞· exp(−λϕ)), (L ⊗m , ‖ . ‖ ∼ ), . . . , (L ⊗m , ‖ . ‖ ∼ ) )
= λ lim
i→∞
I i (ϕ)
for
I i (ϕ) := 1 ( )
m dim X pX Z (OX , ‖ . ‖ ∞· exp(−ϕ)), (L ⊗m , ‖ . ‖ ∼ (i) ), . . . , (L ⊗m , ‖ . ‖ ∼ (i) ) .
} {{ }
dim X times
At this level, we can make the arithmetic intersection product explicit and find
I i (ϕ) =
Z
P()
ϕc 1 (L , ‖ . ‖ (i)
∞) ∧ . . . ∧ c 1 (L , ‖ . ‖ (i)
∞) .
Corollary IV.7.40 therefore yields in view of Proposition 2.12
Z
e 1 (h (L ⊗m ,‖.‖ λ )) ≥ λ· ϕ dτ . ()
P()
The equations (§) and () together imply the assertion.
4.8. Remark. –––– The argument for the case ϕ ∈Ê+·S is due to E. Ullmo.
Cf. [S/U/Z] or [Zh98].
4.9. Theorem (Equidistribution on P()). —– Let P, f , L , and Φ be as described
in 1.13.
Then, for each sequence (x i ) i of closed points on P which is generic and small, the
associated sequence of measures (δ xi ) i converges weakly to the measure τ which is the
zero measure on P()\K and the Haar measure of volume one on K.
Proof. First step. δ xi | P()\K → 0.
By Proposition 3.8, for every ε 1 , ε 2 > 0, there is some non-negative g ε1 ,ε 2
∈ S
such that supp g ε1 ,ε 2
⊆ U ε1 (K ),
□
and
Then, by Proposition 4.7,
max
x∈P() g ε 1 ,ε 2
≤ 1 + ε 2 ,
Z
P()
Z
lim inf
i→∞
P()
g ε1 ,ε 2
dτ = 1 .
g ε1 ,ε 2
dδ xi ≥ 1 .

Sec. 4] EQUIDISTRIBUTION 209
This yields
But
Consequently,
Z
lim sup
i→∞
1 − g ε1 ,ε 2
(x) ≥
P()
(1 − g ε1 ,ε 2
) dδ xi ≤ 0 .
{ 1 if x ∈ P()\U ε1 (K ) ,
−ε 2 if x ∈ U ε1 (K ) .
This shows
lim sup
i→∞
[
δxi
(
P()\U ε1 (K ) ) − ε 2 δ xi
(
Uε1 (K ) )] ≤ 0 .
lim sup
i→∞ δ x i
(
P()\U ε1 (K ) ) ≤ ε 2 .
As the latter is true for every ε 2 > 0, we have
lim δ (
x i P()\U ε1 (K ) ) = 0 .
i→∞
Further, this formula is true for each ε 1 > 0. Hence, (δ xi | P()\K ) i converges
weakly to the zero measure.
Second step. The assertion in general.
Let ϕ ∈ C (P()) be an arbitrary continuous function. As we can interchange
the roles of ϕ and (−ϕ), it suffices to prove
Z
Z
lim inf ϕ dδ xi ≥ ϕ dτ.
i→∞
P()
By Lemma 4.5, we have continuous functions ϕ 1 and ϕ 2 on P() such that
supp ϕ 1 ⊆ P()\K and ϕ 2 ∈Ê+·S.
The result of the first step shows
lim
Z
i→∞
P()
i→∞
P()
P()
ϕ 1 dδ xi = 0 .
Further, by Proposition 4.7,
Z
lim inf ϕ 2 dδ xi ≥
Consequently,
Z
lim inf ϕdδ xi ≥
i→∞
P()
Z
P()
The assertion follows since ε > 0 is arbitrary.
Z
P()
ϕ 2 dτ .
ϕdτ − ε .
4.10. Corollary. –––– Let X ⊂ P be a closed subvariety and (x i ) i a sequence of
closed points on X which is generic and small.
□

210 DISTRIBUTION OF SMALL POINTS [Chap. V
Then, the sequence (δ xi | X ()\K) i converges weakly to the zero measure.
Proof. As there are only finitely many i such that x i ∉ G, let us assume that
the whole sequence (x i ) i is contained in G. Then, we choose some K-invariant
metric on G() and assume that, for some ε > 0, we would have
lim inf
i→∞ δ x i
(
X ()\U ε (K ) ) = δ > 0 .
We will construct another sequence (y i ) i which is generic and small on the
whole of P.
For this, note first that we have
h L
(x) = 1 ( )
h
deg f L f (x) .
Consequently, h L
is invariant under shift by any torsion point
t ∈ K tor =
j∈Æker(m [ j ) .
Further, all the sequences (t·x i ) i fulfill
lim inf
i→∞ δ t·x i
(
P()\U ε (K ) ) = δ .
Finally, K ⊆ G() is assumed to be Zariski dense. Therefore, for each i, the
union S {t·x i } is a Zariski dense subset of P.
t∈K tor
Finally, we make use of the fact there are only countably many proper closed
subvarieties P 0 , P 1 , P 2 , . . . ⊂ P. We can choose a sequence (y i ) i such that, for
every i ∈Æ, one has
y i ∈ [
{t·x i }
and
t∈K tor ,i≥0
y i ∈ P \P 0 \ . . . \P i .
Then,
δ yi
(
P()\U ε (K ) ) > δ/2
for i ≫ 0 and h L
(y i ) → 0. This is a contradiction to Theorem 4.9.
□
4.11. Remark. –––– This corollary shows, in particular, the following.
If X ∩ K = ∅ then
p X Z (L , . . . , L ) > 0 .
In other words, there are no small and generic sequences on X.

CHAPTER VI
CONJECTURES ON THE ASYMPTOTICS OF
POINTS OF BOUNDED HEIGHT
I have no satisfaction in formulas unless I feel their numerical magnitude.
WILLIAM THOMSON 1ST BARON KELVIN,
(Life (1943) by Sylvanus Thompson, p. 827)
1. A heuristic
1.1. –––– Let X be a projective variety overÉ. Then, one of the most natural
questions to ask is
(∗) Does X have finitely many or infinitely manyÉ-rational points?
1.2. –––– If the answer is that #X (É) < ∞ then one might ask for the precise
number. Otherwise, ifX(É) is infinite then one may ask for the asymptotics
of the rational points with respect to a certain height function H on X (É).
(†) What is the asymptotics of the function N X,H given by
for B → ∞?
N X,H (B) := #{x ∈ X (É) | H(x) < B}
1.3. –––– If X is a hypersurface of degree d in P n then there is a statistical
heuristic for the asymptotics of N X,Hnaive .
1.4. Statistical heuristic. –––– Let X be a hypersurface of degree d in P nÉ.
Then,
for some positive constant C.
N X,Hnaive (B) ∼ C ·B n+1−d
“Proof.” On P n , the total number ofÉ-rational points of height < B is ∼B n+1 .
Indeed, these points may be given in the form
(x 0 : . . . : x n )

212 CONJECTURES ON POINTS OF BOUNDED HEIGHT [Chap. VI
for x i ∈such that x i ∈ [−B, B]. There are ∼ B n+1 such (n + 1)-tuples and
the probability that gcd(x 0 , . . . , x n ) = 1 is positive.
Further, X is given by a homogeneous form F of degree d. Its range of values
on [−B, B] n+1 is, a priori, [−DB d , DB d ] for a certain positive constant D.
We assume that the values of F are equally distributed in that range. The value
zero is then hit ∼B n+1−d times.
□
1.5. Remark. –––– This heuristic is definitely an interesting guideline about
which behaviour to expect. It is, however, too naive in oder to work well in
all cases.
1.6. Example (Too few points – p-adic unsolvability). —–
Consider the cubic fourfold X in P 5Égiven by the equation
x 3 + 7y 3 + 49z 3 + 2u 3 + 14v 3 + 98w 3 = 0 .
Then, X (É7) = ∅ which implies X (É) = ∅.
Note that the statistical heuristic would predict cubic growth for the number
ofÉ-rational points x on X such that H naive (x) < B.
1.7. Example (Too few points – The Brauer-Manin obstruction). —–
Consider the cubic surface X in P 3Égiven by the equation
Here,
w(x + w)(12x + w) =
3
∏
i=1
(
x + θ (i) y + (θ (i) ) 2 z ) .
θ := −38 + ∑
i∈(∗ 19 ) 3 ζ i 19 ∈É(ζ 19 )
and θ (i) are the three conjugates of θ inÉ.
As shown in Example III.5.24, X (É) ≠ ∅ but X (É) = ∅. On X, there is a
Brauer-Manin obstruction to the Hasse principle.
In this case, the statistical heuristic would predict linear growth for the number
ofÉ-rational points x on X such that H naive (x) < B.
1.8. Examples (Too many points – Accumulating subvarieties). —–
i) Consider the smooth threefold X ⊂ P 4Égiven by
x 4 + y 4 = z 4 + v 4 + w 4 .
Then, the statistical heuristic predicts linear growth. I.e., there should be ∼CB
rational points of height

Sec. 1] A HEURISTIC 213
ii) Other examples of interest are provided by the diagonal cubic threefolds
V a,1 ∈ P 4Éof the form
V a,1 : ax 3 = y 3 + z 3 + v 3 + w 3 .
Here, quadratic growth is predicted by the statistical heuristic.
However, V a,1 clearly contains the lines given by v = −w and
(x : y : z) = (x 0 : y 0 : z 0 )
for (x 0 : y 0 : z 0 ) aÉ-rational point on the curve
E a : ax 3 = y 3 + z 3 .
This is a twisted Fermat cubic. One has the rational point (0 : 1 : (−1)) ∈ E a (É).
Therefore, E a is an elliptic curve overÉ. The lines described form a cone over
an elliptic curve. There are ∼B 2É-rational points of height < B alone on
this cone.
Examples of twisted Fermat cubics of rank up to 3 have been known to
Ernst S. Selmer as early as 1951 [Sel]. For example, E 6 (É) is of rank one,
generated by (21 : 17 : 37).
1.9. Remark. –––– These examples grew out of our study of diagonal cubic and
quartic threefolds which is described in Chapter VII.
1.10. Geometric interpretation. –––– The exponent n + 1 − d appearing in
the statistical heuristic may be positive, zero, or negative. According to this
distinction, there are three cases.
This distinction into three cases coincides perfectly well with the Kodaira classification
of projective varieties into Fano varieties, varieties of intermediate type,
and varieties of general type. Indeed, the anticanonical divisor (−K ) on a
degree d hypersurface in P n is exactly O(n + 1 − d).
1.11. Remark. –––– The Kodaira classification is a purely geometric one.
It does not make use of any arithmetic information on the projective variety
considered. Only the complex algebraic variety produced by base change
tois exploited.
It is a very remarkable observation that whether a projective variety X is Fano,
of intermediate type, or of general type seemingly has a lot of influence on the
set ofÉ-rational points on X.

214 CONJECTURES ON POINTS OF BOUNDED HEIGHT [Chap. VI
1.12. –––– More concretely, expectations are as follows.
First case. X is a variety of general type.
By definition, this means that the canonical invertible sheaf K is ample.
In the case of a hypersurface, this corresponds to the case that n + 1 − d < 0.
Here, the statistical heuristic is rather illogical. It states that for a large height
bound we expect less points than for a small one. In the limit for B → ∞, we
have B n+1−d → 0.
One would therefore expect only very fewÉ-rational points on X. This is
exactly the content of the conjecture of Lang.
Second case. X is a variety of intermediate type.
In the case of a hypersurface, this corresponds to the case that n + 1 − d = 0.
Here, the statistical heuristic states that there should be a constant number of
points, independently of the height bound B.
One would therefore expect that there are a fewÉ-rational points on X. A more
precise statement is given by the conjecture of Batyrev and Manin.
Third case. X is a Fano variety.
By definition, this means that the anticanonical invertible sheaf K ∨ is ample.
In the case of a hypersurface, this corresponds to the case that n + 1 − d > 0.
Here, one would expect that there are a lot ofÉ-rational points on X. A by far
more precise formulation is given by the conjecture of Manin.
2. The conjecture of Lang
2.1. –––– The conjecture of Langdeals with the case of a variety of general type.
There are actually several versions of it [Lan86].
2.2. Conjecture (Lang). —– Let X be a smooth, projective variety of general type
over a number field K. Then,
i) (Weak Lang conjecture)
the set of rational points X (K ) is not Zariski dense in X.
ii) (Strong Lang conjecture)
Thereis aZariski closed subset Z ⊂ X such that, for any finite field extension L ⊇ K,
one has that X (L)\Z(L) is finite.
2.3. Conjecture (Geometric Lang conjecture). —–
LetX beasmooth, projective variety of generaltype over a field K of characteristic 0.
Then, there is a proper Zariski closed subset Z(X ) ⊂ X, called the Langian exceptional
set, which is the union of all positive dimensional subvarieties which are not
of general type.

Sec. 2] THE CONJECTURE OF LANG 215
2.4. Remark. –––– These conjectures are strongly interrelated. If the geometric
Lang conjecture is true then the weak Lang conjecture implies the strong
Lang conjecture.
2.5. –––– Lang’s conjectures are known to be true when X is a subvariety of
an abelian variety. This was proven by G. Faltings in [Fa91].
Faltings’ 1991 result includes the case that X is a curve of general type. I.e., a
curve of genus g ≥ 2. This particular case of Lang’s conjecture has been popular
for decades as the Mordell conjecture. The Mordell conjecture was proven
by G. Faltings in 1983 [Fa83].
2.6. Examples. –––– If X is a curve then there is no Langian exceptional set Z.
Weak and strong Lang conjecture are therefore equivalent.
This is no longer the case for surfaces of general type.
i) For example, the quintic surface given by
x 5 + y 5 + z 5 + w 5 = 0
in P 3Écontains the line “x = −y, z = −w” on which there are infinitely many
É-rational points.
ii) Another example of a surface of general type containing a projective line is
provided by the Godeaux surface [Bv, Example X.3.4].
iii) Let V 3
a,b be the cubic threefold in P4Égiven by
ax 3 = by 3 + z 3 + v 3 + w 3 .
Then, the moduli space L a,b of the lines onV a,b is a surface of general type [Cl/G,
Lemma 10.13].
If the cubic curve
E a,b : ax 3 + by 3 + z 3 = 0
contains a rational point then, on V a,b , there are the lines given by v = −w
and (x : y : z) = (x 0 : y 0 : z 0 ) for (x 0 : y 0 : z 0 ) ∈ E a,b (É). We call these lines the
obvious lines on V a,b .
Inother words, ifE a,b (É) ≠ ∅thenthesurfaceL a,b , whichisofgeneraltype, containsacopyoftheelliptic
curveE a,b . ThereareinfinitelymanyÉ-rational points
on E a,b , for example for a = 6 and b = 1.
2.7. An experiment. –––– We searched systematically forÉ-rational lines on
the cubic threefolds V a,b for a, b = 1, . . . , 100, a ≥ b. Our method is
described in detail in Chapter VII, Section 4. It guarantees that every line which
contains a point of height

216 CONJECTURES ON POINTS OF BOUNDED HEIGHT [Chap. VI
The results may be interpreted as providing numerical evidence for Lang’s conjecture.
Indeed, aÉ-rational point on L a,b corresponds to aÉ-rational line
on V a,b . Points on the elliptic curves lying on L a,b correspond to the obvious
lines.
In agreement with Lang’s conjecture, only very few non-obvious lines
were found. On all the varieties V a,b for a, b = 1, . . . , 100, a ≥ b together,
there are only 42 non-obvious lines containing a point of height 0, there exists a Zariski open subset X ◦ ⊆ X such that
for B → ∞.
N X ◦ ,H L
(B) = #{x ∈ X ◦ (k) | H L (x) < B} ≪ B r+ε
3.2. –––– Here, H L is the exponential of the height h L defined by L as
introduced in Definition IV.2.10. It is determined only up to a factor which is
bounded above and below by positive constants.
The conjecture is consistent with these changes of the height function since B r+ε
and (CB) r+ε differ by a constant factor.

Sec. 3] THE CONJECTURE OF BATYREV AND MANIN 217
3.3. Remark. –––– This conjecture was first formulated by V. V. Batyrev and
Yu. I. Manin in [B/M]. An excellent presentation may be found in the survey
article [Pe02] by E. Peyre.
3.4. Fact (Varieties of general type). —– The conjecture of Batyrev and Manin
implies the weak Lang conjecture.
Proof. Indeed, if X is a variety of general type then the canonical invertible
sheaf K itself is ample and we may work with L := K .
Then, K + (−1)L = 0 is an effectiveÊ-divisor. Hence, for every ε > 0, the
conjecture of Batyrev and Manin yields that
#{x ∈ X ◦ (k) | H L (x) < B} ≪ B −1+ε
for a suitable Zariski open subset X ◦ ⊂ X. In the limit for B → ∞, this shows
that X ◦ (k) is empty.
The set X (k) is therefore not Zariski dense in X.
3.5. –––– Let X be a smooth Fano variety. Then, the anticanonical invertible
sheaf K ∨ is ample and we may consider an anticanonical height which is defined
by L := K ∨ .
3.6. Fact (Fano varieties). —– For X a smooth Fano variety, the conjecture of
Batyrev and Manin yields that
N X ◦ ,H K ∨(B) = #{x ∈ X ◦ (k) | H K ∨(x) < B} ≪ B 1+ε
□
for a suitable Zariski open subset X ◦ ⊂ X.
Proof. In this situation, K + (−K ) = 0 is an effectiveÊ-divisor.
□
3.7. Remarks. –––– i) In the case of a Fano hypersurface, this assertion fits
perfectly well with the statistical heuristic.
ii) However, for Fano varieties, the conjecture of Manin describes the growth
of N X ◦ ,H K ∨
much more precisely. The most interesting case of the conjecture
of Batyrev and Manin is, therefore, that of a variety of intermediate type.
ii. Varieties of intermediate type. —
3.8. –––– Let X be a smooth, projective, minimal surface of Kodaira dimension
0. Fix an ample invertible sheaf L ∈ Pic(X ).
Then, the conjecture of Batyrev and Manin states that, for every ε > 0, there
exists a Zariski open subset X ◦ ⊂ X such that
N X ◦ ,H L
(B) = #{x ∈ X ◦ (k) | H L (x) < B} ≪ B ε .

)⊗Ê
218 CONJECTURES ON POINTS OF BOUNDED HEIGHT [Chap. VI
Indeed, 12K is linearly equivalent to zero in any of the four cases of the Kodaira
classification. Hence,
[K ] ∈ NS(X
is the class of an effectiveÊ-divisor on X.
3.9. Example. –––– If X is an abelian variety then the conjecture of Batyrev
and Manin is true.
Indeed, if the rank of X (k) is equal to r then
N X ◦ ,H L
(B) ∼ C ·log r/2 B .
3.10. Remark. –––– On the other hand, for K3 surfaces, the conjecture of
Batyrev and Manin is open.
For special K3 surfaces, most notably for Kummer surfaces such that the associated
abelian surface is a product of two elliptic curves, a particular result has
been obtained by D. McKinnon [McK]. If d is the minimal degree of a rational
curve on X then McKinnon shows N X ◦ ,H L
(B) ≪ B 2/d for X ◦ the complement
of the union of all rational curves on X of degree d.
3.11. Observation. –––– There are examples of K3 surfaces over a number
field k which contain infinitely many rational curves defined over k [Bill, Sec. 3].
Let X be such a K3 surface and let C 1 , C 2 , . . . be rational curves on X.
Denote their degrees by d 1 , d 2 , . . . . Assume that the curves are listed in such a
way that the degrees are in ascending order.
Then, on
X ◦
l := X \C 1 \ . . . \C l−1 ,
there are still ≥ c l B 2/d l
k-rational points to be expected. Indeed, X ◦
l contains a
non-empty, open subset of C l .
In other words, there is no way to choose a uniform Zariski open subset X ◦ ⊂ X
such that
N X ◦ ,H L
(B) = #{x ∈ X ◦ (k) | H L (x) < B} ≪ B ε
for every ε > 0. In order to fulfill the conjecture of Batyrev and Manin, one
therefore has to choose X ◦ in dependence of ε.
3.12. Example. –––– Consider the diagonal quartic surface X given in P 3Éby
the equation
x 4 + 2y 4 = z 4 + 4w 4 .

Sec. 3] THE CONJECTURE OF BATYREV AND MANIN 219
On this K3 surface, there are theÉ-rational points (1 : 0 : ±1 : 0) and
(±1 484 801 : ±1 203 120 : ±1 169 407 : ±1 157 520) .
These are the onlyÉ-rational points onX known and the onlyÉ-rational points
of (naive) height less than 10 8 [EJ2, EJ3].
A systematic search forÉ-rational points on the K3 surface X is described in
Chapter XI.
3.13. Remark. –––– In general, not much is known about the arithmetic
of K3 surfaces. Nevertheless, in 1981, F. Bogomolov formulated a very optimistic
conjecture.
3.14. Conjecture (F. Bogomolov, cf. [Bo/T]). —– Let X be a K3 surface over a
number field k. Then, every k-rational point on X lies on rational curve C ⊂ X
(defined over the algebraic closure k).
iii. A somewhat different example. —
3.15. Example. –––– Let X be the blow-up of P 2 in nine points P 1 , . . . , P 9 in
general position.
Denote by E := E 1 + . . . + E 9 the sum of the corresponding nine exceptional
lines. According to the Nakai-Moizhezon criterion, a divisor aL − bE is
ample if and only if a > 3b > 0. Further, we have K = −3L + E.
The conjecture of Batyrev and Manin therefore asserts that, for every ε > 0,
there exists a Zariski open subset X ◦ ⊂ X such that
N X ◦ ,H O(aL−bE)
(B) ≪ B 3/a+ε .
On the other hand, assuming aL − bE to be very ample for simplicity, the
embedding ι: (X
(
։)P
) 2 ֒→ P N is given by homogeneous forms of degree a.
Hence, H O(aL−bE) ι(x) ≤ c · H
a
naive(x) for some constant c which shows the
lower bound
N X ◦ ,H O(aL−bE)
(B) = Ω(B 3/a )
for any Zariski open subset X ◦ ⊂ X.
3.16. –––– On X, there are infinitely many exceptional curves D such that
D 2 = −1 and DK = −1. For every d ∈Æ, there are finitely many of the
type dL − a 1 E 1 − . . . − a 9 E 9 .

220 CONJECTURES ON POINTS OF BOUNDED HEIGHT [Chap. VI
Relative to aL − bE, their degrees are
(dL − a 1 E 1 − . . . − a 9 E 9 )(aL − bE) = da − b(a 1 + . . . + a 9 )
= da − b(3d − 1)
= d(a − 3b) + b .
Since a − 3b > 0, this expression tends to infinity for d → ∞. Only finitely
many of the exceptional curves are of degree < 2 a which is equivalent to ≫B3/a
3
rational points of height

Sec. 4] THE CONJECTURE OF MANIN 221
4.2. Lemma. –––– Let k ⊆be an algebraically closed field of characteristic 0
and X be a smooth, projective variety over k. Assume that X is Fano.
Then, for every prime number l,
i) H 1 ét (X,l) = 0.
ii) The first Chern class induces an isomorphism
∼=
Pic(X −→ H )⊗l
2 ét (X,l(1)) .
Proof. Let first i ∈Æbe arbitrary. By [SGA4, Exp. XVI, Corollaire
1.6], we have, for every k ∈Æ, H i ét (X, µ l k) ∼ = H i ét (X, µ l k). Further,
the comparison theorem [SGA4, Exp. XI, Théorème 4.4] shows
that H i ét (X, µ l k) ∼ = H i (X (),/l k).
On the other hand, by the universal coefficient theorem for cohomology [Sp,
Chap. 5, Sec. 5, Theorem 10],
H i (X (),/l k) ∼ = H i (X (),)⊗/l k⊕Tor 1 (H i+1 k
(X (),),/l k)
∼= H i (X (),)⊗/l k⊕H i+1 (X (),) l k .
Here, the transition maps H i+1 (X (),) l k+1 → H i+1 (X (),) l k are given by
multiplication by l. A finite composition of them is the zero map. This implies
H i ét (X,l(1)) ∼ = lim ←−
H i (X (),)⊗/l
= H i (X . (‡)
(),)⊗l
i) Since H 1 (X (), O X ()) = 0, the long exact cohomology sequence associated
to the exponential sequence yields that H 1 (X (),) = 0. Formula (‡) implies
the claim.
ii) Here, we use both, H 1 (X (), O X ()) = 0 and H 2 (X (), O X ()) = 0.
The long exact cohomology sequence associated to the exponential sequence
then shows that
c 1 : Pic(X ) −→ H 2 (X (),)
is an isomorphism. Tensoring yields isomorphisms
Pic(X )⊗/l k.
and going over to the inverse limit implies the assertion.
k−→ H 2 (X (),)⊗/l
□

222 CONJECTURES ON POINTS OF BOUNDED HEIGHT [Chap. VI
4.3. Remarks. –––– i) The first Chern class in étale cohomology is defined using
the Kummer sequence. Recall that there is the commutative diagram
0
2πi O X
exp O ∗ X
0
exp( 2πi
n ·)
0 µ n O ∗ X
exp( 1 n ·)
(·) n O ∗ X
showing that this agrees with the definition based on the exponential sequence.
ii) The tensor product does in general not commute with inverse limits. However,
for a finitely generated-module A,
k∼ = A⊗l
lim
←− A⊗/l
is the l-adic completion of A [Mat, Theorem 8.7].
=
0
ii. Fixing a particular anticanonical height. —
4.4. Remark. –––– The anticanonical height H K ∨ on X is determined only up
to a certain factor which is bounded above and below by positive constants.
To be able to make any statement on the value of the constant τ, we have to fix
a particular height function.
4.5. –––– For this, according to Definition IV.7.38, it is necessary to choose an
adelic metric ‖ . ‖ = {‖ . ‖ ν } ν∈Val(É) on K ∨ such that (K ∨ , ‖ . ‖) ∈ ˜Pic(X ).
For the remainder of this section, we fix such an adelic metric once and for all.
4.6. Definition. –––– Put
H K ∨(x) := exph (K ∨ ,‖.‖)(x)
for h (K ∨ ,‖ .‖) theabsoluteheightwithrespecttotheintegrablymetrizedinvertible
sheaf (K ∨ , ‖ . ‖) in the sense of Definition IV.7.38.
We will call this height function the anticanonical height defined by the adelic
metric given in 4.5.
4.7. –––– Most height functions occurring in practice are a lot simpler than
the general theory. For this reason, it is probably wise to recall an elementary
particular case.
Choose
i) a projective embedding ι: X → P NÉsuch that K ∨ ∼ = ι ∗ O(d) for some d ∈Æ,
and
ii) a continuous hermitian metric ‖ . ‖ ∞ on K.
∨

Sec. 4] THE CONJECTURE OF MANIN 223
Then, the topological closure of ι(X ) in P Nis an arithmetic variety X which
is a model of X over Spec.
Put
H K ∨(x) := exp h (O(d)|X ,‖ .‖ ∞ )(x)
for h (O(d)|X ,‖.‖ ∞ ) the height function with respect to the hermitian line bundle
(O(d)| X , ‖ . ‖ ∞ ) ∈ ̂Pic C 0 (X ) in the sense of Definition IV.3.12.
4.8. –––– This height function corresponds to the adelic metric on K ∨ which
is given by the following construction. (Cf. Example IV.7.6.)
i) ‖ . ‖ ∞ is part of the data given.
ii) For a prime number p, the metric ‖ . ‖ p is given as follows.
Let x ∈Ép. There is a unique extension x : SpecOÉp → X of x. Then, x∗ O(d)
is a projective O K -module of rank 1. Each l ∈ O(d)(x) induces a rational section
of x ∗ O(d). Put
‖l‖(x) := inf {| a| | a ∈ K, l ∈ a · x ∗ O(d)}.
4.9. Remarks. –––– i) To define the metric ‖ . ‖ p for a particular prime p, a
model L of X over Spec[ 1 ] for p ∤m is sufficient.
m
ii) For every adelic metric ‖ . ‖ = {‖ . ‖ ν } ν∈Val(É) on K ∨ , there exist a model X
of X over Spec[ 1 ] for a certain m ∈Æand an extension of K ∨ to X such
m
that ‖ . ‖ νp is induced by that model for every p ∤m.
iii. The conjecture. —
4.10. –––– The conjecture of Manin deals with the anticanonical height H K ∨
on a Fano variety.
4.11. Conjecture (Manin). —– Let X be a smooth, projective variety overÉ.
Assume that X is Fano.
Then, there exist a positive integer r, a real number τ, and a Zariski open subset
X ◦ ⊆ X such that
for B → ∞.
N X ◦ ,H K ∨(B) = #{x ∈ X ◦ (k) | H K ∨(x) < B} ∼ τB log r B
4.12. Remark. –––– The factor log r B is new in comparison with the statistical
heuristic given in 1.4. It is known for a long time that such a factor is, in general,
necessary. In fact, J. Franke, Yu. I. Manin, and Y. Tschinkel [F/M/T] showed
in 1989 that Manin’s conjecture becomes compatible with direct products of
Fano varieties only when a suitable log r B-factor is added.

224 CONJECTURES ON POINTS OF BOUNDED HEIGHT [Chap. VI
4.13. Remark. –––– At least in the case that X is a surface, it is expected
that r = rkPic(X ) − 1. There are, however, counterexamples to this formula
in dimension three [Ba/T].
On the other hand, in Chapter VII, we will present numerical evidence
for Manin’s conjecture for diagonal cubic and quartic threefolds. For those,
rkPic(X ) = 1 and our experiments indicate that r = 0.
4.14. –––– In [Pe95a], E. Peyre refined Manin’s conjecture by providing an
explicit description for the value of the coefficient τ. Peyre’s constant is a
product of several factors which we will subsequently explain.
5. Peyre’s constant
i. The factor α. —
5.1. Definition (Cf. [Pe95a, Définition 2.4]). —–
Let X be a projective algebraic variety overÉ. Choose an isomorphism
ι: Pic(X )/ Pic(X ) tors
∼ =
−→t .
Identify Pic(X )⊗ÊwithÊt according to ι.
Further, let Λ eff (X ) ⊂ Pic(X )⊗Ê=Êt be the cone generated by the effective
divisors. Consider the dual cone Λ ∨ eff (X ) ⊂ (Êt ) ∨ . Then,
α(X ) := t · vol {x ∈ Λ ∨ eff
| 〈x, −K 〉 ≤ 1 } .
Here, 〈 . , . 〉 denotes the tautological pairing 〈 . , . 〉: (Êt ) ∨ ×Êt →Ê. vol is the
ordinary Lebesgue measure on (Êt ) ∨ .
5.2. Remark. –––– In [Pe/T, Definition 2.5], α(X ) is defined by an integral.
An elementary calculation shows that the two definitions are equivalent.
5.3. Example. –––– Suppose Pic(X ) =. Denote by [L] ∈ Pic(X ) the ample
generator. Let δ ∈Æbe such that [−K ] = δ[L]. Then, α(X ) = 1/δ.
In particular, one has α(X ) = 1 for every smooth cubic surface such that
rkPic(X ) = 1. Indeed, (−K ) 2 = 3 is square-free. Therefore, [−K ] ∈ Pic(X ) is
not divisible.

Sec. 5] PEYRE’S CONSTANT 225
5.4. Example. –––– α(P 1 × P 1 ) = 1/4.
Indeed, Pic(P 1 ×P 1 ) =L 1 ⊕L 2 . The effective cone is generated by L 1 and L 2 .
In the dual space,
Λ ∨ eff (P1 × P 1 ) = {al 1 + bl 2 | a, b ≥ 0} .
Further, −K = 2L 1 + 2L 2 . The condition 〈x, −K 〉 ≤ 1 is therefore equivalent
to 2a + 2b ≤ 1.
The area of the triangle with vertices (0, 0), (1/2, 0), and (0, 1/2) is equal to 1/8.
5.5. Example. –––– Let X be the blow-up of P 2 in aÉ-rational point.
Then, α(X ) = 1/6.
Here, Pic(X ) =L ⊕E. The effective cone is generated by E and L − E.
In the dual space,
Λ ∨ eff (X ) = {al + be | b ≥ 0, a − b ≥ 0} .
Further, −K = 3L−E. The condition 〈x, −K 〉 ≤ 1 is equivalent to 3a−b ≤ 1.
The area of the triangle with vertices (0, 0), (1/3, 0), and (1/2, 1/2) is equal
to 1/12.
5.6. Example. –––– Let X be the blow-up of P 2 in sixÉ-rational points which
form an orbit under Gal(É/É). Then, α(X ) = 4/3.
Again, Pic(X ) =L⊕E. Here, the effective cone is generated by E, L−1/3E,
and 2L − 5/6E. I.e., by E and L − 5/12E.
In the dual space,
Λ ∨ eff (X ) = {al + be | b ≥ 0, a − 5/12b ≥ 0} .
Further, −K = 3L − E. The condition 〈x, −K 〉 ≤ 1 is therefore equivalent
to 3a − b ≤ 1.
The area of the triangle with vertices (0, 0), (1/3, 0), and (5/3, 4) is equal to 2/3.
5.7. Example. –––– Let X be a smooth cubic surface overÉ. Assume the
orbit lengths of the 27 lines under the Gal(É/É)-operation are [1, 10, 16].
Then, α(X ) = 1.
Note that this is the generic case of a cubic surface containing aÉ-rational line.
The computation using GAP discussed in II.8.23 shows that rk Pic(X ) = 2.
Compare the list given in the appendix. We claim that Pic(X ) =K ⊕E
for E theÉ-rational line.

∈
226 CONJECTURES ON POINTS OF BOUNDED HEIGHT [Chap. VI
Indeed, K and E are linearly independent since K 2 = 3 and E 2 = −1. Further,
if aK + bE ∈ Pic(X ) then intersecting with K shows that 3a − b
while intersecting with a line skew to E shows −a ∈. Altogether, a, b ∈.
Write D 1 for the sum of the ten lines meeting E and D 2 for the sum of the sixteen
lines skew to E. As D 1 K = −10 and D 1 E = 10, we have D 1 = −5K − 5E.
Similarly, D 2 K = −16 and D 2 E = 0 imply D 2 = −4K + 4E.
The effective cone is generated by E, D 1 , and D 2 . The calculations show that E
and −K − E form a simpler system of generators.
In the dual space,
Λ ∨ eff (X ) = {ak + be | b ≥ 0, −a − b ≥ 0} .
Further, the condition 〈x, −K 〉 ≤ 1 is equivalent to −a ≤ 1.
The area of the triangle with vertices (0, 0), (−1, 0), and (−1, 1) is equal to 1/2.
5.8. Example. –––– Let X be a smooth cubic surface overÉ. Assume the
orbit lengths of the 27 lines under the Gal(É/É)-operation are [2, 5, 10, 10].
Then, α(X ) = 3/2.
Note that the two lines conjugate to each other are skew. Indeed, if they
were contained in a plane then the third line contained in that plane would be
É-rational. This example describes the generic case of a cubic surface containing
two skew lines which are conjugate to each other over a quadratic number field.
Again, the computation usingGAP discussed in II.8.23 shows that rkPic(X ) = 2.
We claim that Pic(X ) =K ⊕E for E := E 1 + E 2 the sum of the two lines
conjugate to each other.
Indeed, the discriminant of the lattice spanned by K and E is
−2 −2
∣ −2 3∣ = −10 .
As this number is non-zero, we see that K and E are linearly independent.
Since (−10) is a square-free integer, the lattice may not be refined.
Write D 1 for the sum of the five lines meeting E 1 and E 2 , D 2 for the sum of
the ten other lines meeting E 1 or E 2 , and D 3 for the sum of the ten lines not
meeting E at all. Then, D 1 K = −5 and D 1 E = 10 imply that D 1 = −3K −2E.
Similarly, D 2 K = −10 and D 2 E = 10 show that D 2 = −4K − E. Finally,
D 3 K = −10, D 3 E = 0, and D 3 = −2K + 2E.
The effective cone is generated by E, D 1 , D 2 , and D 3 . The calculations yield E
and −3K − 2E as a simpler system of generators.
In the dual space,
Λ ∨ eff (X ) = {ak + be | b ≥ 0, −3a − 2b ≥ 0} .

Sec. 5] PEYRE’S CONSTANT 227
Further, the condition 〈x, −K 〉 ≤ 1 is equivalent to −a ≤ 1.
The area of the triangle with vertices (0, 0), (−1, 0), and (−1, 3/2) is 3/4.
5.9. Remarks. –––– i) Let X be a smooth cubic surface overÉcontaining a
É-rational line. Assume that rk Pic(X ) = 2. Then, α(X ) = 1.
Indeed, the arguments given in Example 5.7 still show that Pic(X ) =K ⊕E.
On the other hand, the Gal(É/É)-orbits of the 27 lines might break into
smaller pieces. For example, assume there is an orbit consisting of k of the
ten lines meeting E. This causes another generator D 1 ′ of the effective cone.
However, we have that D 1 ′ K = −k and D 1 ′ E = k. Therefore,
is simply a scalar multiple of D 1 .
D ′ 1 = − k 2 K − k 2 E = k 10 D 1
Analogously, an orbit consisting of some of the sixteen lines skew to E leads to
a scalar multiple of D 2 . This ensures that Λ ∨ eff
(X ) is the same as in the generic
case described in Example 5.7.
ii) Similarly, let X be any smooth cubic surface overÉcontaining two skew
lines conjugate over a quadratic number field. Assume that rk Pic(X ) = 2.
Then, α(X ) = 3/2.
5.10. Remark. –––– For smooth cubic surfaces in general, the value of α depends
only on the orbit structure of the 27 lines under the Gal(É/É)-operation.
In particular, there are only 350 cases corresponding to the conjugacy classes of
subgroups of W (E 6 ). The examples given above actually cover the lion’s share
of these cases.
In fact, 137 of the 350 conjugacy classes of subgroups of W (E 6 ) lead to Picard
rank one. Then, α(X ) = 1 as shown in Example 5.3.
Further, 133 conjugacy classes yield Picard rank two. The argument of Remark
5.9.i) alone covers 98 of them. Eight further conjugacy classes are treated
by Remark 5.9.ii).
This might indicate that α(X ) may be effectively computed for each of the
350 conjugacy classes of subgroups of W (E 6 ) which could appear as the Galois
groups acting on the 27 lines. This has, however, not yet been done.
In the cases of high Picard rank, the computation of the volume of the resulting
polytope should be the main challenge. For example, László Lovász [Lo]
explains that good algorithms for the exact computation of such volumes are impossible.
His suggestion is to use a Monte-Carlo method instead.
The case of maximal rank has, however, been settled.

228 CONJECTURES ON POINTS OF BOUNDED HEIGHT [Chap. VI
5.11. Example. –––– Let X be a smooth cubic surface over the rationals such
that rk Pic(X ) = 7. I.e., such that each of the 27 lines on X is defined overÉ.
Then, α(X ) = 1
120 .
This has been proven in the Ph.D. thesis of U. Derenthal [Der, Theorem 8.3].
ii. The factor β. —
5.12. Definition. –––– Let X be a projective variety overÉ. Then, β(X ) is
defined as
β(X ) := #H 1( Gal(É/É), Pic(XÉ) ) .
5.13. Remarks. –––– We studied H 1( Gal(É/É), Pic(XÉ) ) in Chapter II. Recall
the following facts.
i) If Pic(XÉ) =then β(X ) = 1.
ii) In the particular case that X is a smooth cubic surface, β(X ) depends only
on the conjugacy class of the subgroup G ⊆ W (E 6 ) defined by the operation
of Gal(É/É) on the 27 lines. Actually, β(X ) depends only on the decomposition
of the 27 lines into Galois orbits.
We computed β(X ) for each of the 350 conjugacy classes of subgroups of W (E 6 ).
The possible values are β(X ) = 1, 2, 3, 4, and 9. More details are given in II.8.23
and the list presented in the appendix.
iii. The value of L at 1. —
5.14. Lemma. –––– Let X be a projective algebraic variety overÉ. Denote by
L( . , χ Pic(XÉ)) the Artin L-function of the Gal(É/É)-representation Pic(XÉ)⊗.
Then, for t the Picard rank of X,
is a real number different from zero.
lim
s→1
(s − 1) t L(s, χ Pic(XÉ))
Proof. The Gal(É/É)-representation Pic(XÉ)⊗contains the trivial representation
t times as a direct summand. Therefore, L(s, χ Pic(XÉ)) = ζ(s) t ·L(s, χ P )
and
lim
s→1
(s − 1) t L(s, χ Pic(XÉ)) = L(1, χ P ) .
Here, ζ denotes the Riemann zeta function and P is a Gal(É/É)-representation
which does not contain trivial components. [Mu-y, Corollary 11.5 and Corollary
11.4] show that L( . , χ P ) has neither a pole nor a zero at 1. □

Sec. 5] PEYRE’S CONSTANT 229
iv. The p-adic measures. —
5.15. Definition. –––– Let p be a prime number and x ∈ X (Ép) be arbitrary.
Choose local coordinates t 1 , . . . , t n on X × SpecÉSpecÉp in a neighbourhood
of x. These define a morphism ofÉp-schemes
ι x : U x −→ A nÉp
from a Zariski open neighbourhood U x of x such that ι(x) = (0, . . . , 0) and ι is
étale at x.
The tensor field (ι −1 ) [ ∂
]
x,Ép ∗ ∂t 1
∧ . . . ∧ ∂
∂t n
is the restriction to the p-adic points
of a section of the anticanonical sheaf K ∨Ép .
In a neighbourhood of x, we define the measure ω p by
[
∥ ∂
∥(ι −1 ) x,Ép ∗ ∧ . . . ∧ ∂ ]∥ ∥∥ · ι
∗
∂t 1 ∂t (dt x,Ép 1 · . . . · dt n ) .
n
Here, each dt i denotes a copy of the Haar measure onÉp normalized in the
usual manner.
5.16. Remark. –––– This definition is independent of the choice of the coordinates
t 1 , . . . , t n . Indeed, for t ′ 1, . . . , t ′ n forming another system of local
coordinates near x, we have
dt ′ 1 · . . . · dt ′ n = det ∂(t′ 1 , . . . , t′ n )
∂(t 1 , . . . , t n ) · dt 1 . . . dt n
and
∂
∂t ′ 1
∧ . . . ∧ ∂
∂t ′ n
= det ∂(t 1, . . . , t n )
∂(t
1 ′, . . . , t′ n ) · ∂
∧ . . . ∧ ∂ .
∂t 1 ∂t n
5.17. Lemma. –––– For a primenumber p, suppose that the metric ‖ . ‖ p is induced
by a model X of X. Then, the measure ω p on X (Ép) is given as follows.
Let a ∈ X (/p k) and put
Then,
U (k)
a := {x ∈ X (Ép) | x ≡ a (mod p k ) } .
ω p (U a (k) ) = lim #{ y ∈ X (/p n) | y ≡ a (mod p k ) }
.
n→∞ p n dim X
Proof. This is simply a more concrete reformulation of the abstract definition.
□
5.18. Corollary. –––– Let p be a prime such that X is smooth over p. Then, for
every a ∈ X (/p),
ω p (U a (1) ) = 1
p . dim X

230 CONJECTURES ON POINTS OF BOUNDED HEIGHT [Chap. VI
Proof. This is a consequence of Hensel’s lemma.
□
5.19. Remark. –––– ω p may as well be described in terms of the affine cone
over X. For example, if X is a hypersurface of degree d in P n then one has
(
ω ) p X (Ép) = 1 − p−k
· lim
1 − p #C (dim X+1)n
−1 X (/p
n)/p
n→∞
for k := n + 1 − d. A proof is given in [Pe/T, Corollary 3.5].
5.20. Definition (The measure τ p on X (Ép)). —– Let p be a prime number.
We define the measure τ p on X (Ép) by
τ p := det ( 1 − p −1 Frob p | Pic(XÉ) I p) · ωp .
Here, Pic(XÉ) I p
denotes the fixed module under the inertia group.
5.21. Remark. –––– In the easiest case that Pic(XÉ) =, one has
det ( )
1 − p −1 Frob p | Pic(XÉ) I 1
p = 1 −
p .
v. The real measure. —
5.22. Definition (The measure τ ∞ nÊ
on X (Ê)). —– Let x ∈ X (Ê) be arbitrary.
Choose local coordinates t 1 , . . . , t n on X × SpecÉSpecÊin a neighbourhood
of x. These define a morphism ofÊ-schemes
ι x : U x −→ A
from a Zariski open neighbourhood U x of x such that ι(x) = (0, . . . , 0) and ι is
étale at x. Consequently, ι x,Ê: U x (Ê) →Ên is a diffeomorphism near x.
[
The tensor field (ιx,Ê) −1 ∂
]
∗ ∂t 1
∧ . . . ∧ ∂
∂t n
is the restriction to the real points of a
section of the anticanonical sheaf K.
∨
In a neighbourhood of x, we define the measure τ ∞ by the differential form
[
∥ ∂
∥(ι −1
x,Ê) ∗ ∧ . . . ∧ ∂ ]∥ ∥∥ · ι
∗
∂t 1 ∂t x,Ê(dt 1 ∧ . . . ∧ dt n ) .
n
5.23. Remark. –––– Again, this definition is independent of the choice of the
coordinates t 1 , . . . , t n . If t ′ 1, . . . , t ′ n form another system of local coordinates
near x then
dt ′ 1 ∧ . . . ∧ dt ′ n = det ∂(t′ 1 , . . . , t′ n )
∂(t 1 , . . . , t n ) · dt 1 ∧ . . . ∧ dt n
and
∂
∂t ′ 1
∧ . . . ∧ ∂
∂t ′ n
= det ∂(t 1, . . . , t n )
∂(t
1 ′, . . . , t′ n) · ∂
∧ . . . ∧ ∂ .
∂t 1 ∂t n

Sec. 5] PEYRE’S CONSTANT 231
5.24. –––– In the situation that X is a hypersurface, the measure τ ∞ may be
described by far more concretely in terms of the Leray measure.
5.25. Definition. –––– Let f ∈Ê[X 0 , . . . , X n ] be a homogeneous polynomial
such that (grad f )(x 0 , . . . , x n ) ≠ 0 for every (x 0 , . . . , x n ) different from
the origin.
Then, the Leray measure on “ f = 0” is given by the formula
1
ω Leray =
‖ grad f ‖ ω hyp .
Here, ω hyp denotes the usual hypersurface measure.
5.26. Lemma. –––– Let f ∈Ê[X 0 , . . . , X n ] be as in Definition 5.25 and
U ⊂Ên+1 be an open subset such that ∂ f
∂x 0
does not vanish on U.
Then, on U ∩“ f = 0”, the Leray measure is given up to sign by the differential form
1
| ∂f/∂x 0 | dx 1 ∧ . . . ∧ dx n .
Proof. This is an immediate consequence of Sublemma 5.27 below.
5.27. Sublemma. –––– Let U ⊂Ên+1 be an open subset and f : U →Êbe a
smooth function such that ∂ f
∂x 0
does not vanish.
Then,thehypersurfacemeasureon“f = 0”isgivenuptosignbythedifferentialform
‖ grad f ‖
| ∂f/∂x 0 | dx 1 ∧ . . . ∧ dx n .
Proof. It is well known that, having resolved the equation
f (x 0 , . . . , x n ) = 0
by x 0 , the hypersurface measure is given by the form
√
1 + (∂x 0 /∂x 1 ) 2 + . . . + (∂x 0 /∂x n ) 2 dx 1 ∧ . . . ∧ dx n .
For every i = 1, . . . , n, the equation f ( )
x 0 (x 1 , . . . , x n ), x 1 , . . . , x n = 0
immediately yields ∂f ∂x 0
∂x 0 ∂x i
+ ∂f
∂x i
= 0 . In other words,
∂x 0
= − ∂f / ∂f
.
∂x i ∂x i ∂x 0
Altogether, we find the differential form
√
1 +
n
∑
i=1
( ∂f
∂x i
/ ∂f
∂x 0
) 2
dx 1 ∧ . . . ∧ dx n
for the hypersurface measure. This is exactly the assertion.
□
□

232 CONJECTURES ON POINTS OF BOUNDED HEIGHT [Chap. VI
5.28. Notation. –––– Let X ⊆ P nÉbe a projective variety. We denote by CX
the affine cone over X.
A metric ‖ . ‖ on O(1)| X () defines a compact subset
N ‖.‖ := {x = (x 0 , . . . , x n ) ∈ CX (Ê) | | x 0 | ≤ ‖X 0 ‖(x), . . . , | x n | ≤ ‖X n ‖(x)}
of the affine cone which is symmetric to the origin. Note that the conditions
| x i | ≤ ‖X i ‖(x) and | x j | ≤ ‖X j ‖(x) are equivalent to each other as long
as x i , x j ≠ 0.
5.29. Examples. –––– i) Consider on O(1)| X () the minimum metric ‖ . ‖ min
from Example IV.3.5. Then,
N ‖ .‖min
= {x = (x 0 , . . . , x n ) ∈ CX (Ê) | | x 0 | ≤ 1, . . . , | x n | ≤ 1}
is a hypercube.
ii) Similarly, for the l 2 -metric,
is the unit ball.
N ‖.‖l 2 = {x = (x 0, . . . , x n ) ∈ CX (Ê) | x 2 0 + . . . + x 2 n ≤ 1}
5.30. Proposition. –––– Let F ∈É[X 0 , . . . , X n ] be a homogeneous polynomial
of degree d and X ⊂ P nÉbe the hypersurface defined by the equation F = 0.
According to the adjunction formula, there is a canonical isomorphism
ι: O(n − d + 1)
∼=
−→ K ∨
X .
The metric ‖ . ‖ on K ∨
X induces a metric ι−1 ‖ . ‖ 1
n+d−1 on O(1)|X (). Let
N := N ι −1 ‖ .‖
be the subset defined by ι −1 ‖ . ‖ 1
n+d−1 .
1
n−d+1
⊂ CX (Ê)
Then, in terms of N and the Leray measure, one may write
τ ∞ (U ) = n − d + 1 Z
ω Leray
2
for every measurable set U ⊆ X (Ê).
Proof. First step: Explicit description of ι.
CU ∩N
The assertion is local in X. We may therefore assume without restriction that
X 0 ≠ 0 and ∂F
∂X 1
≠ 0. Then, t 2 , . . . , t n for t i := X i /X 0 form a local system of
coordinates on X.

Sec. 5] PEYRE’S CONSTANT 233
When putting F = X d 0 f (t 1, . . . , t n ), we find
∂F
= X d−1 ∂f
0 .
∂X 1 ∂t 1
Under the Poincaré residue map [G/H, p. 147], the differential form
∂f
∂t 1
df
f
∧ dt 2 ∧ . . . ∧ dt n = ∂f
∂t 1
dt 1 ∧ . . . ∧ dt n
on P nis mapped to dt 2 ∧ . . . ∧ dt n . Dually, under this correspondence, the
tensor field ∂
∂t 2
∧ . . . ∧ ∂
∂t n
on X () is identified with
f ∂
∧ . . . ∧ ∂ = 1 F ∂
∧ . . . ∧ ∂ .
∂t 1 ∂t n X
∂F 0 ∂t
∂X 1 ∂t n 1
Furthermore, the Euler sequence identifies ∂
∂t 1
∧ . . . ∧ ∂
∂t n
with the global section
X n+1
0 ∈ Γ ( P n , O(n + 1) ) .
Altogether,
ι −1 ( ∂
∂t 2
∧ . . . ∧ ∂
∂t n
)
= 1
∂F
∂X 1
X n 0 .
Second step: The measure τ ∞ of Peyre.
Peyre’s measure τ ∞ on X (Ê) is therefore given by the differential form
1
∥ ∂F ∥ dt 2 ∧ . . . ∧ dt n = | X0 d−1 / ∂F
∂X 1
|·‖X n−d+1
0 ‖dt 2 ∧ . . . ∧ dt n . (§)
∂X 1
X n 0
On the other hand, the subset N ⊂ CX (Ê) is given by
| x 0 | ≤ A := ‖X 0 ‖ .
Therefore, according to Fubini’s theorem,
Z
CU ∩N
ω Leray =
=
=
Z
CU
|x 0 |≤A
Z A
−A
1
| ∂F
x n−1
0
x d−1
0
∂X 1
| dx 0 ∧ dx 2 ∧ . . . ∧ dx n
dx 0 ·
Z
CU
x 0 =1
2
n − d + 1 · An−d+1 ·
1
| ∂F
∂X 1
| dx 2 ∧ . . . ∧ dx n
Z
CU
x 0 =1
1
| ∂F
∂X 1
| dx 2 ∧ . . . ∧ dx n .
Note here that CX is an (n −1)-dimensional cone while the integrand is homogeneous
of degree −(d − 1).

234 CONJECTURES ON POINTS OF BOUNDED HEIGHT [Chap. VI
Consequently,
n − d + 1
2
Z
CU ∩N
ω Leray =
Z
CU
x 0 =1
‖X 0 ‖ n−d+1
dx
| ∂F 2 ∧ . . . ∧ dx n
∂X 1
|
which, according to formula (§), is exactly equal to τ ∞ (U ).
□
5.31. Remarks. –––– i) This result allows a generalization to complete intersections.
ii) If X is a cubic surface then n − d + 1 = 1.
On the other hand, consider the case that X is a hypersurface in P n for n ≥ 4.
Then, Pic(X ) = 1 and α(X ) = 1 . We therefore have that
n−d+1
α(X )·τ ∞ (U ) = 1 Z
ω Leray
2
for every measurable set U ⊆ X (Ê).
vi. The Tamagawa measure. —
CU ∩N
5.32. Definition. –––– The Tamagawa measure τ H on the set X (É) of
points on X is defined to be the product measure
τ H := ∏ τ ν .
ν∈Val(É)
adelic
5.33. Remark. –––– X is projective. Therefore, one has that
X (É) = ∏ X (Éν) .
ν∈Val(É)
5.34. Lemma. –––– The infinite product
(
∏ τ )
ν X (Éν)
ν∈Val(É)
is absolutely convergent. In particular, the infinite product measure ∏ τ ν is well defined.
ν∈Val(É)
Proof. Absolute convergence may not be destroyed by a finite set of factors.
Thus, assume that almost all of the metrics ‖ . ‖ νp of the adelic metric
‖ . ‖ = {‖ . ‖ ν } ν∈Val(É) are induced by a model X of X. Further, we may
restrict the infinite product to all prime numbers p such that X is smooth
over p.
Corollary 5.18 assures that
(
τ ) p X (Ép) = det ( )
1 − p −1 Frob p | Pic(XÉ) I #X (p)
p · . ()
pdim X

Sec. 5] PEYRE’S CONSTANT 235
Further, smoothness over p implies that
Pic(XÉ) I p
= Pic(XÉ) .
We denote the eigenvalues of Frob p on Pic(XÉ) by λ 1 , . . . , λ r . As every
divisor on is actually defined over a finite field extension, all these are roots
Xp
of unity.
Since Pic(XÉ) is a-module of finite rank, the characteristic polynomial
of Frob p is monic with integral coefficients. This shows, if λ is an eigenvalue
then λ = 1 λ is an eigenvalue of Frob p, too.
Clearly, one has
det ( 1 − p −1 Frob p | Pic(XÉ) I p) = (1 − λ1 p −1 ) · . . . · (1 − λ r p −1 )
= 1 − (λ 1 + . . . + λ r )p −1 + E
where
| E| <
( ( k k
p
2)
−2 + . . . + p
k)
−k
for p > 2k.
< k2
p 2 + k3
p 3 + . . .
< 2k2
p 2
In order to determine #X (p), we use the Lefschetz trace formula in étale cohomology
[SGA4 1 , Rapport, Théorème 3.2]. This yields
2
for every prime l ≠ p.
dimX p
#X (p) =
∑
i=0
(−1) i tr ( )
Frob p | H i ét ,Él) (‖)
(Xp
Here, according to the Weil conjectures proven by P. Deligne [Del, Théorème
(1.6)], every eigenvalue of the Frobenius on H i ét (Xp ,Él) is of absolute
value p i/2 .
The theorem on smooth base change implies a comparison theorem even for
the unequal characteristic case [SGA4, Exp. XVI, Corollaire 2.5 and Exp. XV,
Théorème 2.1]. We therefore know that
H i ét (Xp ,Él) ∼ = H i ét (XÉ,Él) .
According to [SGA4, Exp. XVI, Corollaire 1.6 and Exp. XI, Théorème 4.4], the
latter is isomorphic to the usual cohomology H i (X (),Él).

236 CONJECTURES ON POINTS OF BOUNDED HEIGHT [Chap. VI
Let C be a constant such that dim H i (X (),Él) ≤ C for every i ∈Æ. Then,
dim H i ét (Xp ,Él) ≤ C
for every i ∈Æ, too. Further, Lemma 4.2 describes the first and second
cohomologies of a Fano variety more concretely. We have H 1 ét (Xp ,Él) = 0
while the first Chern class induces an isomorphism
Pic(XÉ)⊗Él
∼= H 2 ét (XÉ,Él(1))
∼= H 2 ét (Xp ,Él(1)) .
This isomorphism is compatible with the operation of Frob p . Consequently,
the Frobenius eigenvalues on H 2 ét (Xp ,Él) are λ 1 p, . . . , λ r p.
2 dim X −1
Poincaré duality shows that Hét (Xp ,Él) = 0. Further, by [Del, (2.4)],
Poincaré duality is compatible with the Frobenius eigenvalues. Therefore, the
eigenvalues of the Frobenius on H 2 dimX−2 ,Él) are
ét
(Xp
λ 1 p dimX−1 , . . . , λ r p dimX−1 .
For the number ofp-rational points, this yields, according to (‖),
where
X (p) = p dim X + (λ 1 + . . . + λ r )p dim X −1 + Dp dim X
| D| < C [p −3/2 + p −2 + p −5/2 + . . . ] < Cp −3/2 1
1 − p −1/2 < 4Cp−3/2 .
Altogether,
τ p
(
X (Ép) )
= (1 − (λ 1 + . . . + λ r )p −1 + E) · (1 + (λ 1 + . . . + λ r )p −1 + D)
= 1 + E + D + (λ 1 + . . . + λ r ) 2 p −2 + (E − D)(λ 1 + . . . + λ r )p −1 + ED .
For p > 4C 2 , we certainly have
| E| < 2k2
p < 2C 2
2 p 2
Therefore, τ p
(
X (Ép) ) = 1 + X where
= 4C · C
2
4C
<
p3/2·p1/2 p . 3/2
| X | < 8C
p + C 2
3/2 p + 8C 2 2
16C
+ 2 p5/2 p 3
< 8C + C 2
2C + 8C 2
4C 2 + 16C 2
8C 3
p 3/2

Sec. 5] PEYRE’S CONSTANT 237
= 2 + 17
2 C + 2 C
p 3/2 .
Since 3/2 > 1, the infinite product is absolutely convergent.
□
vii. The constant of E. Peyre. —
5.35. Definition (E. Peyre, [Pe/T, Definition 2.4]). —–
Peyre’s constant or Peyre’s Tamagawa type number is defined as
τ (X ) := α(X )·β(X ) · lim
s→1
(s − 1) t L(s, χ Pic(XÉ)) · τ H
(
X (É)
for t = rk Pic(X ).
Br)
5.36. Remark. –––– Here, X (É) denotes the part which is not
affected by the Brauer-Manin obstruction. The Brauer-Manin obstruction was
discussed in detail in Chapter III. The precise definition of X (É) Br is given
in III.2.5. According to Proposition III.2.3.b.ii), X (É) Br is a closed subset
of X (É) and therefore measurable.
Br ⊆ X (É)
5.37. –––– Thus, there is the following conjecture which refines Conjecture
4.11.
Conjecture (Manin-Peyre). Let X be a smooth, projective variety overÉ.
AssumethatX isFano. DenotebyH K ∨ theanticanonicalheightfunctionintroduced
in Definition 4.6.
a) If dim X ≤ 2 or X ∈ P n is a complete intersection then there exist a real
number τ and a Zariski open subset X ◦ ⊆ X such that
for B → ∞.
N X ◦ ,H K ∨(B) = #{x ∈ X ◦ (k) | H K ∨(x) < B} ∼ τB log rk Pic(X )−1 B
b) Suppose that, for X,thereexist a realnumber τ and aZariski opensubset X ◦ ⊆ X
such that
for B → ∞.
N X ◦ ,H K ∨(B) = #{x ∈ X ◦ (k) | H K ∨(x) < B} ∼ τB log rk Pic(X )−1 B
Then, τ = τ (X ) is Peyre’s constant.

238 CONJECTURES ON POINTS OF BOUNDED HEIGHT [Chap. VI
5.38. Remarks (Some motivation). —– i) Morally, the conjecture of Manin in
the refined form due to Peyre states the following. The (conjectural) constant
describing the growth of the number ofÉ-rational points on X is equal to a
regularized product over the densities of the p-adic points on X together with
the density of the real points.
That there is such a connection is actually not very surprising. For instance,
a low density of p-adic points on X for a particular prime p implies strong
congruence conditions forÉ-rational points.
ii) More precise results have been obtained by the classical circle method [Bir].
For complete intersections in a very high-dimensional projective space, the
circle method provides an asymptotic formula for the number ofÉ-rational
points on X and an error term. The assumptions on the dimension of the
ambient projective space are rather restrictive. They are needed in order to
ensure that the provable error term is smaller than the main term.
The coefficient of the main term is a product of p-adic densities together with a
factor corresponding to the archimedean valuation. Unlike the p-adic densities,
the latter factor does not coincide in general with E. Peyre’s factor τ ∞
(
X (Ê) ) .
Rather, it is an integral over the Leray measure (at least in the case that X is
a hypersurface).
By Proposition 5.30, the factor corresponding to the archimedean valuation
may be rewritten in the form α(X ) · τ ∞
(
X (Ê) ) . Therefore, one sees that the
coefficient of the main term provided by the circle method is equal to
Here,
(
∏τ ) (
p X (Ép) · α(X ) · τ ) (
∞ X (Ê) = α(X ) · τ H X (É) p
τ p
(
X (Ép) ) =
) .
(
1 − 1 )
#{ y ∈ X (/p n) | y ≡ a (mod p k ) }
· lim
.
p n→∞ p n dim X
The (1 − 1 ) are convergence-generating factors.
p
In the cases where the circle method is applicable, one always has Pic(X ) ∼ =.
It seems very natural to use (1 − 1 p )t for t = rk Pic(XÉ) as the convergencegenerating
factors for the general case.
In the definition of Peyre’s constant, this is done and more than that. Indeed, the
Gal(É/É)-representation Pic(XÉ) I p ⊗contains the trivial representation
t times. Denote the complementary summand by P. Then, the factors used in
Definition 5.20 may be decomposed as
det ( (
)
1 − p −1 Frob p | Pic(XÉ) I p = 1 − 1 t
· det
p) ( 1 − p −1 Frob p | P ) .

Sec. 5] PEYRE’S CONSTANT 239
The factors det ( 1 − p −1 Frob p | P ) form the Euler product for
1/L(s, χ P )
at s = 1. The value of the function L at 1 is introduced into Definition 5.35
only in order to cancel these factors out.
iii) The definition of the factor α(X ) is somehow more complicated and, may
be, more mysterious than those of the other parts. Some motivation for the
appearance of such a factor is given by the circle method.
There is, however, another point which is at least as important. The definition
of α(X ) implies that the conjecture of Manin-Peyre is compatible with direct
products of Fano varieties.
iv) For a smooth cubic surface of the “first case” of Colliot-Thélène, Kanevsky,
and Sansuc, the assertion of Theorem III.6.4.c) implies that
(
τ ) H X (É) .
On the other hand, β(X ) = 3 such that one might want to simplify Peyre’s
formula to
) . (∗∗)
Br) = 1 3 ·τ H(
X (É)
τ (X ) := α(X ) · lim
s→1
(s − 1) t L(s, χ Pic(XÉ)) · τ H
(
X (É)
Clearly, this is also true in all cases when β(X ) = 1.
Formula (∗∗) is, however, wrong in general. For example, when X is a smooth
cubic surface on which there is a Brauer-Manin obstruction to the Hasse principle
then we need τ (X ) = 0 and this is not provided by the simplified formula.
Furthermore, in [Pe/T], E. Peyre and Y. Tschinkel reported strong numerical
evidencefor Peyre’sformulaincasessimilar toExampleIII.5.36. I.e., ifβ(X ) = 3
but the Brauer-Manin obstruction does not exclude any adelic point on X then
there are three times moreÉ-rational points on X than naively expected.
5.39. Remark (Known cases). —– As indicated above, Manin’s conjecture in
the refined form due to Peyre is established for smooth complete intersections
of multidegree d 1 , . . . , d n in the case that the dimension of V is very large
compared to d 1 , . . . , d n [Bir]. This is the classical circle method.
Further, Conjecture 5.37 is proven for projective spaces and quadrics. Finally,
there are a number of further particular cases in which Manin’s conjecture
is known to be true. See, e.g., the survey given in [Pe02, Sec. 4].
5.40. Remark (Numerical evidence). —– R. Heath-Brown [H-B92a] as well as
E. Peyre and Y. Tschinkel [Pe/T] reported numerical evidence for Conjecture
5.37 for isolated examples of smooth cubic surfaces.

240 CONJECTURES ON POINTS OF BOUNDED HEIGHT [Chap. VI
In Chapter VII, we present numerical evidence for Manin’s conjecture in the case
of the threefolds V e
a,b given by axe = by e + z e + v e + w e in P 4Éfor e = 3 and 4.
In Chapter IX, we test Manin’s conjecture numerically for general cubic surfaces.
5.41. Remark. –––– It is not all trivial that the conditionally convergent Euler
product
1
det ( 1 − p −1 Frob p | P )
∏
p
really converges versus L(1, χ P ). The point is that the value L(1, χ P ) is actually
defined by analytic continuation.
5.42. Remark. –––– The idea behind Peyre’s constant is in fact an old one.
The main term of the asymptotic formula provided by the circle method was
used before, for example, by D. R. Heath-Brown in his experimental investigations
on cubic surfaces where weak approximation fails [H-B92a].
Actually, Heath-Brown’s point of view was a little different. He considered
the factors (1 − 1 p )t as convergence-generating factors growing out of the circle
method. The resulting conditionally convergent infinite product was treated
like a definition. The value of the function L at 1 appeared, too, but only as
part of a numerical method to speed up convergence.
E. Peyre’s approach is most likely inspired by considerations like those
in [H-B92a]. It has, however, the potential to work in much more generality.

PART C
NUMERICAL EXPERIMENTS

CHAPTER VII
POINTS OF BOUNDED HEIGHT ON CUBIC
AND QUARTIC THREEFOLDS ∗
. . . , one by one, or all at once.
W. S. GILBERT AND A. SULLIVAN: The Yeomen of the Guard (1888)
1. Introduction — Manin’s Conjecture
i. Summary. —
1.1. –––– For the families ax 3 = by 3 + z 3 + v 3 + w 3 , a, b = 1, . . . , 100, and
ax 4 = by 4 +z 4 +v 4 +w 4 , a, b = 1, . . . , 100, of projective algebraic threefolds,
we test numerically the conjecture of Manin (in the refined form due to Peyre,
cf. Conjecture VI.5.37) about the asymptotics of points of bounded height on
Fano varieties.
This includes searching for points, computing the Tamagawa number, and detecting
the accumulating subvarieties. The goal of this chapter is describe these
computations as well some background on the geometry of cubic and quartic
threefolds.
ii. Manin’s conjecture. —
1.2. –––– Let X be a projective algebraic variety overÉ. We fix an embedding
ι: X → P nÉ. In this situation, recall there is the well-known naive
height H naive : X (É) →Êgiven by
H naive (P) := max
i=0,...,n | x i| .
Here, (x 0 : . . . : x n ) := ι(P) ∈ P n (É) where the projective coordinates are
integers satisfying gcd(x 0 , . . . , x n ) = 1.
(∗) This chapter is a revised and slightly extended version of the article: The Asymptotics of
Points of Bounded Height on Diagonal Cubic and Quartic Threefolds, in: Algorithmic number
theory, Lecture Notes in Computer Science 4076, Springer, Berlin 2006, 317–332, joint with
A.-S. Elsenhans.

244 POINTS OF BOUNDED HEIGHT ON THREEFOLDS [Chap. VII
1.3. –––– It is of interest to ask for the asymptotics of the number ofÉ-rational
points on X of bounded naive height. This applies particularly to Fano varieties
as those are expected to have many rational points (at least after a finite extension
of the ground-field).
Simplest examples of Fano varieties are complete intersections in P nÉof a multidegree
(d 1 , . . . , d r ) such that d 1 + . . . + d r ≤ n. In this case, the conjecture
of Manin discussed in general in Chapter VI reads as follows.
1.4. Conjecture. –––– Let X ⊆ P nÉbe a non-singular complete intersection of
multidegree (d 1 , . . . , d r ). Assumedim X ≥ 3andk := n+1−d 1 − . . . −d r > 0.
Then, there exists a Zariski open subset X ◦ ⊆ X such that
#{x ∈ X ◦ (É) | H naive (x) k < B} ∼ τB
(∗)
for a constant τ.
1.5. Example. –––– Let X ⊂ P 4Ébe a smooth hypersurface of degree 4.
Conjecture 1.4 predicts ∼ τB rational points of height < B. However, the
hypersurface x 4 + y 4 = z 4 + v 4 + w 4 contains the line given by x = z, y = v,
and w = 0 on which there is quadratic growth, already. This explains the
necessity of the restriction to a Zariski open subset X ◦ ⊆ X.
1.6. Remark. –––– Conjecture 1.4 is proven for P n , linear subspaces, and
quadrics. Further, it is established [Bir] in the case that the dimension of X
is very large compared to d 1 , . . . , d r . The general version (Conjecture VI.5.37)
is known to be true in a number of further particular cases. A rather complete
list may be found in the survey article [Pe02, Sec. 4].
In this chapter, we present numerical evidence for Conjecture 1.4 in the case of
the varieties X e a,b given by axe = by e + z e + v e + w e in P 4Éfor e = 3 and 4.
1.7. Remark. –––– By the Noether-Lefschetz Theorem, the assumptions made
on X imply that Pic(XÉ) ∼ =[Ha70, Corollary IV.3.2]. This is no longer true
in dimension two. See Remark 1.14.ii) for more details.
iii. The constant. —
1.8. –––– Conjecture 1.4 is compatible with results obtained by the classical
circle method (e.g. [Bir]). Motivated by this, E. Peyre provided a description
of the constant τ expected in (∗). We formulated the definition for the general
case in VI.5.35.
In the situation considered here, Pic(XÉ) ∼ =implies that β(X ) = 1 and there
is no Brauer-Manin obstruction on X. Peyre’s constant is therefore equal to the

Sec. 1] INTRODUCTION — MANIN’S CONJECTURE 245
Tamagawa-type number
τ (X ) := α(X ) · ∏ (1 − 1 ) ω p p(X (Ép)) .
p∈È∪{∞}
In this formula, α(X ) := 1/k for k ∈Æsuch that O(−K ) ∼ = O(k).
The measure ω p is given in local p-adic analytic coordinates x 1 , . . . , x d by
∂
∥ ∧ . . . ∧
∂ ∥ ∥∥∥p
dx 1 . . . dx d .
∂x 1 ∂x d
Here, each dx i denotes a Haar measure onÉp which is normalized in the
∂
usual manner.
∂x 1
∧ . . . ∧ ∂
∂x d
is a section of O(−K ).
1.9. –––– For p finite, one has a natural model X ⊆ P of X given by the
np
defining equation. This induces the metric ‖ . ‖ p on O(k). It is almost immediate
from the definition that
(
ω ) #X (/p m)
p X (Ép) = lim .
m→∞ p m dim X
1.10. Remark. –––– There is another description of ω p of interest for finite p.
One has
(
ω ) p X (Ép) = 1 − p−k
lim
1 − p #CX (/p m)/p (d+1)m .
−1 m→∞
Here, CX denotes the affine cone over X . For a proof, see [Pe/T, Corollary
3.5].
1.11. –––– The hermitian metric ‖ . ‖ ∞ corresponding the naive height H naive
is given by ‖ . ‖ ∞ := ‖ . ‖ k min on O(−K ) ∼ = O(k). Here, ‖ . ‖ min is the hermitian
metric on O(1) defined by
‖x i ‖ min := inf
j=0,...,n | x i/x j | .
The factor ω ∞
(
X (É∞) ) may then be described as follows.
1.12. Lemma. –––– If X ⊂ P n is a hypersurface defined by the equation f = 0
then
(
α(X ) · ω ) ∞ X (Ê) = 1 Z
ω Leray .
2
The Leray measure ω Leray on
f (x 0 ,...,x n )=0
|x 0 |,...,|x n |≤1
{(x 0 , . . . , x n ) ∈Ên+1 | f (x 0 , . . . , x n ) = 0}

246 POINTS OF BOUNDED HEIGHT ON THREEFOLDS [Chap. VII
1
is related to the usual hypersurface measure by the formula ω Leray =
‖ grad f ‖ ω hyp.
On the other hand, one may also write
1
ω Leray =
| ∂ f
∂x i
(x 0 , . . . , x n )| dx 0 ∧ . . . ∧ ̂dx i ∧ . . . ∧ dx n .
Proof. The equivalence of the two descriptions of the Leray measure is shown
in Lemma VI.5.26. The main assertion is proven in Proposition VI.5.30. □
1.13. Remark. –––– As the description of the constant τ given here grew out
of a definition given in canonical terms, it is no surprise that τ is invariant
under scaling.
This can also be seen directly. When we multiply f by a prime number p then
τ p gets multiplied by a factor of p. On the other hand, τ ∞ gets multiplied by a
factor of 1/p and all the other factors remain unchanged.
1.14. Remark. –––– There are several ways to generalize Conjecture 1.4.
These are presented in detail in Chapter VI. For completeness, let us recall
the following facts.
i) One may consider more general heights corresponding to the tautological
invertible sheaf O(1). This includes to
a) replace the minimum metric by an arbitrary continuous hermitian metric
on O(1). This would affect the domain of integration for the factor at infinity.
b) multiply H naive (x) with a function that depends on the reduction of x modulo
some N ∈Æ. This augments Conjecture 1.4 by an equidistribution statement.
ii) Instead of complete intersections, one may consider arbitrary projective Fano
varieties X. Then, H k naive needs to be replaced by a height defined by the
anticanonical sheaf O(−K X ).
If Pic(XÉ) ̸∼ =then the description of the constant C gets more complicated in
several ways. First, there is an additional factor β := #H 1( Gal(É/É), Pic(XÉ) ) .
Further, instead of (1 − 1 ), one has to use more complicated convergencegenerating
factors. (Cf. Remark VI.5.38.) Finally, the Tamagawa measure has
p
to be taken not of the full variety X (É) but of the subset which is not affected
by the Brauer-Manin obstruction.
If Pic(X ) ̸∼ =already overÉthen the right hand side of (∗) has to be replaced
by CB log t B. For the exponent of the log-term, there is the expectation that
t = rk Pic(X ) − 1. There are, however, examples [Ba/T] in dimension three in
which the exponent is larger.
The definition of α is rather complicated, in general. The factor α depends on
the structure of the effective cone in Pic(X ) and on the position of (−K X ) in it.
(Cf. Definition VI.5.1).

Sec. 2] COMPUTING THE TAMAGAWA NUMBER 247
2. Computing the Tamagawa number
i. Counting points over finite fields. —
2.1. –––– We consider the projective varieties X e a,b
given by
ax e = by e + z e + v e + w e
in P . We assume a, b ≠ 0 (and p ∤e) in order to avoid singularities. Observe
that, even for large p, these are at most e 2 varieties up to obviousp-
4p
isomorphism as∗ p consists of no more than e cosets modulo (∗ p ) e .
It follows from the Weil conjectures, proven by P. Deligne [Del, Théorème (8.1)],
that
#X e a,b (p) = p 3 + p 2 + p + 1 + E e a,b
where the error-term E e a,b may be estimated by | Ee a,b | ≤ C e p 3/2 .
Here, C 3 = 10 and C 4 = 60 as dim H 3 (X 3 ,Ê) = 10 for every smooth cubic
threefoldanddim H 3 (X 4 ,Ê) = 60for everysmoothquarticthreefoldinP 4 ().
These dimensions result from the Weak Lefschetz Theorem together with
F. Hirzebruch’s formula [Hi, Satz 2.4] for the Euler characteristic which actually
works in much more generality.
2.2. Remark. –––– Suppose e = 3 and p ≡ 2 (mod 3). Then,
#X 3 a,b (p) = #X 1 a,b (p)
as gcd(p − 1, 3) = 1. Similarly, for e = 4 and p ≡ 3 (mod 4), one has
gcd(p − 1, 4) = 2 and
#X 4 a,b (p) = #X 2 a,b (p).
In these cases, the error term vanishes and
#X e a,b (p) = p 3 + p 2 + p + 1.
In the remaining cases p ≡ 1 (mod 3) for e = 3 and p ≡ 1 (mod 4) for e = 4,
our goal is to compute the number ofp-rational points on X e a,b . As X e a,b ⊆ P4 ,
there would be an obvious O(p 4 )-algorithm. We can do significantly better
than that.
2.3. Definition. –––– Let K be a field and let x ∈ K n and y ∈ K m be two vectors.
Then, their convolution z := x∗y ∈ K n+m−1 is defined to be z k := ∑ x i y j .
i+j=k

248 POINTS OF BOUNDED HEIGHT ON THREEFOLDS [Chap. VII
2.4. Theorem (FFT convolution). —– Let n = 2 l and K be a field which contains
the 2n-th roots of unity. Then, the convolution x ∗ y of two vectors x, y of
length ≤ n can be computed in O(n logn) steps.
Proof. The idea is to apply the Fast Fourier Transform (FFT) [Fo, Satz 20.3].
The connection to the convolution is shown in [Fo, Satz 20.2] or [C/L/R,
Theorem 32.8].
□
Theorem 2.4 is the basis for the following algorithm.
2.5. Algorithm (FFT point counting on X e a,b
). —–
i) Initialize a vector x[0 . . . p] with zeroes.
ii) Let r run from 0 to p − 1 and increase x[r e mod p] by 1.
iii) Calculate ỹ := x ∗ x ∗ x by FFT convolution.
iv) Normalize by putting y[i] := ỹ[i] + ỹ[i + p] + ỹ[i + 2p] for each
i ∈ { 0, . . . , p − 1 }.
v) Initialize N as zero.
vi) (Counting points with first coordinate ≠ 0)
Let j run from 0 to p − 1 and increase N by y[(a − bj 4 ) mod p].
vii) (Counting points with first coordinate 0 and second coordinate ≠ 0)
Increase N by y[(−b) mod p].
viii) (Counting points with first and second coordinate 0)
Increase N by (y[0] − 1)/(p − 1).
ix) Return N as the number of allp-valued points on X e a,b .
2.6. Remark. –––– For the running-time, step iii) is dominant. Therefore, the
running-time of Algorithm 2.5 is O(p log p).
To count, for fixed e and p,p-rational points on X e a,b
with varying a and b,
one needs to execute the first four steps only once. Afterwards, one may
perform steps v) through ix) for all pairs (a, b) of elements from a system
of representatives for∗ p /(∗ p ) e . Note that steps v) through ix) alone are of
complexity O(p).
2.7. –––– We ran this algorithm for all primes p ≤ 10 6 and stored the cardinalities
in a file.
2.8. Remark. –––– There is a formula for #X e a,b (p) in terms of Jacobi sums.
A skilful manipulation of these sums should lead to another efficient algorithm
which serves the same purpose as Algorithm 2.5.

Sec. 2] COMPUTING THE TAMAGAWA NUMBER 249
ii. The local factors at finite places. —
2.9. –––– We are interested in the Euler product
τ e a,b,fin := ∏ −
p∈È(1 1 )
p
lim
m→∞
#X e
a,b
(/pm)
.
p 3m
2.10. Lemma. –––– a) (Good reduction)
If p ∤ abe then the sequence (#X e
a,b (/p m)/p 3m ) m∈Æis constant.
b) (Bad reduction)
i) If p divides ab but not e then the sequence (#X e
a,b (/p m)/p 3m ) m∈Æbecomes
stationary as soon as p m divides neither a nor b.
ii) If p = 2 and e = 4 then the sequence (#X e
a,b (/p m)/p 3m ) m∈Æbecomes
stationary as soon as 2 m does not divide 8a or 8b.
iii) If p = 3 and e = 3 then the sequence (#X e
a,b (/p m)/p 3m ) m∈Æbecomes
stationary as soon as 3 m divides neither 3a nor 3b.
iii. An estimate. —
2.11. Theorem. –––– For every pair (a, b) of integers such that a, b ≠ 0, the
Euler product τ e a,b,fin
is convergent.
Proof. Cf. Lemma VI.5.34 where this is proven in more generality.
Let p be a prime bigger than | a|, | b|, and e. Then, the factor at p is
τ p := (1 − 1 p )(1 + p + p2 + p 3 + D p p 3/2 )/p 3
where |D p | ≤ C e for C 3 = 10 and C 4 = 60, respectively. As sums are easier to
estimate than products, we take a look at the logarithm,
log τ p = D p
p 3/2 + O(p−5/2 ) .
Taking the logarithm, we consider ∑ p log τ p . In the case e = 3, the sum is
effectively over the primes p = 1 (mod 3). If e = 4 then summation extends
over all primes p = 1 (mod 4). In either case, we take a sum over one-half of

250 POINTS OF BOUNDED HEIGHT ON THREEFOLDS [Chap. VII
all primes. This leads to the following estimate,
[ ]
Ce
∑ | log τ p | ≤ ∑
p≥N
p≥N
p + 3/2 O(p−5/2 ) ∼ C Z ∞
e
2
≤
N
C e
2 logN
1
t 3/2 logt dt
Z ∞
N
t −3/2 dt =
C e
√ N logN
.
2.12. Remark. –––– We are interested in an∣explicit upper bound for
∣∣∣∣ ∣ ∑ log τ p .
p≥10 6
Using Taylor’s formula, one gets
∣ ∣ ∑
D ∣∣∣∣
p
log τ p − ∑ ≤ 10 −8 .
p
p≥10 6 p≥10 3/2 6
Since D p
is zero for p ≡ 3 (mod 4) (or p ≡ 2 (mod 3)), the sum should be compared
with log(ζ K (3/2)). Here, ζ K is the Dedekind zeta function of K =É(i)
p 3/2
orÉ(ζ 3 ), respectively. This yields
∑
p≥10 6
p≡1(mod 4)
√
1
≤ log
p3/2 (1 − 2 −3/2 ) −1/2 · ∏
p≡3(mod 4)
ζÉ(i)(3/2)
(1 − p −3 ) −1/2 · ∏(1 − p −3/2 ) −1
p

Sec. 2] COMPUTING THE TAMAGAWA NUMBER 251
numbers had been precomputed using FFT point counting (Algorithm 2.5).
The algorithm below is based on the fact that the vast majority of the factors
actually do not need to be computed. They are available from a list.
2.15. Algorithm (Compute an approximate value for τ 3 (τ 4 a,b,fin a,b,fin
)). —–
i) Let p run over all prime numbers such that p ≡ 2 (mod 3) (p ≡ 3 (mod 4))
and p ≤ N and calculate the product of all values of (1 − 1/p 4 ).
ii) Compute the factor corresponding to p = 3 (p = 2) by Lemma 2.10.b).
iii) Let p run over all prime numbers such that p ≡ 1 (mod 3) (p ≡ 1 (mod 4))
and p ≤ N . Calculate the product of the factors described below.
If p|ab then the corresponding factor is given by Lemma 2.10.b). Otherwise,
compute the e-th power residue-symbols of a and b and look up the
precomputed factor for thisp-isomorphism class of varieties in the list.
iv) Multiply the two products from steps i) and iii) and the factor from step ii)
with each other. Correct the product by taking the bad primes p ≡ 2 (mod 3)
(p ≡ 3 (mod 4)) into consideration.
2.16. Remark. –––– When we meet a bad prime p, we have to count
/p m-valued points on X e
a,b
. This is done by an algorithm which is very
similar to Algorithm 2.5.
2.17. –––– We used Algorithm 2.15 to compute the Euler products τ 3 a,b,fin
and τ 4 a,b,fin for a, b = 1, . . . , 100. We did all calculations for N = 106 .
Note that step i) had to be done only once for e = 3 and once for e = 4.
v. The factor at the infinite place. —
2.18. –––– In order to get an approximation for the integrals, we tried to use
some standard methods from numerical analysis [Kr, Chapter 9], particularly
the Gauß-Legendre formula. A direct application of this method is known to
work well as long as the integrand is smooth enough and the dimension of the
domain of integration is not too big.
2.19. –––– For the quartic X 4 a,b
, we have the integral
over
ω ∞
(
X
4
a,b (Ê) ) = 1
4 4 √ a
ZZZZ
R
1
dy dz dv dw
(by 4 + z 4 + v 4 + w 4 )
3/4
R := {(y, z, v, w) ∈Ê4 | |y|, |z|, |v|, |w| ≤ 1 and |by 4 + z 4 + v 4 + w 4 | ≤ a} .
The integrand is singular in one point. We used a simple substitution to make
it sufficiently smooth for numerical integration.

252 POINTS OF BOUNDED HEIGHT ON THREEFOLDS [Chap. VII
On the other hand, for the cubic X 3 a,b
, we have to consider
ω ∞
(
X
3
a,b (Ê) ) = 1
6 3 √ a
ZZZZ
R
1
dy dz dv dw
(by 3 + z 3 + v 3 + w 3 )
2/3
for R := {(y, z, v, w) ∈Ê4 | |y|, |z|, |v|, |w| ≤ 1 and |by 3 +z 3 +v 3 +w 3 | ≤ a} .
The difficulty here is the handling of the singularity of the integrand. It is located
in the zero set of by 3 + z 3 + v 3 + w 3 in R which is a cone over a cubic surface.
Since (by 3 +z 3 +v 3 +w 3 ) −2/3 is a homogeneous function, it is enough to integrate
over the boundary of R. This reduces the problem to several three-dimensional
integrals of functions having a two-dimensional singular locus. If a ≥ b + 3
then R is a cube and the boundary of R is easy to describe. We restricted
our attention to this case. We smoothed the singularities by separation of
Puiseux expansions and substitutions. The resulting integrals were treated by
the Gauß-Legendre formula [Kr].
3. On the geometry of diagonal cubic threefolds
i. Surfaces on a hypersurface in P 4 . —
3.1. Lemma. –––– Let X ⊂ P 4 be any smooth hypersurface. Then, every (reduced
but possibly singular) surface S ⊂ X is a complete intersection X ∩ H d with a
hypersurface H d ⊂ P 4 .
Proof. By the Noether-Lefschetz Theorem, we have Pic(X ) ∼ =. The surface
S is a Weil divisor on X. Hence, O(S) = O(d) ∈ Pic(X ) for a
certain d > 0. The restriction Γ(P 4 , O(d)) → Γ(X, O(d)) is surjective as
H 1 (P 4 , O X (d − degX)) = 0 [Ha77, Theorem III.5.1.b)].
□
ii. Elliptic Cones. —
3.2. –––– Let X ⊂ P 4 () be the diagonal cubic threefold given by the equation
x 3 + y 3 + z 3 + v 3 + w 3 = 0. Fix ζ ∈such that ζ 3 = 1. Then, for every
point (x 0 : y 0 : z 0 ) on the elliptic curve F : x 3 + y 3 + z 3 = 0, the line given by
(x : y : z) = (x 0 : y 0 : z 0 ) and v = −ζw is contained in X. All these lines
together form a cone CF over F the cusp of which is (0 : 0 : 0 : −ζ : 1). C F is
a singular model of a ruled surface over an elliptic curve. This shows, there are
no other rational curves contained in C F .

Sec. 3] ON THE GEOMETRY OF DIAGONAL CUBIC THREEFOLDS 253
By permuting coordinates, one finds a total of thirty elliptic cones of that type
within X. The cusps of these cones are usually named Eckardt points [Mu-e,
Cl/G]. We call the lines contained in one of these cones the obvious lines lying
on X. It is clear that there are an infinite number of lines on X running through
each of the thirty Eckardt points (1 : −1 : 0 : 0 : 0), (1 : 0 : −1 : 0 : 0),
. . . , (0 : 0 : 0 : 1 : −1), (1 : −e 2πi/3 : 0 : 0 : 0), . . . , (0 : 0 : 0 : 1 : −e −2πi/3 ).
3.3. Proposition. (cf. [Mu-e, Lemma 1.18]). —– Let X ⊂ P 4 be the diagonal
cubic threefold given by theequation x 3 +y 3 +z 3 +v 3 +w 3 = 0. Then, through each
pointP ∈ X differentfromthethirtyEckardtpointstherearepreciselysixlinesonX.
Proof. Let P = (x 0 : y 0 : z 0 : v 0 : w 0 ). A line l through
P and another point Q = (x : y : z : v : w) is parametrized by
(s : t) ↦→ ((sx 0 + tx) : . . . : (sw 0 + tw)). Comparing coefficients at s 2 t, st 2 ,
and t 3 , we see that the condition that l lies on X may be expressed by the three
equations below.
x 2 0x + y 2 0y +z 2 0z + v 2 0v + w 2 0w = 0 (†)
x 0 x 2 + y 0 y 2 + z 0 z 2 + v 0 v 2 + w 0 w 2 = 0 (‡)
x 3 + y 3 + z 3 + v 3 + w 3 = 0 (§)
The first equation means that Q lies on the tangent hyperplane H P at P while
equation (§) just encodes that Q ∈ X. By Lemma 3.6, H P ∩ X is an irreducible
cubic surface.
On the other hand, the quadratic form q on the left hand side of equation (‡)
is of rank at least 3 as P is not an Eckardt point. Therefore, q is not just the
product of two linear forms. In particular, q| HP ≢ 0.
As H P ∩X is irreducible, Z(q| HP ) and H P ∩X do not have a component in common.
By Bezout’s theorem, their intersection in H P is a curve of degree 6. □
3.4. Remark. –––– It may happen that some of the six lines coincide. Actually,
it turns out that a line appears with multiplicity > 1 if and only if it is
obvious [Mu-e, Lemma 1.19]. In particular, for a general point P the six lines
through it are different from each other.
Under certain exceptional circumstances it is possible to write down all six
lines explicitly. For example, if P = ( 3 √ −4 : 1 : 1 : 1 : 1) then the line
( 3 √ −4t : (t + s) : (t + is) : (t − s) : (t − is)) through P lies on X. Permuting the
three rightmost coordinates yields all six lines.
3.5. Remark. –––– The following lemma is a special case of Zak’s theorem [Za,
Corollary 1.8] which is more elementary and completely sufficient for our purposes.
We present it here for convenience of the reader.

254 POINTS OF BOUNDED HEIGHT ON THREEFOLDS [Chap. VII
3.6. Lemma. –––– Let X ⊂ P 4 be the diagonal cubic threefold given by the equation
x 3 +y 3 +z 3 +v 3 +w 3 = 0. Then, for any hyperplane H ⊂ P 4 , the intersection
H ∩ X is irreducible and has at most finitely many singular points.
Proof. The intersection of X with a hyperplane H is singular precisely in those
points where H is tangent to X.
The tangent hyperplane H at (x 0 : y 0 : z 0 : v 0 : w 0 ) ∈ X is given by
x 2 0 x + y2 0 y + z2 0 z + v2 0 v + w2 0w = 0. From this formula, we see that H is
tangent to X at all points of the form (±x 0 : ±y 0 : ±z 0 : ±v 0 : ±w 0 ) which
happen to lie on X and at no others. By consequence, H ∩ X admits only a
finite number of singular points.
Every irreducible component of H ∩ X is a hypersurface in the projective
3-space H . Two such components would intersect in a curve of singular points.
Therefore, H ∩ X is necessarily irreducible.
□
3.7. Remark. –––– For a general point p ∈ X, the intersection of its tangent
hyperplane with X admits exactly one singular point. Indeed, let
p = (x 0 : y 0 : z 0 : v 0 : w 0 ). If (−x 0 : y 0 : z 0 : v 0 : w 0 ) ∈ X then x 0 = 0.
If (−x 0 : −y 0 : z 0 : v 0 : w 0 ) ∈ X then x 0 + ζy 0 = 0 for ζ a third root of
unity or ζ = 1. At this point, up to permutation of coordinates, the list of all
possibilities is already complete. Multi-tangent hyperplanes are caused only by
points lying on a finite arrangement of hyperplanes.
4. Accumulating subvarieties
i. The detection ofÉ-rational lines on the cubics. —
4.1. –––– On a cubic threefold X 3 a,b
, quadratic growth is predicted for the
number ofÉ-rational points of bounded height. Lines are the only curves with
such a growth rate.
The moduli space of the lines on a cubic threefold is well-understood. It is a
surface of general type [Cl/G, Lemma 10.13]. Nevertheless, we do not know
of a method to find allÉ-rational lines on a given cubic threefold, explicitly.
For that reason, we use the algorithm below which is an irrationality test for
the six lines through a given point (x 0 : y 0 : z 0 : v 0 : w 0 ) ∈ X 3 a,b (É).
4.2. Algorithm (Test the six lines through a given point for irrationality). —–
i) Let p run through the primes from 3 to N .
For each p, solve the system of equations (†), (‡), (§) (adapted to X 3 a,b ) in5 p .
If the multiples of (x 0 , y 0 , z 0 , v 0 , w 0 ) are the only solutions then output that there
is noÉ-rational line through (x 0 : y 0 : z 0 : v 0 : w 0 ) and terminate prematurely.

Sec. 4] ACCUMULATING SUBVARIETIES 255
ii) If the loop comes to its regular end then output that the point is suspicious.
It could possibly lie on aÉ-rational line.
4.3. Remark. –––– We use a very naive O(p)-algorithm to solve the system
of equations overp. If, say, x 0 ≠ 0 then it is sufficient to consider quintuples
such that x = 0. We parametrize the projective plane given by (†). Then, we
compute all points on the conic given by (†) and (‡). For each such point, we
compute the cubic form on the left hand side of (§). When a non-trivial solution
is found, we stop immediately.
We carried out the irrationality test on everyÉ-rational point found on any
of the cubics except for the points lying on an obvious line. We worked
with N = 600. It turned out that suspicious points are rare and that, at least in
our sample, each of them actually lies on aÉ-rational line.
The lines found. We found only 42 non-obviousÉ-rational lines on all of the
cubics X 3 a,b
for 100 ≥ a ≥ b ≥ 1 together. Among them, there are only five
essentially different ones. We present them in the table below. The list might
be enlarged by two, as X21,6 3 and X 22,5 3 may be transformed into X 48,21 3 and X 40,22 3 ,
respectively, by an automorphism of P 4 . Further, each line has six pairwise
different images under the obvious operation of the group S 3 .
TABLE 1. Sporadic lines on the cubic threefolds
a b Smallest point Point s.t. x = 0
19 18 (1 : 2 : 3 : -3 : -5) (0 : 7 : 1 : -7 : -18)
21 6 (1 : 2 : 3 : -3 : -3) (0 : 9 : 1 : -10 : -15)
22 5 (1 : -1 : 3 : 3 : -3) (0 : 27 : -4 : -60 : 49)
45 18 (1 : 1 : 3 : 3 : -3) (0 : 3 : -1 : 3 : -8)
73 17 (1 : 5 : -2 : 11 : -15) (0 : 27 : -40 : 85 : -96)
4.4. Remark. –––– It is a priori unnecessary to search for accumulating surfaces,
at least if we assume the conjectures of Batyrev/Manin and Lang formulated in
Chapter VI.
First of all, only rational surfaces are supposed to accumulate that many rational
points that it could be seen through our asymptotics of O(B 2 ). Indeed, a surface
which is abelian or bielliptic may not have more than O(log t B) points of
height < B. Non-rational ruled surfaces accumulate points in curves, anyway.
Further, it is expected (Conjecture VI.3.1) that K3 surfaces, Enriques surfaces,
and surfaces of Kodaira dimension one may have no more than O(B ε ) points
of height < B outside a finite union of rational curves. For surfaces of general
type, finally, expectations are even stronger (Conjecture VI.2.2).

256 POINTS OF BOUNDED HEIGHT ON THREEFOLDS [Chap. VII
A rational surface S is, up to exceptional curves, the image of a rational
map ϕ: P 2 X ⊂ P 4 . There is a birational morphism ε: P → P 2 such
that ϕ := ϕ ◦ ε is a morphism of schemes. ε is given by a sequence of blowingups
[Bv, Theorem II.11]. ϕ is defined by the linear system |dH − E| where
d := deg ϕ, H is a hyperplane section, and E is the exceptional divisor. On the
other hand, K := K P = −3H + E. Therefore, if d ≥ 3 then
H naive (ϕ(p)) = H dH −E (p) = H 3H −
3
d E(p)d/3 ≥ c · H −K (p) d/3
for p ∉ supp(E). Manin’s conjecture implies there are O((B log t B) 3 d ) = o(B 2 )
points of height < B on the Zariski-dense subset ϕ(P \ supp(E)) ⊆ S.
It remains to show that there are no rational maps ϕ: P 2
X of degree
d ≤ 2. Indeed, under this assumption, deg ϕ(P 2 ) ≤ 4. This implies,
by virtue of Lemma 3.1, that ϕ(P 2 ) is necessarily a hyperplane section X ∩ H .
Lemma 3.6 shows that X ∩ H contains only finitely many singular points.
It is, however, well known that cubic surfaces in 3-space which are the image
of P 2 under a quadratic map have a singular line [Bv, Corollary IV.8].
ii. The detection ofÉ-rational conics on the quartics. —
4.5. –––– On a quartic threefold, linear growth is predicted for the number of
É-rational points of bounded height. The assumption b > 0 ensures that there
are noÊ-rational lines contained in X 4 a,b
. The only other curves with at least
linear growth one could think about are conics.
We do not know of a method to find allÉ-rational conics on a given quartic
threefold, explicitly. Worse, we were unable to create an efficient routine to test
whether there is aÉ-rational conic through a given point. The resulting system
of equations seems to be too complicated to handle.
Conics through two points. A conic Q through (x 0 : y 0 : z 0 : v 0 : w 0 )
and (x 1 : y 1 : z 1 : v 1 : w 1 ) may be parametrized in the form
(s : t) ↦→ ((λx 0 s 2 + µx 1 t 2 + xst) : . . . : (λw 0 s 2 + µw 1 t 2 + wst))
for some x, y, z, v, w, λ, µ ∈. The condition that Q is contained in X 4 a,b leads
to a system G of seven equations in x, y, z, v, w, and λµ. The phenomenon
that λ and µ do not occur individually is explained by the fact that they are not
invariant under the automorphisms of P 1 which fix 0 and ∞.
4.6. Algorithm (Test for conic through two points). —– i) Let p run through
the primes from 3 to N .

Sec. 4] ACCUMULATING SUBVARIETIES 257
In the exceptional case that G could allow a solution such that p|x, y, z, v, w but
p 2 ∤ λµ, do nothing. Otherwise, solve G in6 p . If (0, 0, 0, 0, 0, 0)is the only solution
then output that there is noÉ-rational conic through (x 0 : y 0 : z 0 : v 0 : w 0 )
and (x 1 : y 1 : z 1 : v 1 : w 1 ) and terminate prematurely.
ii) If the loop comes to its regular end then output that the pair is suspicious.
It could possibly lie on aÉ-rational conic.
4.7. –––– To solve the system G in6 p , we use an O(p)-algorithm. Actually,
comparison of coefficients at s 7 t and st 7 yields two linear equations in x, y,
z, v, and w. We parametrize the projective plane I given by them. Comparison
of coefficients at s 6 t 2 and s 2 t 6 leads to a quadric O and an equation
λµ = q(x, y, z, v, w)/M with a quadratic form q overand an integer
M ≠ 0. The case p|M sends us to the next prime immediately. Otherwise,
we compute all points on the conic I ∩ O. For each of them, we
test the three remaining equations. When a non-trivial solution is found, we
stop immediately.
Conics through three points. Three points P 1 , P 2 , and P 3 on X 4 a,b
define a projective
plane P. The points together with the two tangent lines P∩ T P1 and P∩ T P2
determine a conic Q, uniquely. It is easy to transform this geometric insight
into a formula for a parametrization of Q. We then need a test whether a conic
given in parametrized form is contained in X 4 a,b
. This part is algorithmically
simple but requires the use of multiprecision integers.
Detecting conics. For each quartic X 4 a,b
, we tested every pair ofÉ-rational
points of height < 100 000 for a conic through them. The existence of a
conic through (P, Q) is equivalent to the existence of a conic through (gP, gQ)
for g ∈ (/2) 4 ⋊S 3 ⊆ Aut(X 4 a,b
). This reduces the running time by a factor of
about 96. Further, pairs already known to lie on the same conic were excluded
from the test.
For each pair (P, Q) found suspicious, we tested the triples (P, Q, R) for R running
through theÉ-rational points of height < 100 000, until a conic was found.
Due to the symmetries, one finds several conics at once. For each conic detected,
all points on it were marked as lying on this conic.
Actually, there were a few pairs found suspicious through which no conic could
be found. In any of these cases, it was easy to prove by hand that there is actually
noÊ-rational conic passing through the two points. This means, we detected
every conic which meets at least two of the rational points of height < 100 000.
Concerning our programming efforts, this was the most complex part of the
entire project.

258 POINTS OF BOUNDED HEIGHT ON THREEFOLDS [Chap. VII
The conics found. Up to symmetry, we found a total of 1 664É-rational conics
on all of the quartics X 4 a,b
for 1 ≤ a, b ≤ 100 together.
Among them, 1 538 are contained in a plane of type z = v+w and Yx−Xy = 0
for (X, Y, t) a rational point on the genus one curve aX 4 − bY 4 = 2t 2 . Further,
there are 93 conics which are slight modifications of the above with y
interchanged with z, v, or w. This is possible if b is a fourth power.
There is a geometric explanation for the occurrence of these conics. The hyperplane
given by z = v + w intersects X 4 a,b
in a surface S with the two singular
points (0 : 0 : −1 : e ±2πi/3 : e ∓2πi/3 ). The linear projection π : S P 1 to the
first two coordinates is undefined only in these two points. Its fibers are plane
quartics which split into two conics as (v+w) 4 +v 4 +w 4 = 2(v 2 +vw+w 2 ) 2 . After
resolution of singularities, the two conics become disjoint. ˜S is a ruled surface
over a twofold cover of P 1 ramified in the four points such that ax 4 − by 4 = 0.
I.e., over a curve of genus one.
In the case a is twice a square, a different sort of conics comes from the equations
v = z + Dy and w = Ly when (L, D) is a point on the affine genus three curve
C b : L 4 + b = D 4 . Here, by 4 + z 4 + v 4 + w 4 becomes twice a full square
after the substitutions. This explains why this particular intersection of X 4 a,b
with a plane splits into two conics. We found 28 conics of this type. C b has
aÉ-rational point for b = 5, 15, 34, 39, 65, 80, and 84. The conics actually
admit aÉ-rational point for a = 2, 18, 32, and 98.
The remaining five conics are given as follows. For a = 3, 12, 27, or 48
and b = 10, intersect with the plane given by v = y + z and w = 2y + z.
For a = 17 and b = 30, put v = 2x + y and w = x + 3y + z.
4.8. Remark. –––– Again, it is not necessary to search for accumulating surfaces.
Here, rational maps ϕ: P 2 X ⊂ P 4 such that deg ϕ ≤ 3 need to be
taken into consideration. We claim, such a map is impossible.
If deg ϕ = 3 then we had ϕ: (λ : µ : ν) ↦→ (K 0 (λ, µ, ν) : . . . : K 4 (λ, µ, ν))
where K 0 , . . . , K 4 are cubic forms defined overÉ. K 0 = 0 defines a plane cubic
which has infinitely many real points, automatically. As the image of ϕ
is assumed to be contained in X 4 a,b , we have that K 0(λ, µ, ν) = 0 implies
K 1 (λ, µ, ν) = . . . = K 4 (λ, µ, ν) = 0 for λ, µ, ν ∈Ê. By consequence,
K 1 , . . . , K 4 are divisible by K 0 (or by a linear factor of K 0 in the case it is
reducible) and ϕ is not of degree three.
For deg ϕ ≤ 2, we had deg ϕ(P 2 ) ≤ 4 such that ϕ(P 2 ) = X ∩ H is a hyperplane
section. Lemma 3.6 shows it has at most finitely many singular points.
On the other hand, a quartic in P 3 which is the image of a quadratic map from P 2
is a Steiner surface. It is known [Ap, p. 40] to have one, two, or (in generic case)
three singular lines.

Sec. 5] RESULTS 259
5. Results
i. A technology to find solutions of Diophantine equations. —
5.1. –––– In Chapter XI (cf. [EJ2] and [EJ3]), we describe a modification
of D. Bernstein’s [Be] method to search efficiently for all solutions of naive
height < B of a Diophantine equation of the particular form
f (x 1 , . . . , x n ) = g(y 1 , . . . , y m ) .
The expected running-time of our algorithm is O(B max{n,m} ). Its basic idea is
as follows.
5.2. Algorithm (Search for solutions of a Diophantine equation). —–
i) (Writing)
Evaluate f on all points of the cube {(x 1 , . . . , x n ) ∈n | |x i | < B} of
dimension n. Store the values within a hash table H .
ii) (Reading)
Evaluate g on all points of the cube {(y 1 , . . . , y m ) ∈m | |y i | < B}. For each
value, start a search in order to find out whether it occurs in H . When a
coincidence is detected, reconstruct the corresponding values of x 1 , . . . , x n and
output the solution.
5.3. Remark. –––– In the case of a variety X e a,b
, the running-time is obviously
O(B 3 ). We decided to store the values of z e +v e +w e into the hash table.
Afterwards, we have to look up the values of ax e − by e .
In this form, the algorithm would lead to a program in which almost the entire
running-time is consumed by the writing part. Observe, however, the following
particularity of our method. When we search on up to O(B) threefolds,
differing only by the values of a and b, simultaneously, then the running-time
is still O(B 3 ).
5.4. Remark (Running times). —– We worked with B = 5 000 for the cubics
and B = 100 000 for the quartics. In either case, we dealt with all threefolds
arising for a, b = 1, . . . , 100, simultaneously.
The by far largest portion of the running time was spent on point search on
the quartics. All in all, this took around 155 days of CPU time. This is
approximately only three times longer than searching on a single threefold
had lasted. Searching for conics was done within 32 days. In comparison with
this, the corresponding computations for the cubics could be done in a negligible
amount of time. The reason for this is simply that for the cubics the search

260 POINTS OF BOUNDED HEIGHT ON THREEFOLDS [Chap. VII
bound was by far lower. A program with integrated line detection took us
approximately ten days.
To compute Peyre’s constants, the precomputation was actually the main part.
Point counting using FFT took 100 hours for the cubics and 100 hours for
the quartics. For the final computation, the running-time was a quarter of an
hour for either exponent.
ii. The results for the cubics. —
5.5. –––– We counted allÉ-rational points of height less than 5 000 on the
threefolds X 3 a,b where a, b = 1, . . . , 100 and b ≤ a. Note that X 3 a,b ∼ = X 3 b,a .
Points lying on one of the elliptic cones or on a sporadicÉ-rational line
in X a,b were excluded from the count. The smallest number of points found is
3 930 278 for (a, b) = (98, 95). The largest numbers of points are 332 137 752
for (a, b) = (7, 1) and 355 689 300 in the case that a = 1 and b = 1.
On the other hand, for each threefold X 3 a,b
whereas a, b = 1, . . . , 100
and b + 3 ≤ a, we calculated the expected number of points and the quotients
# { points of height < B found } / # { points of height < B expected }.
Let us visualize the quotients by two histograms.
250
250
200
200
150
150
100
100
50
50
0 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02 0 0.88 0.9 0.92 0.94 0.96 0.98 1 1.02
FIGURE 1. Distribution of the quotients for B = 1 000 and B = 5 000.

Sec. 5] RESULTS 261
The statistical parameters are listed in the table below.
TABLE 2. Parameters of the distribution in the cubic case
B = 1 000 B = 2 000 B = 5 000
mean value 0.98179 0.98854 0.99383
standard deviation 0.01274 0.00823 0.00455
iii. The results for the quartics. —
5.6. –––– We counted allÉ-rational points of height less than 100 000 on
the threefolds X 4 a,b
where a, b = 1, . . . , 100. It turns out that on 5 015 of
these varieties, there are noÉ-rational points occurring at all as the equation
is unsolvable inÉp for some small p. In this situation, Manin’s conjecture is
true, trivially.
For the remaining varieties, the points lying on a knownÉ-rational conic in X a,b
were excluded from the count. Table 3 shows the quartics sorted by the numbers
of points remaining.
TABLE 3. Numbers of points of height < 100 000 on the quartics.
a b # points # not on conic # expected
(by Manin-Peyre)
29 29 2 2 135
58 87 288 288 272
58 58 290 290 388
87 87
. .
386
.
386
.
357
.
34 1 9938976 5691456 5 673000
17 64 5708664 5708664 5 643000
1 14 7205502 6361638 6 483000
3 1 12657056 7439616 7 526000
We see that the variation of the quotients is higher than in the cubic case.

262 POINTS OF BOUNDED HEIGHT ON THREEFOLDS [Chap. VII
20
15
10
5
0 0.7 0.8 0.9 1 1.1 1.2 1.3
FIGURE 2. Distribution of the quotients for B = 100 000.
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
FIGURE 3. Distribution of the quotients for B = 1 000 and B = 10 000.
The statistical parameters are listed in the table below.

Sec. 5] RESULTS 263
TABLE 4. Parameters of the distribution in the quartic case
B = 1 000 B = 10 000 B = 100 000
mean value 0.9853 0.9957 0.9982
standard deviation 0.3159 0.1130 0.0372
iv. Interpretation of the result. —
5.7. –––– The results suggest that Manin’s conjecture should be true for the
two families of threefolds considered. In the cubic case, the standard deviation
is by far smaller than in the case of the quartics. This, however, is not very
surprising as on a cubic there tend to be much more rational points than on a
quartic. This makes the sample more reliable.
5.8. Remark. –––– The data we collected might be used to test the sharpening
of the asymptotic formula (∗) suggested by Sir P. Swinnerton-Dyer [S-D05].
5.9. Question. –––– Our calculations seem to indicate that the number of rational
points often approaches its expected value from below. Is that more than
an accidental effect?

CHAPTER VIII
ON THE SMALLEST POINT ON A
DIAGONAL QUARTIC THREEFOLD ∗
In the course of your work, you will from time to time encounter the situation
where the facts and the theory do not coincide. In such circumstances,
young gentlemen, it is my earnest advice to respect the facts.
IGOR IVANOVICH SIKORSKY
1. A computer experiment
1.1. –––– Let X ⊆ P nÉbe a Fano variety defined overÉ. If X (Éν) ≠ ∅ for
every ν ∈ Val(É) then it is natural to ask whether X (É) ≠ ∅ (Hasse’s principle).
Further, it would be desirable to have an a-priori upper bound for the height
of the smallestÉ-rational point on X as this would allow to effectively decide
whether X (É) ≠ ∅ or not.
When X is a conic, Legendre’s theorem on zeroes of ternary quadratic forms
proves the Hasse principle and, moreover, yields an effective bound for the
smallest point. For quadrics of arbitrary dimension, the same is true by an
observation due to J. W. S. Cassels [Cas55]. Further, there is a theorem of
C. L. Siegel [Si73, Satz 1] which provides a generalization to hypersurfaces
defined by norm equations. For more general Fano varieties, no theoretical
upper bound is known for the height of the smallestÉ-rational point. Some of
these varieties fail the Hasse principle.
In this chapter, we present some theoretical and experimental results concerning
the height of the smallestÉ-rational point on quartic hypersurfaces in P 4É.
1.2. –––– There is the conjecture, due to Yu. I. Manin, that the number ofÉ-rational
points of anticanonical height < B on a Fano variety X is asymptotically
equal to τB log rkPic(X )−1 B, for B → ∞. A detailed description of Manin’s
conjecture is given in Chapter VI.
(∗) This chapter is a revised version of the article: On the smallest point on a diagonal quartic
threefold, J. Ramanujan Math. Soc. 22(2007), 189–204, joint with A.-S. Elsenhans.

266 ON THE SMALLEST POINT ON A DIAGONAL QUARTIC THREEFOLD [Chap. VIII
In the particular case of a quartic threefold, the anticanonical height is the
same as the naive height. Further, rkPic(X ) = 1 and, finally, the coefficient
τ ∈Êequals the Tamagawa-type number τ (X ) introduced by E. Peyre (Definition
VI.5.35).
Thus, one expects ∼τ (X )B points of height < B. Assuming equidistribution,
the height of the smallest point should be ∼ 1
τ
. Being a bit optimistic, this
(X )
might lead to the expectation that m(X ), the height of the smallestÉ-rational
point on X, is always less than C
τ
for a certain absolute constant C.
(X )
1.3. –––– To test this expectation, we computed the Tamagawa number and
ascertained the smallestÉ-rational point for each of the quartic threefolds
X (−a,b)
4
⊂ P 4Égiven by ax 4 = by 4 + z 4 + v 4 + w 4 for a, b = 1, . . . , 1000.
On 516 820 of these varieties, there are noÉ-rational points as the equation is
unsolvable inÉp for p = 2, 5, or 29. (Note that, for each prime p different
from 2, 5, or 29, there are p-adic points already on the Fermat quartic given
by z 4 + v 4 + w 4 = 0.) On each of the remaining varieties,É-rational points
were found. In other words, there are no counterexamples to the Hasse principle
in this family.
The computation of Tamagawa numbers is explained in Chapter VII, Section 2.
The methods to systematically search for solutions of Diophantine equations
we applied are described in Chapter VII, Section 5.
The results are summarized by the diagrams below.
a, b = 1 . . . 100 a, b = 1 . . . 1000
FIGURE 1. Height of smallest point versus Tamagawa number
It is apparent from the diagrams that the experiment agrees with the expectation
above. The slope of a line tangent to the top right of each of the scatter
plots is indeed near (−1). However, we show in Section 2 that, in general, the
inequality m(X ) < C
τ
does not hold. The following remains a logical possibility.
(X )

Sec. 1] A COMPUTER EXPERIMENT 267
1.4. Question. –––– For every ε > 0, does there exist a constant C (ε) such
that, for each quartic threefold,
m(X ) <
C (ε)
τ (X ) 1+ε ?
1.5. Peyre’s constant. –––– Recall from the previous chapter that, in our case,
E. Peyre’s Tamagawa-type number may be defined as an infinite product
τ (X ) := ∏ τ ν (X ) .
ν∈Val(É)
Indeed, for a smooth quartic threefold X, one has O(−K ) = O(1). In particular,
α(X ) = 1.
For a prime number p, the definition of the local factor may be simplified to
(
τ p (X ) := 1 − 1 )
#X (/p
· m)
lim
p m→∞ p 3m
for X the model in P ngiven by the equation defining X.
τ ∞ (X ) is described in full generality in Definition VI.5.22. In the case of
the diagonal quartic threefold X (a 0, ... ,a 4 )
given by a 0 x 4 0 + . . . + a 4 x 4 4 = 0
(a 0 < 0, a 1 , . . . , a 4 > 0) in P 4É, this yields
τ ∞ (X (a 0, ... ,a 4 ) ) = 1
4 4 √
|a0 |
ZZZZ
where the domain of integration is
R
1
(a 1 y 4 + a 2 z 4 + a 3 v 4 + a 4 w 4 ) 3/4 dy dz dv dw (∗)
R := { (y, z, v, w) ∈ [−1, 1] 4 | | a 1 y 4 + a 2 z 4 + a 3 v 4 + a 4 w 4 | ≤ |a 0 | } .
1.6. –––– In the case of diagonal quartic threefolds, there is an estimate for
1
m(X ) in terms of τ (X ). Namely,
τ
admits a fundamental finiteness property.
(X )
More precisely, in Section 3, we will show the following results.
Theorem. Leta = (a 0 , . . . , a 4 )beavectorsuchthata 0 , . . . , a 4 ∈,a 0 < 0, and
a 1 , . . . , a 4 > 0. Denote by X a the quartic in P 4Égiven by a 0 x 4 0+ . . . +a 4 x 4 4 = 0.
Then, for each ε > 0 there exists a constant C (ε) > 0 such that
1
τ (X a ) ≥ C (ε) · H( 1
a 0
: . . . :
1
a 4
) 1
4
−ε
.
Corollary (Fundamental Finiteness). For each B > 0, there are only finitely many
quartics X a : a 0 x 4 0 + . . . + a 4 x 4 4 = 0 in P 4Ésuch that a 0 < 0, a 1 , . . . , a 4 > 0,
and τ (X a ) > B.

268 ON THE SMALLEST POINT ON A DIAGONAL QUARTIC THREEFOLD [Chap. VIII
Corollary (An inefficient search bound). There exists a monotonically decreasing
function S : (0, ∞) → [0, ∞), the search bound, satisfying the following condition.
Let X a be the quartic threefold given by the equation a 0 x 4 0 + . . . + a 4 x 4 4 = 0.
Assume a 0 < 0, a 1 , . . . , a 4 > 0, and X a (É) ≠ ∅. Then, X a admits aÉ-rational
point of height ≤ S(τ (X a )).
Proof. One may simply put S(t) := max min H(P).
τ (X a )≥t P∈X a (É) X a (É)≠∅
□
In other words, we have m(X a ) ≤ S(τ (X a )) as soon as X a (É) ≠ ∅.
2. A negative result
For a ∈Æ, let X (−a) ⊂ P 4Ébe given by ax 4 = y 4 + z 4 + v 4 + w 4 and let
m(X (−a) ) := min { H(x : y : z : v : w) | (x : y : z : v : w) ∈ X (−a) (É)}
be the smallest height of aÉ-rational point on X (−a) . We compare m(X (−a) )
with the Tamagawa type number τ (−a) := τ (X (−a) ).
2.1. Notation. –––– For a prime number p and integers y, z, . . . , not all of
which are equal to zero, we write gcd p
(y, z, . . . ) for the largest power of p
dividing all of the y, z, . . . .
2.2. Theorem. –––– There is no constant C such that
for all a ∈Æ.
m(X (−a) ) < C
τ (−a)
Proof. The proof consists of several steps.
First step. For a ≥ 4, one has τ (−a)
∞ = 1 4√ a
I where I is an integral independent
of a.
This follows immediately from formula (∗) above.
Second step. For the height of the smallest point, we have m(X (−a) ) ≥ 4 √ a
4 .
|x| ≥ 1 yields y 4 + z 4 + v 4 + w 4 ≥ a and max{|y|, |z|, |v|, |w|} ≥ 4 √ a
4 .
Third step. There are two positive constants C 1 and C 2 such that, for all a ∈Æ,
C 1 < ∏
τ (−a)
p prime
p>13,p∤a
p < C 2 .

Sec. 2] A NEGATIVE RESULT 269
For a prime p of good reduction, Hensel’s lemma shows
(
τ p (X (−a) ) = 1 − 1 )
· #X (−a) (p)
.
p p 3
Further, for the number of points on a non-singular variety over a finite
field, there are excellent estimates provided by the Weil conjectures, proven
by P. Deligne. In our situation, [Del, Théorème (8.1)] may be directly applied.
It shows #X (−a) (p) = p 3 + p 2 + p + 1 + E (−a) with an errorterm
|E (−a) | ≤ 60p 3/2 . Note that dim H 3 (X,Ê) = 60 for every smooth quartic
threefold X in P 4[Hi, Satz 2.4].
Consequently,
1 −
60(1 − 1/p)
p 3/2
− 1 p 4 ≤ τ (−a)
p
≤ 1 +
60(1 − 1/p)
p 3/2 − 1 p 4 .
Here, the left hand side is positive for p > 13. The infinite product over all
1 − 60(1−1/p) − 1 (respectively 1 + 60(1−1/p) − 1 ) is convergent.
p 3/2 p 4 p 3/2 p 4
Fourth step. There is a sequence (a i ) i∈Æof
i∈Æ
natural numbers such that
( )
∏ τ (−a i)
p
p prime
p≤13 or p|ai
is unbounded.
Let C ∈Êbe given. We will show that
∏
τ p
(−a)
p prime
p≤13 or p|a
> C
when a := p 1 · . . . · p r is a product of sufficiently many different primes
p i ≡ 3 (mod 4) fulfilling a ≡ 1 (mod M ) for M := 16 · 3 · 5 · 7 ·11·13.
Let p be a prime such that p|a. We made sure that p 4 ∤a. Then, for any
point (x : y : z : v : w) ∈ X (−a) (/p n), the assumption p | y, z, v, w
would imply p 4 |ax 4 and p|x. Therefore, gcd p
(y, z, v, w) = 1. Further,
y 4 + z 4 + v 4 + w 4 ≡ 0 (mod p).
As p ≡ 3 (mod 4), the number of solutions of that congruence is the same as
that of y 2 + z 2 + v 2 + w 2 ≡ 0 (mod p). Since p remains prime in[i], this
quadratic form is a direct sum of two norm forms. Its number of zeroes in4 p
is therefore equal to 1 + (p − 1)(p + 1) 2 .
gcd p
(y, z, v, w) = 1 implies that Hensel’s lemma is applicable. It shows
#X (−a) (/p n) = pn · (p − 1)(p + 1) 2 p 3(n−1)
(p − 1)p n−1 = p 3n−2 (p + 1) 2 .

270 ON THE SMALLEST POINT ON A DIAGONAL QUARTIC THREEFOLD [Chap. VIII
Hence, τ (−a)
p
= (1 − 1 p ) (p+1)2
p 2
∏
p prime
p≤13 or p|a
τ (−a)
p =
= 1 + 1 − 1 − 1 . We may consequently write
p p 2 p 3
r (
1 + 1 − 1 − 1 )
·
p i pi
2 p
∏τ (−a)
i
3 p .
∏
i=1
The second product is over p = 2 and a finite number of primes of good reduction.
The value of the product depends only on the residue of a modulo
M = 16·3·5·7·11·13by virtue of Lemma VII.2.10.a) and b.ii). In particular,
our assumption a ≡ 1 (mod M ) implies
T := ∏
τ p
(−a)
p prime
p≤13
is a constant. It is clear that T > 0 as the equation x 4 = y 4 + z 4 + v 4 + w 4
admits a non-zero solution in/mfor any m > 1.
Itremains to show that there exists a set {p 1 , . . . , p r } of primes p i ≡ 3 (mod 4)
such that p i > 13, r (
1 + 1 − 1 − 1 )
≥ C p i pi
2 pi
3 T , (†)
∏
i=1
and p 1 · . . . · p r ≡ 1 (mod M ).
1
Condition (†) is easily satisfied as the series ∑ p≡3 (mod 4) diverges. We find a
p
set {p 1 , . . . , p s } of prime numbers p i > 13 of the form p i ≡ 3 (mod 4) such
− 1 p 3 i
p prime
p≤13
) ≥
C
T . Enlarging {p 1, . . . , p s } makes that product
(
that ∏ s i=1 1 +
1
p i
− 1 pi
2
even bigger. We may therefore assume p 1 · . . . · p s ≡ 3 (mod 4).
The numbers M and p 1 · . . . · p s are relatively prime. By Dirichlet’s prime number
theorem, there exists a prime p s+1 , larger than each of the p 1 , . . . , p s , such
that p 1 · . . . · p s+1 ≡ 1 (mod M ). This shows, in particular, p s+1 ≡ 3 (mod 4).
The assertion follows.
Conclusion. The four steps together show that m(X (−a) )·τ (−a) is unbounded.
□
3. The fundamental finiteness property
i. An estimate for the factors at the finite places. —
3.1. Notation. –––– i) For a prime number p and an integer x ≠ 0, we
put x (p) := p ν p(x) . Note x (p) = 1/‖x‖ p for the normalized p-adic valuation.
ii) By putting ν(x) := min
ξ∈p
to/p r.
x=(ξ mod p r )
ν(ξ), we carry the p-adic valuation fromp over
Note that any 0 ≠ x ∈/p rhas the form x = ε·p ν(x) where ε ∈ (/p r) ∗ is
a unit. Clearly, ε is unique only in the case ν(x) = 0.

Sec. 3] THE FUNDAMENTAL FINITENESS PROPERTY 271
3.2. Definition. –––– For (a 0 , . . . , a 4 ) ∈5 , r ∈Æ, and ν 0 , . . . , ν 4 ≤ r, put
S (r)
ν 0 , ... ,ν 4 ;a 0 , ... ,a 4
:= {(x 0 , . . . ,x 4 ) ∈ (/p r) 5 |
ν(x 0 ) = ν 0 , . . . ,ν(x 4 ) = ν 4 ; a 0 x 4 0 + . . . + a 4 x 4 4 = 0 ∈/p r} .
For the particular case ν 0 = . . . = ν 4 = 0, we will write
I.e.,
Z (r)
a 0 , ... ,a 4
:= S (r)
0, ... ,0;a 0 , ... ,a 4
.
Z (r)
a 0 , ... ,a 4
= {(x 0 , . . . , x 4 ) ∈ [(/p r) ∗ ] 5 | a 0 x 4 0 + . . . + a 4 x 4 4 = 0 ∈/p r} .
We will use the notation z (r)
a 0 , ... ,a 4
:= #Z (r)
a 0 , ... ,a 4
.
3.3. Sublemma. –––– If p k |a 0 , . . . , a 4 and r > k then we have
z (r)
a 0 , ... ,a 4
= p 5k · z (r−k)
a 0 /p k , ... ,a 4 /p k .
Proof. Since a 0 x 4 0 + . . . + a 4x 4 4 = pk (a 0 /p k · x 4 0 + . . . + a 4/p k · x 4 4 ), there is a
surjection
ι: Z a (r)
0 , ... ,a 4
−→ Z (r−k) ,
a 0 /p k , ... ,a 4 /p k
given by (x 0 , . . . , x 4 ) ↦→ ((x 0 mod p r−k ), . . . , (x 4 mod p r−k )). The kernel of
the homomorphism of modules underlying ι is (p r−k/pr) 5 . □
3.4. Lemma. –––– Assume gcd p
(a 0 , . . . , a 4 ) = p k . Then, there is an estimate
z (r)
a 0 , ... ,a 4
≤ 8p 4r+k .
Proof. Suppose first that k = 0. This means, one of the coefficients is prime
to p. Without restriction, assume p ∤a 0 .
For any (x 1 , . . . , x 4 ) ∈ (/p r) 4 , there appears an equation of the form
a 0 x 4 0 = c. For p odd, it cannot have more than four solutions in (/p r) ∗
as this group is cyclic. On the other hand, in the case p = 2, we have
(/2 r) ∗ ∼ =/2 r−2×/2and up to eight solutions are possible.
The general case follows directly from Sublemma 3.3. Indeed, if k < r then
z (r)
a 0 , ... ,a 4
= p 5k · z (r−k)
a 0 /p k , ... ,a 4 /p k ≤ p 5k · 8p 4(r−k) = 8p 4r+k .
On the other hand, if k ≥ r then the assertion is completely trivial since
z (r)
a 0 , ... ,a 4
= #Z (r)
a 0 , ... ,a 4
< p 5r ≤ p 4r+k < 8p 4r+k .
□

272 ON THE SMALLEST POINT ON A DIAGONAL QUARTIC THREEFOLD [Chap. VIII
3.5. Remark. –––– The proof shows that in the case p ≠ 2 the same inequality
is true with coefficient 4 instead of 8. If p ≡ 3 (mod 4) then one could even
reduce the coefficient to 2. Unfortunately, these observations do not lead to a
substantial improvement of our final result.
3.6. Lemma. –––– Let r ∈Æand ν 0 , . . . , ν 4 ≤ r. Then,
#S ν (r)
0 , ... ,ν 4 ;a 0 , ... ,a 4
= z(r) · ϕ(p r−ν p 4ν 0a 0 , ... ,p 4ν 0
4a ) · . . . · ϕ(p
r−ν 4)
4
.
ϕ(p r ) 5
Proof. As p 4ν 0 a0 x 4 0 + . . . + p4ν 4 a4 x 4 4 = a 0(p ν 0 x0 ) 4 + . . . + a 4 (p ν 4 x4 ) 4 , we have
a surjection
π : Z (r) −→ S (r)
p 4ν 0a 0 , ... ,p 4ν 4a 4
ν 0 , ... ,ν 4 ;a 0 , ... ,a 4
,
given by (x 0 , . . . , x 4 ) ↦→ (p ν 0 x0 , . . . , p ν 4 x4 ).
For i = 0, . . . , 4, consider the mapping ι:/p r→/pr, x 0 ↦→ p ν i
x 0 .
If ν i = r then ι is the zero map. All ϕ(p r ) = (p − 1)p r−1 units are mapped
to zero. Otherwise, observe that ι is p ν i
: 1 on its image. Further, ν(ι(x)) = ν i
if and only if x is a unit. By consequence, π is (K (ν 0)
· . . . · K (ν4) ) : 1 when
we put K (ν) := p ν for ν ≠ r and K (r) := (p − 1)p r−1 . Summarizing, we could
have written K (ν) := ϕ(p r )/ϕ(p r−ν ). The assertion follows.
□
3.7. Corollary. –––– Let (a 0 , . . . , a 4 ) ∈ (\{0}) 5 . Then, for the local factor
:= τ p (X (a 0, ... ,a 4 ) ), one has
τ (a 0, ... ,a 4 )
p
τ (a 0, ... ,a 4 )
p
r
r→∞
∑
ν 0 , ... ,ν 4 =0
= lim
Proof. Remark VI.5.19 shows that
τ (a 0, ... ,a 4 )
p
Lemma 3.6 yields the assertion.
z (r) · ϕ(p r−ν 0
) · . . . · ϕ(p r−ν 4
)
p 4ν 0a 0 , ... ,p 4ν 4a 4
.
p 4r · ϕ(p r ) 5
r
#S ν
r→∞
∑
(r)
0 , ... ,ν 4 ;a 0 , ... ,a 4
ν 0 , ... ,ν 4 =0
= lim
p 4r .
3.8. Proposition –––– Let (a 0 , . . . , a 4 ) ∈ (\{0}) 5 . Then, for each ε such
that 0 < ε < 1 , one has
4
(
τ (a 0, ... ,a 4 ) 1
)( 1
) 4 (
p ≤ 8
· a
(p)
1 − 1 1 − 1 0
· . . . · a (p) ) 1−ε (
4
3
a (p) ) ε.
4
p 1−4ε p ε
Proof. By Lemma 3.4,
z (r)
p 4ν 0a 0 , ... ,p 4ν 4a 4
/p 4r ≤ 8 gcd p
(p 4ν 0
a 0 , . . . , p 4ν 4
a 4 )
= 8 gcd ( p 4ν 0
a (p)
0 , . . . , p4ν 4
a (p)
4
)
.
□

Sec. 3] THE FUNDAMENTAL FINITENESS PROPERTY 273
(
(p)) Writing k i := ν p (a i ) = ν p a i , we see
z (r)
p 4ν 0a 0 , ... ,p 4ν 4a 4
/p 4r ≤ 8 gcd(p 4ν 0+k 0
, . . . , p 4ν 4+k 4
)
= 8p min{4ν 0+k 0 , ... ,4ν 4 +k 4 } .
We estimate the minimum by a weighted arithmetic mean with weights 1−ε
, and ε,
1−ε
4 , 1−ε
, 1−ε
4 4
min{3ν 0 + k 0 , . . . , 3ν 3 + k 3 }
This shows
≤ 1 − ε
3
· (3ν 0 + k 0 ) + 1 − ε · (3ν 1 + k 1 )
3
+ 1 − ε · (3ν 2 + k 2 ) + ε(3ν 3 + k 3 )
3
= (1 − ε)(ν 0 + ν 1 + ν 2 ) + 3εν 3
+ 1 − ε (k 0 + k 1 + k 2 ) + εk 3 .
3
z (r) /p 4r ≤ 8p (1−ε)(ν 0+ ... +ν 3 )+4εν 4 + 1−ε
p 4ν 0a 0 , ... ,p 4ν 4 (k 0+ ... +k 3 )+εk 4
4a 4
(a
= 8p (1−ε)(ν 0+ ... +ν 3 )+4εν4 (p)
·
0
· . . . · a (p) ) 1−ε (
4
3
a (p) ) ε.
4
We may therefore write
τ (a 0, ... ,a 4 )
p
≤ 8 ( a (p)
0
· . . . · a (p) ) 1−ε (
4
3
a (p)
4
r
r→∞
∑
ν 0 , ... ,ν 4 =0
· lim
) ε
p (1−ε)(ν 0+ ... +ν 3 )+4εν 4 · ϕ(p
r−ν 0) · . . . · ϕ(p
r−ν 4)
.
ϕ(p r ) 5
Here, the term under the limit is precisely the product of four copies of the
finite sum r
p
∑
(1−ε)ν · ϕ(p r−ν r−1
) 1
=
ν=0
ϕ(p r )
∑
ν=0
(p ε ) + p 1
ν p − 1 (p ε ) r
and one copy of the finite sum
r
p
∑
4εν · ϕ(p r−ν r−1
)
=
ν=0
ϕ(p r )
∑
ν=0
1
(p 1−4ε ) + p
ν
1
p − 1 (p 1−4ε ) . r
For r → ∞, geometric series do appear while the additional summands tend
to zero.
□
4 ,
3.9. Remark. –––– Unfortunately, the constants
(
C p (ε) 1
)( 1
) 4
:= 8
1 − 1 1 − 1 p 1−4ε p ε

274 ON THE SMALLEST POINT ON A DIAGONAL QUARTIC THREEFOLD [Chap. VIII
have the property that the product ∏ p C p
(ε) diverges. On the other hand, we
have at least that C p
(ε) is bounded for p → ∞, say C p (ε) ≤ C (ε) .
3.10. Lemma. –––– Let C > 1 be any constant. Then, for each ε > 0, one has
∏
p prime
p|x
for a suitable constant c (depending on ε).
C ≤ c · x ε
Proof. This follows directly from [Nat, Theorem 7.2] together with [Nat,
Section 7.1, Exercise 7].
□
3.11. Proposition –––– For each ε > 0, there exists a constant c such that
∏ τ (a 0, ... ,a 4 )
p ≤ c · | a 0 · . . . · a 4 | 4 1 · ∏
p prime
for all (a 0 , . . . , a 4 ) ∈ (\{0}) 5 .
p prime
min ‖a i‖ 1 4 −ε
p
i=0, ... ,4
Proof. As already noticed in the third step of the proof of Theorem 2.2, the
product over all primes of good reduction is bounded by consequence of the
Weil conjectures. It, therefore, remains to show that
∏ τ (a 0, ... ,a 4 )
p ≤ c · |a 0 · . . . · a 4 | 4 1 · ∏
p prime
p|2a 0 ... a 4
p prime
For this, we may assume that ε is small, say ε < 1 4 .
Then, by Proposition 3.8, we have at first
min ‖a i‖ 1 4 −ε
i=0, ... ,4
p .
τ (a 0, ... ,a 4 )
p
≤ C (ε)
p
= C (ε)
p
·
·
(a(p)
0
· . . . · a (p) ) 1
4 − ε5
3
· (a (p)
4 ) 4 5 ε
(a(p)
0
· . . . · a (p) ) 1
3 a(p) 4
− ε5
4
· (a (p)
4 )− 1 4 +ε .
Here, the indices 0, . . . , 4 are interchangeable. Hence, it is even allowed to
write
τ (a 0, ... ,a 4 )
p
≤ C (ε)
p
= C (ε)
p
·
·
(a(p)
0
· . . . · a (p)
4
(a(p)
0
· . . . · a (p)
4
) 1
4
− ε5 · (max
) 1
4
− ε5 · min
i
i
)
a (p) −
1
4
+ε
i
‖a i ‖ 1 4 −ε
p .
Now, we multiply over all prime divisors of 2a 0 · . . . · a 4 . Thereby, on the
right hand side, we may twice write the product over all primes since the two

Sec. 3] THE FUNDAMENTAL FINITENESS PROPERTY 275
rightmost factors are equal to one for p ∤a 0 · . . . · a 4 , anyway.
∏ τ (a 0, ... ,a 4 )
p
p prime
p|2a 0 ... a 4
≤ ∏ C p
(ε)
p prime
p|2a 0 ... a 4
(
(p)
· ∏ a
0
· . . . · a (p) ) 1
4
− ε 5
4
· ∏
p prime
= ∏ C p (ε) · | a 0 · . . . · a 4 | 4 1 − ε 5 · ∏
p prime
p|2a 0 ... a 4
p prime
p prime
min ‖a 4
i‖ 1 −ε
p
i=0, ... ,4
min ‖a i‖ 1 4 −ε
p
i=0, ... ,4
when we observe that ∏ p a (p) = | a|. Further, we have C p
(ε)
Lemma 3.10,
∏ C (ε) ≤ c · |2a 0 · . . . · a 4 | ε 5 .
p prime
p|2a 0 ... a 4
We finally estimate 2 ε 5 by a constant. The assertion follows.
≤ C (ε) and, by
ii. A bound for the factor at the infinite place. — We want to estimate the integral
τ (a 0, ... ,a 4 )
∞ := τ ∞ (X (a 0, ... ,a 4 ) ) described in 1.5.
3.12. Lemma. –––– There exist two constants C 1 and C 2 such that
τ (a 0, ... ,a 4 )
∞
⎧
1
(
⎨ 4√ C1 min(|a 0 |, 2a 4 ) 1/4) , if |a 0 | ≤ a 4 ,
|a0 |a
≤
1·...·a 4
(
1
⎩ 4√ C1 min(|a 0 |, 2a 4 ) 1/4 + C 2 a 1/4
|a0 |a
4
log min(|a 0|,3a 1 )) a 1·...·a 4 4
, otherwise,
for all (a 0 , . . . , a 4 ) ∈Ê5 satisfying a 1 ≥ a 2 ≥ a 3 ≥ a 4 ≥ 1 and a 0 ≤ −1.
Proof. A linear substitution shows
τ (a 0, ... ,a 4 )
∞ = 1 4 ·
where
ZZZZ
1
√
| a0 |a 1 · . . . · a 4
4
1
dy dz dv dw
(y 4 + z 4 + v 4 + w 4 )
3/4
R (0)
R (0) := {(y, z, v, w) ∈ [−a 1/4
1
, a 1/4
1
] × · · · × [−a 1/4
4
, a 1/4
4
]
| |y 4 + z 4 + v 4 + w 4 | ≤ |a 0 |} .
The integrand is non-negative. We cover R (0) ⊆ R 1 ∪ R 2 by two sets as follows,
R 1 := {(y, z, v, w) ∈Ê4 | y 4 + z 4 + v 4 + w 4 ≤ min(| a 0 |, 2a 4 )} ,
R 2 := {(y, z, v, w) ∈Ê4 | a 4 ≤ y 4 + z 4 + v 4 ≤ min(| a 0 |, 3a 1 ) and
□
w ∈ [−a 1/4
4
, a 1/4
4
]} ,
and estimate. In the case | a 0 | ≤ a 4 , the domain of integration is covered by R 1
alone and we may omit R 2 , completely.

276 ON THE SMALLEST POINT ON A DIAGONAL QUARTIC THREEFOLD [Chap. VIII
By homogeneity, we have
ZZZZ
1
(y 4 + z 4 + v 4 + w 4 ) dy dz dv dw = ω 3/4 1 ·
R 1
min(| a 0 |,2a 4 ) 1/4
Z
0
1
r 3 · r3 dr
= ω 1 · min(| a 0 |, 2a 4 ) 1/4
where ω 1 is the three-dimensional hypersurface measure of the l 4 -unit hypersphere
S 1 := {(y, z, v, w) ∈Ê4 | y 4 + z 4 + v 4 + w 4 = 1} .
Further,
ZZZZ
1
dy dz dv dw
(y 4 + z 4 + v 4 + w 4 )
3/4
R 2
ZZZ
≤ 2a 1/4 1
4
dy dz dv
(y 4 + z 4 + v 4 )
3/4
R 3
where
R 3 := {(y, z, v) ∈Ê3 | a 4 ≤ y 4 + z 4 + v 4 ≤ min(| a 0 |, 3a 1 )} .
The latter integral may be treated in much the same way as the one above.
We see
ZZZ
1
(y 4 + z 4 + v 4 ) dy dz dv = ω 3/4 2 ·
R 3
min(|a 0 |,3a 1 ) 1/4
Z
a 1/4
4
1
r 3 · r2 dr
where ω 2 is the usual two-dimensional hypersurface measure of the l 4 -unit sphere
Finally,
min(| a 0 |,3a 1 ) 1/4
Z
a 1/4
4
S 2 := {(y, z, v) ∈Ê3 | y 4 + z 4 + v 4 = 1} .
1
r 3 · r2 dr = log min(| a 0|, 3a 1 ) 1/4
a 1/4
4
= 1 4 log min(| a 0|, 3a 1 )
a 4
. □
3.13. Proposition –––– For every ε > 0, there exists a constant C such that
τ (a 0, ... ,a 4 )
∞
≤ C · | a 0 · . . . · a 4 | − 1 4 +ε · min
i=0, ... ,4 ‖a i‖ 1 4
∞
for each (a 0 , . . . , a 4 ) ∈5 satisfying a 0 < 0 and a 1 , . . . , a 4 > 0.
Proof. We assume without restriction that a 1 ≥ . . . ≥ a 4 . There are two cases
to be distinguished.

Sec. 3] THE FUNDAMENTAL FINITENESS PROPERTY 277
First case. | a 0 | ≤ a 4 .
Then, by Lemma 3.12, we have
τ (a 0, ... ,a 4 )
∞ ≤ | a 0 · . . . · a 4 | − 1 4 · C1 min{| a 0 |, 2a 4 } 1 4
= C 1 · | a 0 · . . . · a 4 | − 1 4 · | a0 | 1 4
= C 1 · | a 0 · . . . · a 4 | − 1 4 · min
i=0, ... ,4 ‖a i‖ 1 4
∞ .
Second case. | a 0 | > a 4 .
Here, Lemma 3.12 shows
τ (a 0, ... ,a 4 )
∞ ≤ | a 0 · . . . · a 4 | − 1 4
≤ | a 0 · . . . · a 4 | − 1 4
= | a 0 · . . . · a 4 | − 1 4 · | a4 | 1 4
(
C 1 min{|a 0 |, 2a 4 } 1 4 + C2 a 1/4
(
C 1 (2a 4 ) 1 4 + C2 a 1/4
4
log | a 0|
(
C 1 2 1 4 + C2 log | a 0|
a 4
)
4
log min{|a 0|, 3a 1 }
)
a 4
)
a 4
(
= | a 0 · . . . · a 4 | − 4 1 · min ‖a i‖ 1 4
∞ · C 1 2 4 1 + C2 log | a )
0|
i=0, ... ,4 a 4
≤ | a 0 · . . . · a 4 | − 1 4 · min
i=0, ... ,4 ‖a i‖ 1 4
∞ · (C
1 2 1 4 + C2 log | a 0 · . . . · a 4 | )
≤ | a 0 · . . . · a 4 | − 1 4 · min
i=0, ... ,4 ‖a i‖ 1 4
∞ · (C
3 · | a 0 · . . . · a 4 | ) ε
.
□
iii. The Tamagawa number. —
3.14. Proposition –––– For every ε > 0, there exists a constant C > 0 such that
) 1
1 4
a 4
1
τ ≥ C · H( 1
a 0
: . . . :
(a 0, ... ,a 4 )
| a 0 · . . . · a 4 | ε
for each (a 0 , . . . , a 4 ) ∈5 satisfying a 0 < 0 and a 1 , . . . , a 4 > 0.
Proof. By Proposition 3.13, we have
τ (a 0, ... ,a 4 )
∞
On the other hand, by Theorem 3.11,
≤ C 1 · | a 0 · . . . · a 4 | − 1 4 + ε 2 · min
i=0, ... ,4 ‖a i‖ 1 4
∞ .
∏ τ (a 0, ... ,a 4 )
p ≤ C 2 · | a 0 · . . . · a 4 | 4 1 · ∏
p prime
p prime
min ‖a i‖ 1 4 − ε 2
i=0, ... ,4
p .

278 ON THE SMALLEST POINT ON A DIAGONAL QUARTIC THREEFOLD [Chap. VIII
It follows that
τ (a 0, ... ,a 4 )
≤ C 3·|a 0 · . . . · a 4 | ε 2 · ∏
and
p prime
1
τ (a 0, ... ,a 4 )
≥ 1 C 3
·
= 1 C 3
·
= 1 C 3
·
[ ]
min ‖a 4
i‖ 1 p · min ‖a i‖ 1 −
ε
4
∞ · ∏ min ‖a 2
i‖ p
i=0, ... ,4 i=0, ... ,4 i=0, ... ,4
[
∏
p prime
] −
1
min ‖a 4
i‖ p ·
i=0, ... ,4
| a 0 · . . . · a 4 | ε 2 · ∏
p prime
∥ 1 ∏ max ∥ 1 ∥∥ 4
i=0, ... ,4
a i
· max
p
p prime
| a 0 · . . . · a 4 | ε 2 · ∏
p prime
H ( 1
a 0
: . . . :
[
| a 0 · . . . · a 4 | ε 2 · ∏
p prime
p prime
[ ] −
1
min ‖a 4
i‖ ∞
i=0, ... ,4
[
min
‖a ] −
ε
i‖ 2 p
i=0, ... ,4 ∥ 1 ∥ 1 ∥∥ 4
a i
∞
i=0, ... ,4
[
max
i=0, ... ,4 a(p) i
) 1
1 4
a 4
It is obvious that max
i=0, ... ,4 a(p) i ≤ | a (p)
0
· . . . · a (p)
4 | and
This shows
max
i=0, ... ,4 a(p) i
∏ | a (p)
0
· . . . · a (p)
4 | = | a 0 · . . . · a 4 |.
p prime
] ε
2
] ε
2
.
1
τ ≥ 1 H ( ) 1
1 1
a
·
0
: . . . : 4
a 4 (a 0, ... ,a 4 )
C 3 | a 0 · . . . · a 4 | ε 2 · | a 0 · . . . · a 4 | ε 2
= 1 · H( ) 1
1 1
a 0
: . . . : 4
a 4
. □
C 3 | a 0 · . . . · a 4 | ε
3.15. Lemma. –––– Let (a 0 : . . . : a 4 ) ∈ P 4 (É) be any point such that
a 0 ≠ 0, . . . , a 4
∈
≠ 0. Then,
H(a 0 : . . . : a 4 ) ≤ H( 1 1
a 0
: . . . :
a 4
) 4 .
Proof. First, observe that (a 0 : . . . : a 4 ) ↦→ ( )
1
1
a 0
: . . . :
a 4
is a welldefined
map. Hence, we may assume without restriction that a 0 , . . . , a 4
and gcd(a 0 , . . . , a 4 ) = 1. This yields H(a 0 : . . . : a 4 ) = max | a i|.
i=0, ... ,4
On the other hand, ( 1 1
a 0
: . . . :
a 4
) = (a 1 a 2 a 3 a 4 : . . . : a 0 a 1 a 2 a 3 ). Consequently,
H ( 1
a 0
: . . . :
1
a 4
) ≤ [ max
i=0,...,4 | a i|] 4 = H(a 0 : . . . : a 4 ) 4 .

Sec. 3] THE FUNDAMENTAL FINITENESS PROPERTY 279
From this, the asserted inequality emerges when the roles of a i and 1 a i
are interchanged.
□
3.16. Corollary. –––– Let a 0 , . . . , a 4 ∈such that gcd(a 0 , . . . , a 4 ) = 1. Then,
| a 0 · . . . · a 4 | ≤ H ( 1
a 0
: . . . :
1
a 4
) 20.
Proof. Observe that | a 0 · . . . · a 4 | ≤ max | a i| 5 = H(a 0 : . . . : a 4 ) 5 and apply
i=0, ... ,4
Lemma 3.15.
□
3.17. Theorem. –––– For each ε > 0, there exists a constant C (ε) > 0 such that,
for all (a 0 , . . . , a 4 ) ∈5 satisfying a 0 < 0 and a 1 , . . . , a 4 > 0,
1
τ ≥ C (ε) · H( ) 1
1 1
(a 0, ... ,a 4 ) a 0
: . . . : 4
−ε
a 4
.
Proof. We may assume that gcd(a 0 , . . . , a 4 ) = 1. Then, by Proposition 3.14,
1
τ ≥ C · H( ) 1
1 1
a 0
: . . . : 4
a 4
.
(a 0, ... ,a 4 )
|a 0 · . . . · a 4 | ε 20
(
Corollary 3.16 yields | a 0 · . . . · a 4 | ε 20 ≤ H 1
)
1 ε.
a 0
: . . . :
a 4
3.18. Corollary (Fundamental finiteness). —– For each B > 0, there are only
finitely many quartics X (a 0, ... ,a 4 ) : a 0 x 4 0 + . . . + a 4 x 4 4 = 0 in P 4Ésuch that a 0 < 0,
a 1 , . . . , a 4 > 0, and τ (a 0, ... ,a 4) > B.
Proof. This is an immediate consequence of the comparison to the naive height
established in Theorem 3.17.
□
□

CHAPTER IX
GENERAL CUBIC SURFACES ∗
Anyone who considers arithmetical methods of producing random digits is,
of course, in a state of sin.
JOHN VON NEUMANN (1951)
1. Rational points on cubic surfaces
1.1. –––– The arithmetic of cubic surfaces is a fascinating subject. To a large
extent, it was initiated by the work of Yu. I. Manin, particularly by his fundamental
and influential book on Cubic Forms [Man].
In this chapter, we study the distribution of rational points on general cubic
surfaces overÉ. The main problems are
i) Existence ofÉ-rational points,
ii) Asymptotics ofÉ-rational points,
iii) The height of the smallest point.
i. Existence of rational points. —
1.2. –––– Let X be an algebraic variety defined overÉ. Recall that the
Hasse principle is said to hold for X if
X (É) = ∅ ⇐⇒ ∃ ν ∈ Val(É): X (Éν) = ∅ .
For quadrics in P nÉ, the Hasse principle holds by the famous Theorem of Hasse-
Minkowski.
It is, however, well-known that for smooth cubic surfaces overÉthe Hasse principle
does not hold, in general. In all known examples, this is explained by the
Brauer-Manin obstruction which was discussed in detail in Chapter III.
(∗) This chapter is a revised version of the article: Experiments with general cubic surfaces, to
appear in: The Manin Festschrift, joint with A.-S. Elsenhans.

282 GENERAL CUBIC SURFACES [Chap. IX
ii. Asymptotics of rational points. —
1.3. –––– On the asymptotics of rational points of bounded height, there is
Manin’s conjecture which was described in Chapter VI.
1.4. Conjecture (Manin). —– Let X be an arbitrary Fano variety overÉand H
be an anticanonical height on X. Then, there exist a dense, Zariski open subset
X ◦ ⊆ X and a constants r and τ such that
#{x ∈ X ◦ (É) | H(x) < B} ∼ τB log r B
(∗)
for B → ∞.
In the case that X is a surface, it is expected that r = rkPic(X ) − 1.
1.5. Remark (Known cases). —– Conjecture 1.4 is established for smooth
complete intersections of multidegree d 1 , . . . , d n in the case that the dimension
of X is very large compared to d 1 , . . . , d n [Bir]. Further, it is proven for projective
spaces and quadrics. Finally, there are a number of further special cases
in which Manin’s conjecture is known to be true. See, e.g., [Pe02, Sec. 4].
In Chapter VII, numerical evidence for Conjecture 1.4 is presented in the case
of the threefolds X e a,b given by axe = by e + z e + v e + w e in P 4Éfor e = 3 and 4.
1.6. Remark (Peyre’s constant). —– Motivated by results obtained by the classical
circle method, E. Peyre refined Manin’s conjecture by a conjectural value for
the leading coefficient τ. We formulated a general description of Peyre’s constant
in Definition VI.5.35.
1.7. –––– We are interested in smooth cubic surfaces. Let us explain
Peyre’s constant when X is the smooth surface in P 3Édefined by a homogeneous
cubic polynomial f ∈[X 0 , X 1 , X 2 , X 3 ].
Consider the anticanonical height which is the same as the naive height. Assume
that rkPic(X ) = 1 and H 1( Gal(É/É), Pic(XÉ) ) = 0.
Then, Peyre’s constant is equal to the Tamagawa type number τ given by
where
τ p =
τ := ∏ τ p
p∈È∪{∞}
(
1 − 1 )
#X (/p m)
· lim
p m→∞ p 2m

Sec. 1] RATIONAL POINTS ON CUBIC SURFACES 283
for p finite and
τ ∞ = 1 2
Z
x∈[−1,1] 4
f (x)=0
1
dS .
‖(grad f )(x)‖ 2
Here, X ⊂ P 3is the integral model of X defined by the polynomial f .
dS denotes the usual hypersurface measure on the cone CX (Ê), considered as a
hypersurface inÊ4 .
1.8. Remarks. –––– i) The assumption that rk Pic(X ) = 1 guarantees that there
are no log-factors in the conjectural asymptotics.
ii) According to Example VI.5.3, the assumption rkPic(X ) = 1 is sufficient
for α(X ) = 1. On the other hand, H 1( Gal(É/É), Pic(XÉ) ) = 0 guarantees
that there is no Brauer-Manin obstruction to weak approximation. Further,
β(X ) = 1 and both additional factors do not need to be considered.
In general, the definition of Peyre’s constant is more complicated, even for
smooth cubic surfaces.
1.9. Remark. –––– According to Definition VI.5.35, the local factors τ p
are equipped with an additional factor det ( 1 − p −1 Frob p | p) I .
Here, Pic(XÉ) I p
denotes the fixed module under the inertia group. For compensation,
another factor is added to Peyre’s constant, the value L(1, χ P ) of the
Artin-L-function associated to the Gal(É/É)-representation P ⊂ Pic(XÉ)⊗
complementary to the trivial summand.
Inour case, therepresentationP turnsouttobeirreducibleofranksix. Thefinite
quotient of Gal(É/É) acting on P is isomorphic to W (E 6 ) of order 51 840.
We do not even make an attempt to compute the exact value of L. Instead, we
work with the conditionally convergent series described.
iii. The smallest point. —
1.10. –––– It would be desirable to have an a-priori upper bound for the height
of the smallestÉ-rational point on X as this would allow to effectively decide
whether X (É) ≠ ∅ or not.
When X is a conic, Legendre’s theorem on zeroes of ternary quadratic forms
yields an effective bound for the smallest point. For quadrics of arbitrary dimension,
the same is true by an observation due to J. W. S. Cassels [Cas55].
Further, there is a theorem of C. L. Siegel [Si73, Satz 1] which provides a generalization
to hypersurfaces defined by norm equations. This certainly includes
some special cubic surfaces but, in general, no theoretical upper bound is known
for the height of the smallestÉ-rational point on a cubic surface.

284 GENERAL CUBIC SURFACES [Chap. IX
1.11. Remark. –––– If one had an error term [S-D05] for (∗) uniform over all
cubic surfaces X of Picard rank 1 then this would imply that the height m(X )
of the smallestÉ-rational point is always less than
C
τ
for certain constants
(X ) α
α > 1 and C > 0.
The investigations on quartic threefolds made in Chapter VIII (cf. [EJ5]) indicate
that one might have even m(X ) < C (ε) for any ε > 0. The same inequality
τ (X ) 1+ε
is suggested by the experiments with diagonal cubic surfaces which we will
describe in the next chapter.
Assuming equidistribution, one would expect that the height of the smallestÉ-rational
point on X should be even ∼ 1
τ . An inequality of the
(X )
form m(X ) < C
τ
is, however, known to be wrong for diagonal quartic threefolds.
(Cf. Theorem VIII.2.2.) Under the Generalized Riemann Hypothesis,
(X )
Theorem X.5.3 will show that m(X ) < C
τ
is wrong for diagonal cubic surfaces,
(X )
too.
iv. The results. —
1.12. –––– We consider two families of cubic surfaces which are produced by
a random number generator. For each of these surfaces, we do the following.
i) We verify that the Galois group acting on the 27 lines is equal to W (E 6 ).
This implies that rk Pic(X ) = 1 and H 1( Gal(É/É), Pic(XÉ) ) = 0.
ii) We compute E. Peyre’s constant τ (X ).
iii) Up to a certain bound for the anticanonical height, we count allÉ-rational
points on the surface X.
Thereby, we establish the Hasse principle for each of the surfaces considered.
Further, we test numerically the conjecture of Manin, in the refined form due to
E. Peyre, on the asymptotics of points of bounded height. Finally, we study the
behaviour of the height of the smallestÉ-rational point versus E. Peyre’s constant.
This means, we test the estimates formulated in Remark 1.11.
2. Background
i. The 27 lines. —
2.1. –––– A non-singular cubic surface defined overÉcontains exactly 27 lines.
The symmetries of the configuration of the 27 lines respecting the intersection
pairing are given by the Weyl group W (E 6 ). The order of the group #W (E 6 )
is 51 840.

Sec. 2] BACKGROUND 285
These are standard facts which may be found in many places in the literature.
We discussed them before in Chapter II, 8.21 ff. Observe the list of the 350 conjugacy
classes of W (E 6 ) generated using GAP which is given in the appendix.
2.2. Remark. –––– The group W (E 6 ) admits the following properties.
i) W (E 6 ) has 3 4·5 = 405 2-Sylow subgroups.
ii) W (E 6 ) has 2 5·5 = 160 3-Sylow subgroups.
iii) W (E 6 ) has 2 4·3 4 = 1296 5-Sylow subgroups.
2.3. Fact. –––– LetX beasmoothcubic surface definedoverÉand let K bethe field
of definition of the 27 lines on X. Then K is a Galois extension ofÉ. The Galois
group Gal(K/É) is a subgroup of W (E 6 ).
Proof. This was shown in Chapter II, 8.21.
□
2.4. Remark. –––– W (E 6 ) contains a subgroup U of index two which is
isomorphic to the simple group of order 25 920. It is of Lie type B 2 (3),
i.e. U ∼ = Ω 5 (3) ⊂ SO 5 (3). In U, the stabilizer of two non-intersecting lines
is the alternating group A 5 .
2.5. Remark. –––– On the other hand, the operation of W (E 6 ) on the 27 lines
gives rise to a transitive permutation representation ι: W (E 6 ) → S 27 . It turns
out that the image of ι is contained in the alternating group A 27 . We will call
an element σ ∈ W (E 6 ) even if σ ∈ U and odd, otherwise. This should not be
confused with the sign of ι(σ) ∈ S 27 which is always even.
2.6. Remark. –––– The operation of the group W (E 6 ) on the 27 lines satisfies
the properties below.
i) It is transitive.
ii) Fix a line l 1 . Then, the remaining 26 lines split into two classes. There are
10 lines which intersect l 1 and 16 lines which do not. The stabilizer of l 1 acts
transitively on each class.
iii) Choose a second line l 2 which does not intersect l 1 . Then, there are precisely
five lines g 1 , . . . , g 5 intersecting both l 1 and l 2 . The stabilizer of l 1
and l 2 is isomorphic to S 5 . An isomorphism is given by the operation on the
lines g 1 , . . . , g 5 .
In particular, the group order is indeed 27·16·5! = 51 840.
ii. The Brauer-Manin obstruction. —
2.7. –––– For Fanovarieties, allknownobstructionsagainsttheHasseprinciple
are explained by the following observation.

286 GENERAL CUBIC SURFACES [Chap. IX
2.8. Observation (Manin). —– Let π : X → SpecÉbe a non-singular variety
overÉ. Choose an element α ∈ Br(X ) from its Brauer group.
Then, anyÉ-rational point x ∈ X (É) gives rise to an adelic point (x ν ) ν ∈ X (AÉ)
satisfying the condition
ev(α, x) :=
∑ inv(α| xν ) = 0 .
ν∈Val(É)
Here, inv: Br(Éν) →É/(respectively inv: Br(Ê) → 1 2/) denotes the canonical
isomorphism.
Proof. This is the Brauer-Manin obstruction which was studied in detail in Chapter
III. The assertion was proven in Proposition III.2.3.b.iv).
□
2.9. –––– Recall the following facts.
i) Proposition III.2.3.a.ii) shows that inv(α| xν ) depends continuously
on x ν ∈ X (Éν). Further, by Proposition III.2.3.b.ii), for every proper variety
X overÉand each α ∈ Br(X ), there exists a finite set S ⊂ Val(É) such that
inv(α| xν ) = 0 for every ν ∉ S and x ν ∈ X (Éν).
This implies that the Brauer-Manin obstruction, if present, is an obstruction to
the principle of weak approximation.
ii) For certain varieties X, it happens, however, that the Brauer-Manin obstruction
is not an obstruction to the Hasse principle. The point is that it possibly
excludes only part of the adelic points.
This is true, e.g., for the cubic surface given in Example III.5.25. Likewise for
all diagonal cubic surfaces of the “first case” according to the classification
of J.-L. Colliot-Thélène, D. Kanevsky, and J.-J. Sansuc [CT/S]. See Theorem
III.6.4.
iii) It is obvious that altering α ∈ Br(X ) by some Brauer class π ∗ ρ for ρ ∈ Br(É)
does not change the obstruction defined by α. By consequence, it is only the
factor group Br(X )/π ∗ Br(É) which is relevant for the Brauer-Manin obstruction.
As was shown in Proposition II.8.11 and Remark II.8.13, the latter
is canonically isomorphic to H 1( Gal(É/É), Pic(XÉ) ) . In particular,
if H 1( Gal(É/É), Pic(XÉ) ) = 0 then there is no Brauer-Manin obstruction
on X.
2.10. –––– For a smooth cubic surface X, the geometric Picard group Pic(XÉ)
is generated by the classes of the 27 lines on XÉ. Its first cohomology group
can be described in terms of the Galois action on these lines. Indeed, by
Proposition II.8.18, there is a canonical isomorphism
H 1( Gal(É/É), Pic(XÉ) ) ∼ = Hom
(
(NF ∩ F0 )/NF 0 ,É/) . (†)

Sec. 3] THE GALOIS OPERATION ON THE 27 LINES 287
Here, F ⊂ Div(XÉ) is the group generated by the 27 lines, F 0 ⊂ F denotes
the subgroup of principal divisors, and N is the norm map under the operation
of Gal(É/É)/H , H being the stabilizer of F.
2.11. Remark. –––– Consider the particular case when the Galois group acts
transitively on the 27 lines. Then, (†) shows that H 1( Gal(É/É), Pic(XÉ) ) = 0.
In particular, there is no Brauer-Manin obstruction in this case.
It is expected that the Hasse principle holds for all cubic surfaces such that
H 1( Gal(É/É), Pic(XÉ) ) = 0. A statement which is very close to the
Hasse principle was conjectured by J.-L. Colliot-Thélène and J.-J. Sansuc [CT/S,
Conjecture C].
3. The Galois operation on the 27 lines
Let X be a smooth cubic surface defined overÉand let K be the field of
definition of the 27 lines on X. By Fact 2.3, K/Éis a Galois extension and
the Galois group G := Gal(K/É) is a subgroup of W (E 6 ). For general cubic
surfaces, G is actually equal to W (E 6 ). To verify this for particular examples,
the following lemma is useful.
3.1. Lemma. –––– Let H ⊆ W (E 6 ) be a subgroup which acts transitively on the
27 lines and contains an element of order five. Then, either H is the subgroup
U ⊂ W (E 6 ) of index two or H = W (E 6 ).
Proof. H ∩ U still acts transitively on the 27 lines and still contains an element
of order five. Thus, we may suppose H ⊆ U.
Assume that H U. Denote by k the index of H in U. The natural action
of U on the set of cosets U/H yields a permutation representation i : U → S k .
As U is simple, i is necessarily injective. In particular, since #U ∤ 8!, we see
that k > 8. Let us consider the stabilizer H ′ ⊂ H of one of the lines. As H acts
transitively, it follows that #H ′ = #H = #U . We distinguish two cases.
27
= 960
27·k k
First case: k > 16. Then, k ≥ 20 and #H ′ ≤ 48. This implies that the
5-Sylow subgroup is normal in H ′ . Its conjugate by some σ ∈ H therefore depends
only on σ ∈ H /H ′ . By consequence, the number n of 5-Sylow subgroups
in H is a divisor of #H /#H ′ = 27. Sylow’s congruence n ≡ 1 (mod 5) yields
that n = 1.
Let H 5 ⊂ H be the 5-Sylow subgroup. Then, ι(H 5 ) ⊂ S 27 is generated by a
product of disjoint 5-cycles leaving at least two lines fixed. It is, therefore, not
normal in the transitive group ι(H ). This is a contradiction.
Second case: 9 ≤ k ≤ 16. We have k | 960. On the other hand, the assumption
5 | #H implies 5 ∤k. This shows, there are only two possibilities, k = 12

288 GENERAL CUBIC SURFACES [Chap. IX
and k = 16. As, in U, there is no subgroup of index eight or less, H ⊂ U
must be a maximal subgroup. In particular, the permutation representation
i : U → S k is primitive.
Primitive permutation representations of degree up to 20 have been classified
already in the late 19th century. It is well known that no group of order 25 920
allows a faithful primitive permutation representation of degree 12 or 16 [Sim,
Table 1].
□
3.2. Remark. –––– The subgroups of the simple group U have been completely
classified by L. E. Dickson [Di04] in 1904. There are 114 conjugacy classes of
proper subgroups H ≠ {e} in U. It would not be complicated to deduce the
lemma from Dickson’s list.
3.3. –––– Let the smooth cubic surface X be given by a homogeneous equation
f = 0 with integral coefficients. We want to compute the Galois group G.
An affine part of a general line l can be described by four coefficients a, b, c, d
via the parametrization
l: t ↦→ (1 : t : (a + bt) : (c + dt)) .
l is contained in S if and only if it intersects S in at least four points. This implies
that
f (l(0)) = f (l(∞)) = f (l(1)) = f (l(−1)) = 0
is a system of equations for a, b, c, d which encodes that l is contained in S.
By a Gröbner base calculation in SINGULAR, we compute a univariate polynomial
g of minimal degree belonging to the ideal generated by the equations. If g
is of degree 27 then the splitting field of g is equal to the field K of definition
of the 27 lines on X. We then use van der Waerden’s criterion [Poh/Z, Proposition
2.9.35]. This means, we factor g modulo various primes. If all irreducible
factors are distinct then the degrees of the factors describe the cycle structure of
an element in the Galois group of g.
If the various factorizations are incompatible [Coh, Section 3.5.2] then we get
the information that g is irreducible. The Galois group G then acts transitively
on the 27 lines. Furthermore, if we find an element of order five and an element
which is not contained in the index two subgroup then G is actually equal
to W (E 6 ). More precisely, our algorithm works as follows.
3.4. Algorithm (Verifying G = W (E 6 )). —– Given the equation f = 0 of a
smooth cubic surface, this algorithm verifies G = W (E 6 ).

Sec. 3] THE GALOIS OPERATION ON THE 27 LINES 289
i) Compute a univariate polynomial 0 ≠ g ∈[d] of minimal degree such that
g ∈ ( f (l(0)), f (l(∞)), f (l(1)), f (l(−1))) ⊂É[a, b, c, d]
where l: t ↦→ (1 : t : (a + bt) : (c + dt)).
If g is not of degree 27 then terminate with an error message. In this case, the
coordinate system for the lines is not sufficiently general. If we are erroneously
given a singular cubic surface then the algorithm will fail at this point.
ii) Factor g modulo all primes below a given limit. Ignore the primes dividing
the leading coefficient of g.
iii) If one of the factors is multiple then go to the next prime immediately.
Otherwise, check whether the decomposition type corresponds to one of the
cases listed below,
A := {(9, 9, 9)} , B := {(1, 1, 5, 5, 5, 5, 5), (2, 5, 5, 5, 10)} ,
C := {(1, 4, 4, 6, 12), (2, 5, 5, 5, 10), (1, 2, 8, 8, 8)} .
iv) If each of the cases occurred for at least one of the primes then output the
message “The Galois group is equal to W (E 6 ).” and terminate.
Otherwise, output “Can not prove that the Galois group is equal to W (E 6 ).”
3.5. Remark. –––– The cases above are functioning as follows.
a) Case B shows that the order of the Galois group is divisible by five.
b) Cases A and B together guarantee that g is irreducible. Therefore,
by Lemma 3.1, A and B prove that G contains the index two subgroup
U ⊂ W (E 6 ).
c) Case C is a selection of the most frequent odd conjugacy classes in W (E 6 ).
3.6. Remark. –––– One could replace cases B and C by their common element
(2, 5, 5, 5, 10). This would lead to a simpler but less efficient algorithm.
3.7. Remark. –––– Actually, a decomposition type as considered in step iii)
does not always represent a single conjugacy class in W (E 6 ). Two elements
ι(σ), ι(σ ′ ) ∈ S 27 might be conjugate in S 27 via a permutation τ ∉ ι(W (E 6 )).
For example, as is easily seen using GAP, the decomposition type (3, 6, 6, 6, 6)
falls into three conjugacy classes two of which are even and one is odd (cf. Remark
2.4). However, all the decomposition types searched for in Algorithm 3.4
do represent single conjugacy classes.

290 GENERAL CUBIC SURFACES [Chap. IX
3.8. Remark. –––– Since we expect G = W (E 6 ), we can estimate the probability
of each case by the Chebotarev density theorem. Case A has a probability
of 1 . This is the lowest value among the three cases.
9
3.9. Remark. –––– As we do not use the factors of g explicitly, it is enough
to compute their degrees and to check that each of them occurs with multiplicity
one. This means, we only have to compute gcd(g(X ), g ′ (X ))
and gcd(g(X ), X pd −X ) inp[X ] for d = 1, 2, . . . , 13 [Coh, Algorithms 3.4.2
and 3.4.3].
4. Computation of Peyre’s constant
i. The Euler product. —
4.1. –––– We want to compute the product over all τ p . For a finite place p,
we have
(
τ p = 1 − 1 )
X (/p m)
· lim .
p m→∞ p 2m
If the reduction is smooth then the sequence under the limit is constant
Xp
by virtue of Hensel’s Lemma. Otherwise, it becomes stationary after finitely
many steps.
We approximate the infinite product over all τ p by the finite product taken over
the primes less than 300. Numerical experiments show that the contributions of
larger primes do not lead to a significant change. (Compare the values calculated
for the concrete example in Section 6.)
ii. The factor at the infinite place. —
4.2. –––– We want to compute
τ ∞ = 1 2
where the domain of integration is given by
Z
R
1
‖ grad f ‖ 2
dS
R = {(x, y, z, w) ∈ [−1, 1] 4 | f (x, y, z, w) = 0} .
Here, dS denotes the usual hypersurface measure on R, considered as a hypersurface
inÊ4 . Thus, τ ∞ is given by a three-dimensional integral.
Since f is a homogeneous polynomial, we may reduce to an integral over the
boundary of R which is a two-dimensional domain. In our particular case, we

Sec. 4] COMPUTATION OF PEYRE’S CONSTANT 291
have deg f = 3. Then, a direct computation leads to
Z
Z
τ ∞ =
dA +
‖( ∂ f
‖( ∂ f
Z
+
1
, ∂ f , ∂ f )‖ R 0 ∂y ∂z ∂w 2
1
‖( ∂ f , ∂ f , ∂ f )‖ R 2 ∂x ∂y ∂w 2
where the domains of integration are
Z
dA +
1
, ∂ f , ∂ f )‖ R 1 ∂x ∂z ∂w 2
1
‖( ∂ f , ∂ f , ∂ f )‖ R 3 ∂x ∂y ∂z 2
R i = {(x 0 , x 1 , x 2 , x 3 ) ∈ [−1, 1] 4 | x i = 1 and f (x 0 , x 1 , x 2 , x 3 ) = 0} .
dA denotes the two-dimensional hypersurface measure on R i , considered as a
hypersurface inÊ3 .
We therefore have to integrate a smooth function over a compact part of a
smooth two-dimensional submanifold inÊ3 . To do this, we approximate the
domain of integration by a triangular mesh.
4.3. Algorithm (Generating a triangular mesh). —–
Given the equation f = 0 of a smooth surface inÊ3 , this algorithm constructs
a triangular mesh approximating the part of the surface which is contained in a
given cube.
i) We split the cube into eight smaller cubes and iterate this procedure a predefined
number of times, recursively. During recursion, we exclude those
cubes which obviously do not intersect the manifold. To do this, we estimate
‖grad f ‖ 2 on each cube.
ii) Then, each resulting cube is split into six simplices.
iii) For each edge of each simplex which intersects the manifold, we compute
an approximation of the point of intersection. We use them as the vertices of
the triangles to be constructed. This leads to a mesh consisting of one or two
triangles per simplex.
4.4. Remark. –––– Similar algorithms are popular in the computer graphics
community.
4.5. –––– The next step is to compute the contribution of each triangle ∆ i to
the integral. For this, we use some adaption of the midpoint rule.
We approximate the integrand g by its value g(C i ) at the barycenter C i of
the triangle. Note that this point usually lies outside the surface given by f = 0.
Algorithm 4.3 guarantees only that the three vertices of each facet are contained
in that surface. We map it to the manifold along the flow of the gradient
field grad f .
dA
dA

292 GENERAL CUBIC SURFACES [Chap. IX
The product g(C i )A(∆ i ) seems to be a reasonable approximation of the contribution
of ∆ i to the integral. We correct by an additional factor, the cosine of the
angle between the normal vector of the triangle and the gradient vector grad f
at the barycenter C. We calculated the contribution of each triangle by some
adaption of the midpoint rule. The problem is our algorithm makes sure for
each facet that its three vertices are contained in the manifold R i . This means
that its barycenter is usually outside R i . We map it to the manifold along the
flow of the gradient field grad f .
4.6. Remark. –––– In our application, these correctional factors are close to 1
and seem to converge versus 1 when the number of recursions is growing.
This is, however, not a priori clear. H. A. Schwarz’s cylindrical surface [Schw]
constitutes a famous example of a sequence of triangulations where the triangles
become arbitrarily small and the factors are nevertheless necessary for
correct integration.
We use the method described above to approximate the value of τ ∞ . In Algorithm
4.3, we work with six recursions.
4.7. Remark. –––– Our method of numerical integration is a combination of
standard algorithms for 2.5-dimensional mesh generation and two dimensional
integration which are described in the literature [Hb].
On a triangle, we integrate linear functions correctly. This indicates that the
method should converge of second order. The facts that we work with the
area of a linearized triangle and that the barycenters C i are located in a certain
distance from the manifold generate errors of the same order.
5. Numerical Data
i. The computations carried out. —
5.1. –––– A general cubic surface is described by twenty coefficients. With current
technology, it is impossible to study all cubic surfaces with coefficients
below a reasonable bound. For that reason, we decided to work with coefficient
vectors provided by a random number generator. Our first sample consists
of 20 000 surfaces with coefficients randomly chosen in the interval [0, 50].
The second sample consists of 20 000 surfaces with randomly chosen coefficients
from the interval [−100, 100].
These limits were, of course, chosen somewhat arbitrarily. There is, at least,
some reason not to work with too large limits as this would lead to low values

Sec. 5] NUMERICAL DATA 293
of τ. (The reader might want to compare Theorem X.6.21 where this is rigorously
proven in a different situation.) Low values of τ are undesirable as they
require high search bounds in order to satisfactorily test Manin’s conjecture.
We verified explicitly that each of the surfaces studied is smooth. For this,
we inspected a Gröbner base of the ideal corresponding to the singular locus.
The computations were done in SINGULAR.
Then, using Algorithm 3.4, and a prime limit of 1000, we proved that, for each
surface, the full Galois group W (E 6 ) acts on the 27 lines. The largest prime used
was 457. This means that all our examples are general from the Galois point
of view. By consequence, their Picard ranks are equal to 1. Further, according
to Remark 2.11, the Brauer-Manin obstruction is not present on any of the
surfaces considered.
Almost as a byproduct, we verified that no two of the 40 000 surfaces are isomorphic.
Actually, when running part ii) of Algorithm 3.4, we wrote the
decomposition types found into a file. Primes at which the algorithm failed
were labeled by a special marker. A program, written in C, ran in an iterated
loop over all pairs of surfaces and looked for a prime at which the decomposition
types differ. The largest prime needed to distinguish two surfaces was 73.
We counted allÉ-rational points of height less than 250 on the surfaces of the
first sample. It turns out that, on two of these surfaces, there are noÉ-rational
points occurring as the equation is unsolvable inÉp for some small p. In this
situation, Manin’s conjecture is true, trivially.
On each of the remaining surfaces, we found at least oneÉ-rational point.
228 examples contained less than ten points. On the other hand, 1213 examples
contained at least one hundredÉ-rational points. The largest number of points
found was 335.
For the second sample, the search bound was 500. Again, on two of these
surfaces, there are noÉ-rational points occurring as the equation is unsolvable in
a certainÉp. There were 202 examples containing between one and nine points.
1857 examples contained at least one hundredÉ-rational points. The largest
number of points found was 349.
To find theÉ-rational points, we used a 2-adic search method which works
as follows. Let a cubic surface X be given. Then, in a first step, we determined
on X all points defined over/512(respectively/1024). Then, for each
of the points found we checked which of its lifts to P 3 () actually lie on X.
This leads to an O(B 3 )-algorithm which may be efficiently implemented in C.
Furthermore, using the method described in Section 4, we computed an approximation
of Peyre’s constant for each surface.

294 GENERAL CUBIC SURFACES [Chap. IX
5.2. Remark. –––– To search for theÉ-rational points, there are algorithms
which are asymptotically faster. For example, Elkies’ method which is O(B 2 )
and implemented in Magma. A practical comparison shows, however, that
Elkies’ method is not yet faster for our relatively low search bounds.
ii. The density results. —
5.3. –––– For each of the surfaces considered we calculated the quotient
#{ points of height < B found } / #{ points of height < B expected }.
Let us visualize the distribution of the quotients by some histograms.
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
0.5 1 1.5 2 2.5 0
0.5 1 1.5 2 2.5
First sample, B =125 First sample, B =250
FIGURE 1. Distribution of the quotients for the first sample

Sec. 5] NUMERICAL DATA 295
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
0.5 1 1.5 2 2.5 0
0.5 1 1.5 2 2.5
Second sample, B =250 Second sample, B =500
FIGURE 2. Distribution of the quotients for the second sample
Some statistical parameters are as follows.
TABLE 1. Parameters of the distribution for the first sample
search bound 125 250
mean 0.99993 0.99887
standard deviation 0.23558 0.16925
TABLE 2. Parameters of the distribution for the second sample
search bound 250 500
mean 1.00093 0.99943
standard deviation 0.22527 0.16158
iii. The results for the smallest point. —
5.4. –––– For each of the surfaces in our samples, we determined the height
m(X ) of its smallest point. We visualize the behaviour of m(X ) in the diagrams
below.
At the first glance, it looks very natural to consider the distribution of the values
of m(X ) versus the Tamagawa type number τ (X ). In view of the inequalities
asked for in the introduction, it seems, however, to be better to make a slight
modification and plot the product m(X )τ (X ) instead of m(X ) itself.

296 GENERAL CUBIC SURFACES [Chap. IX
m
1000
m
1000
100
100
10
10
1
0.001 0.01 0.1 1 10
First sample
τ
1
0.001 0.01 0.1 1 10
Second sample
τ
FIGURE 3. The smallest height of a rational point versus the Tamagawa number
5.5. Conclusion –––– Our experiments suggest that, for general cubic surfaces
X overÉ, the following assertions hold.
i) There are no obstructions to the Hasse principle.
ii) Manin’s conjecture is true in the form refined by E. Peyre.
Further, it is apparent from the diagrams in Figure 3 that the experiment agrees
with the expectation for the heights of the smallest points formulated in Remark
1.11. Indeed, for both samples, the slope of a line tangent to the top right
of the scatter plot is near (−1). This indicates that even the strong form of the
estimate should be true. I.e., m(X ) < C (ε)
τ (X ) 1+ε for any ε > 0.
5.6. Running Times –––– The largest portion of the running time was spent on
the calculation of the Euler products. It took 20 days of CPU time to calculate
all 40 000 Euler products for p < 300.
For comparison, we estimated all the integrals, using six recursions, within
36 hours. Further, it took eight days to systematically search for all points of
height less than 500 on the surfaces of the second sample. Search for points of
height less than 250 on the surfaces of the first sample took only one day.
When running Algorithm 3.4, the lion’s share of the time was used for the
computation of the univariate degree 27 polynomials. This took approximately
seven days of CPU time.
In comparison with that, all other parts were negligible. It took only twelve
minutes to ensure that all 40 000 surfaces are smooth. The C program verifying
that no two of the surfaces are isomorphic to each other ran approximately
80 seconds.

Sec. 6] A CONCRETE EXAMPLE 297
6. A concrete example
i. The Example. —
6.1. Example –––– Let us conclude the article by some results on the particular
cubic surface X given by
x 3 + 2xy 2 + 11y 3 + 3xz 2 + 5y 2 w + 7zw 2 = 0 . (‡)
Example (‡) was not among the surfaces produced by the random number generator.
Our intention is just to present the output of our algorithms in a specific
(and not too artificial) example and, most notably, to show the intermediate
results of Algorithm 3.4.
A Gröbner base calculation inMagma shows that X has bad reduction at p = 2,
3, 7, 23, and 22 359 013 270 232 677. The idea behind that calculation is the same
as described above for the verification of smoothness. The only difference is
that we consider Gröbner bases overinstead ofÉ.
6.2. The Galois group –––– The first step of Algorithm 3.4 works well on X.
I.e., the polynomial g is indeed of degree 27. Its coefficients become rather large.
The absolutely largest one is that of d 13 . It is equal to 38 300 982 629 255 010.
The leading coefficient of g is 5 3 · 7 12 . We find case A at p = 373. The common
decomposition type (2, 5, 5, 5, 10) of the cases B and C occurs at p = 19, 31, 59,
61, 191, 199, and 223.
Consequently, X is an explicit example of a smooth cubic surface overÉ
admitting the property that the Galois group which acts on the 27 lines is equal
to W (E 6 ).
6.3. Remark. –––– ThefirstsuchexampleshavebeenconstructedbyT. Ekedahl
[Ek, Theorem 2.1].
6.4. Remark. –––– Our example (‡) is different from Ekedahl’s. Indeed, in
Ekedahl’s examples, the Frobenius Frob 11 acts on the 27 lines as an element
of the conjugacy class C 15 ⊂ W (E 6 ) (in Sir P. Swinnerton-Dyer’s numbering).
In our case, however, the first two steps of Algorithm 3.4 show that Frob 11
yields the decomposition type (1, 1, 1, 1, 1, 2, 4, 4, 4, 4, 4). This corresponds to
the class C 18 [Man,кIV, §9,Ìн]. Note that Ekedahl’s examples, as
well as ours, have good reduction at p = 11.

298 GENERAL CUBIC SURFACES [Chap. IX
ii. The computation of Peyre’s constant. —
6.5. –––– As an approximation of the Euler product, we get
∏ τ p ≈ 0. 729 750 .
p

Sec. 6] A CONCRETE EXAMPLE 299
TABLE 4. Points on X of height ≤ 20
Point Height Point Height
( 0 : 0 : 0 : 1) 1 ( 4 : -6 : -13 : 2) 13
( 0 : 0 : 1 : 0) 1 ( 5 : 5 : 0 : -14) 14
( 1 : 2 : -3 : -2) 3 (10 : -14 : 12 : 11) 14
( 3 : -2 : -3 : 2) 3 ( 0 : 7 : -16 : 7) 16
( 0 : 7 : -6 : -7) 7 (16 : -8 : 3 : -4) 16
( 0 : 4 : -3 : 8) 8 ( 6 : -9 : 16 : 3) 16
( 5 : -5 : 0 : 8) 8 (12 : 7 : -6 : 17) 17
( 2 : -8 : 8 : 7) 8 (14 : -9 : -2 : 17) 17
(10 : -5 : 0 : -1) 10 ( 6 : -3 : -18 : 7) 18
( 0 : 5 : 0 : -11) 11 ( 9 : -6 : -1 : 18) 18
( 8 : 6 : -11 : -8) 12 ( 3 : 6 : -19 : -6) 19
(12 : -6 : -4 : -3) 12 ( 8 : 7 : -4 : 19) 19
( 9 : -12 : 9 : 10) 12
iv. The field of definition of the 27 lines. —
6.7. –––– Having done the Gröbner base calculation in Algorithm 3.4.i),
the 27 lines may be computed at high precision. This allows to find the 45 triangles
on X,explicitly. We calculated a degree 45 resolvent G of the degree 27 polynomial
g the zeroes of which are all the sumsa i1 +a i2 +a i3 for l i1 , l i2 , l i3 representing
three lines which form a triangle. Here, l i : t ↦→ (1 : t : (a i +b i t) : (c i +d i t))
denote parametrizations of the 27 lines. As G ∈[X ], our floating point
calculation is in fact exact.
6.8. Proposition. –––– The unique quadratic subfield in the field K of definition of
the 27 lines on X isÉ(√
−23 · 22 359 013 270 232 677
)
.
Proof. K is unramified at all places of good reduction of X. This leaves us with
only 2 6 − 1 = 63 possibilities for the quadratic subfieldÉ( √ d). To exclude 62
of them is algorithmically easy.
Indeed, for a good prime p, there is a way to compute ( d
p)
without knowledge
of d. We factor the degree 45 resolvent G modulo p. If p divides the
leading coefficient or there are multiple factors then we get no answer. Otherwise,
( d
p) = ±1 depending on whether the decomposition type found is even
or odd in S 45 .
It turns out that it is sufficient to do this for p = 13, 17, 19, 29, 31, and 53.
□

300 GENERAL CUBIC SURFACES [Chap. IX
6.9. Proposition. –––– The field extension K/Éis ramified exactly at p = 2, 3, 7,
23, and 22 359 013 270 232 677.
Proof. It remains to verify ramification at p = 2, 3, and 7. For that, we
computed in Magma the p-adic factorization of g. The decomposition types are
(3, 24) for p = 2 and 3 and (1, 1, 1, 4, 4, 4, 4, 8) for p = 7.
Let Z p be the decomposition field of p. If p were unramified then Gal(K/Z p )
would be a cyclic group, i.e. Gal(K/Z p ) = 〈σ〉 for some σ ∈ W (E 6 ). On the
other hand, on the 27 lines, the orbit structure under the operation of Gal(K/Z p )
isthesameasunder theoperationofGal(Ép/Ép). Thereis, however, noelement
in W (E 6 ) which yields the decomposition type (3, 24) or (1, 1, 1, 4, 4, 4, 4, 8)
[Man,кIV, §9,Ìн].
□
6.10. Remark. –––– This shows that there is no integral model of X which is
smooth over p = 2, 3, 7, 23, or 22 359 013 270 232 677.

CHAPTER X
ON THE SMALLEST POINT ON A
DIAGONAL CUBIC SURFACE ∗
All life is an experiment. The more experiments you make the better.
RALPH WALDO EMERSON (1842)
1. Introduction
1.1. –––– In Chapter VIII, we presented some theoretical and experimental
results concerning the height of the smallestÉ-rational point on diagonal quartic
threefolds in P 4É. In this chapter, we will consider diagonal cubic surfaces in P 3É.
It will turn out that the analogies to the case treated before are enormous.
However, there are a few points indicating that the case of a diagonal cubic
surface is technically more complicated.
An obvious difference is that the Picard rank of a quartic threefold is always
equal to one. By consequence, H 1( Gal(É/É), Pic(XÉ) ) is automatically the
trivial group. Both these observations are wrong for diagonal cubic surfaces.
Hence, the factors α and β in the definition of Peyre’s constant (Definition
VI.5.35)will not always be the same and need to be considered. Further, one
has to expect that a Brauer-Manin obstruction is present.
1.2. –––– The conjecture of Manin states that the number ofÉ-rational points
of anticanonical height < B on a Fano variety X is asymptotically equal to
τB log rkPic(X )−1 B for B → ∞.
In the particular case of a cubic surface, the anticanonical height is the same
as the naive height. Further, the coefficient τ ∈Êequals the Tamagawa-type
number τ (X ) introduced by E. Peyre.
Thus, one expects at least ∼τ (X )B points of height

302 ON THE SMALLEST POINT ON A DIAGONAL CUBIC SURFACE [Chap. X
this might lead to the expectation that m(X ), the smallest height of aÉ-rational
point on X, is always less than C
τ
for a certain absolute constant C.
(X )
1.3. –––– To test this expectation, we computed the Tamagawa number and
ascertained the smallestÉ-rational point for each of the cubic surfaces given by
ax 3 + by 3 + 2z 3 + w 3 = 0
for a = 1, . . . , 3000 and b = 1, . . . , 300.
Thereby, we restricted our considerations to the cases that
i) a and b are odd,
ii) there exists an odd prime p dividing a but not b such that 3 ∤ ν p (a), or
iii) there exists an odd prime p dividing b but not a such that 3 ∤ν p (b).
This guarantees that we are in the “first case” according to the classification of
J.-L. Colliot-Thélène [CT/K/S] and his coworkers. In this case, the effect of
the Brauer-Manin obstruction is clear. Precisely two thirds of the adelic points
are excluded.
The results are summarized by the diagram below.
FIGURE 1. Height of smallest point versus Tamagawa number
It is apparent from the diagram that the experiment agrees with the expectation
above. The slope of a line tangent to the top right of the scatter plot is
indeed near (−1).
However, we will show in Section 5 that, as in the threefold case, the inequality
m(X ) < C
τ
does not hold in general. The following remains a logical possibility.
(X )

Sec. 1] INTRODUCTION 303
1.4. Question. –––– For every ε > 0, does there exist a constant C (ε) such
that, for each cubic surface,
m(X ) <
C (ε)
τ (X ) 1+ε ?
1.5. Peyre’s constant. –––– Recall that, in Definition VI.5.35, E. Peyre’s Tamagawa-type
number was defined as
τ (X ) := α(X )·β(X ) · lim
s→1
(s − 1) t L(s, χ Pic(XÉ)) · τ H
(
X (É)
for t = rk Pic(X ).
The factor β(X ) is simply defined as
β(X ) := #H 1( Gal(É/É), Pic(XÉ) ) .
Br)
α(X ) is given as follows. Let Λ eff (X ) ⊂ Pic(X )⊗Êbe the cone generated by
the effective divisors. Identify Pic(X )⊗ÊwithÊt via a mapping induced by an
∼=
isomorphism Pic(X ) −→t . Consider the dual cone Λ ∨ eff (X ) ⊂ (Êt ) ∨ . Then,
α(X ) := t · vol {x ∈ Λ ∨ eff | 〈x, −K 〉 ≤ 1 } .
L( · , χ Pic(XÉ)) denotes the Artin L-function of the Gal(É/É)-representation
Pic(XÉ) ⊗which contains the trivial representation t times as a direct summand.
Therefore, L(s, χ Pic(XÉ)) = ζ(s) t · L(s, χ P ) and
lim
s→1
(s − 1) t L(s, χ Pic(XÉ)) = L(1, χ P )
where ζ denotes the Riemann zeta function and P is a representation which
does not contain trivial components. [Mu-y, Corollary 11.5 and Corollary 11.4]
show that L(s, χ P ) has neither a pole nor a zero at s = 1.
Finally, τ H is the Tamagawa measure on the set X (É) of adelic points on X
and X (É) Br ⊆ X (É) denotes the part which is not affected by the Brauer-
Manin obstruction.
1.6. The Tamagawa measure. –––– As X is projective, we have
X (É) = ∏ X (Éν).
ν∈Val(É)
τ H is defined to be a product measure τ H := ∏ τ ν .
ν∈Val(É)
For a prime number p, the local measure τ p on X (Ép) is given as follows.
Let X ⊆ P 3be the model of X given by the defining cubic equation.
For a ∈ X (/p k), put
U (k)
a := {x ∈ X (Ép) | x ≡ a (mod p k ) } .

304 ON THE SMALLEST POINT ON A DIAGONAL CUBIC SURFACE [Chap. X
Then,
τ p (U (k)
a ) := det(1 − p−1 Frob p |Pic(XÉ) I p
)
#{ y ∈ X (/p
· m) | y ≡ a (mod p k ) }
lim
.
m→∞ p m dim X
Here, Pic(XÉ) I p
denotes the fixed module under the inertia group.
τ ∞ is described in Definition VI.5.22. In the case of a cubic surface defined by
the equation f = 0, this yields
τ ∞ (U ) = 1 Z
ω Leray
2
CU
|x 0 |, ... ,|x 3 |≤1
for U ⊂ X (Ê). Here, ω Leray is the Leray measure on the cone CX (Ê) associated
to the equation f = 0.
The Leray measure is related to the usual hypersurface measure by the formula
ω Leray = 1
‖ grad f ‖ ω hyp.
1.7. The results. –––– In the case of diagonal cubic surfaces, there is an estimate
1
for m(X ) in terms of τ (X ). Namely,
τ
admits a fundamental finiteness property.
More precisely, in Section 6, we will show the following
(X )
theorem.
Theorem. Let a = (a 0 , . . . , a 3 ) ∈ (\{0}) 4 be a vector. Denote by X a the cubic
surface in P 3Égiven by a 0 x 3 0 + . . . + a 3 x 3 3 = 0. Then, for each ε > 0 there exists a
constant C (ε) > 0 such that
1
τ (X a ) ≥ C (ε) · H( 1
a 0
: . . . :
1
a 3
) 1
3
−ε
.
Corollary (Fundamental finiteness). For each T > 0, there are only finitely many
diagonal cubic surfaces X a : a 0 x 3 0 + . . . + a 3 x 3 3 = 0 in P 3Ésuch that τ (V a ) > T .
Corollary (An inefficient search bound). There exists a monotonically decreasing
function F : (0, ∞) → [0, ∞), the search bound, satisfying the following condition.
Let X a be the cubic surface given by the equation a 0 x 3 0 + . . . + a 3 x 3 3 = 0.
Assume X a (É) ≠ ∅. Then, X a admits aÉ-rational point of height ≤F (τ (X a )).
Proof. One may simply put F (t) := max min H(P).
τ (X a )≥t P∈X a (É) X a (É)≠∅
□
In other words, we have m(X a ) ≤ F (τ (X a )) as soon as X a (É) ≠ ∅.

Sec. 2] THE FACTORS α AND β 305
2. The factors α and β
2.1. –––– Recall that on a smooth cubic surface X over an algebraically closed
field, there are exactly 27 lines. For the Picard group, which is isomorphic to7 ,
the classes of these lines form a system of generators.
2.2. Notation. –––– i) ThesetL ofthe27linesisequippedwith theintersection
product 〈 , 〉: L ×L → {−1, 0, 1}. The pair (L , 〈 , 〉) is the same for all
smooth cubic surfaces. It is well known from Chapter II, 8.21 that the group
of permutations of L respecting 〈 , 〉 is isomorphic to W (E 6 ). We fix such
an isomorphism.
Denote by F ⊂ Div(X ) the group generated by the 27 lines and by F 0 ⊂ F the
subgroup of principal divisors. Then, F is equipped with an operation of W (E 6 )
such that F 0 is a W (E 6 )-submodule. We have Pic(X ) ∼ = F/F 0 .
ii) If X is a smooth cubic surface overÉthen Gal(É/É) operates canonically
on the set L X of the 27 lines on XÉ. Fix a bijection i X : L X −→ L
∼ =
respecting the intersection pairing. This induces a group homomorphism
ι X : Gal(É/É) → W (E 6 ). We denote its image by G ⊂ W (E 6 ).
2.3. Lemma. –––– There is a constant C such that, for all smooth cubic surfaces X
overÉ,
1 ≤ β(X ) ≤ C.
Proof. By definition, β(X ) = #H 1( Gal(É/É), Pic(XÉ) ) . Using the notation
just introduced, we may write H 1( Gal(É/É), Pic(XÉ) ) = H 1 (G, F/F 0 ).
Note that this cohomology group is always finite. Indeed, since G is a finite
group and F/F 0 is a finite[G]-module, the description via the standard
complex shows it is finitely generated. Further, it is annihilated by #G.
H 1 (G, F/F 0 ) depends only on the subgroup G ⊂ W (E 6 ) occurring. For that,
there are finitely many possibilities. This implies the claim.
□
2.4. Remark. –––– A more precise consideration (cf. Proposition II.8.18) yields
a canonical isomorphism
H 1( Gal(É/É), Pic(XÉ) ) ∼ = Hom
(
(NF ∩ F0 )/NF 0 ,É/) .
Here, N is the norm map under the operation of G.
As an application of this, one may inspect the 350 conjugacy classes of subgroups
of W (E 6 ) using GAP. See Chapter II, 8.23, or the complete list of the values
of H 1( Gal(É/É), Pic(XÉ) ) given in the appendix. The calculations show that
the lemma is actually true for C = 9.

306 ON THE SMALLEST POINT ON A DIAGONAL CUBIC SURFACE [Chap. X
2.5. Lemma. –––– There are positive constants C 1 and C 2 such that, for all smooth
cubic surfaces X overÉsatisfying X (É) ≠ ∅,
C 1 ≤ α(X ) ≤ C 2 .
Proof. Again, we claim that α(X ) is completely determined by the group
G ⊂ W (E 6 ). Thus, suppose that we do not have the full information available
about what surface X is but are given the group G only.
The assumption X (É) ≠ ∅ makes sure that Pic(X ) ∼ = Pic(XÉ) G [K/T, Remark
3.2.ii)]. We may therefore write Pic(X ) ∼ = (F/F 0 ) G . The effective cone
Λ eff (X ) ⊂ Pic(X )⊗∼ = (F/F0 ) G ⊗is generated by the symmetrizations
of the classes l 1 , . . . , l 27 of the 27 lines in F. In particular, it is determined
by G, completely. Further, we have K = − 1(l 9 1 + . . . + l 27 ). These data are
sufficient to compute α(X ) according to its very definition.
□
2.6. Remark. –––– Here, we do not know the optimal values of C 1 and C 2 in
explicit form. α(X ) has not yet been computed in all cases.
2.7. Remark. –––– In the experiment, we work entirely with cubic surfaces of
Picard rank one overÉ. This may easily be seem from Theorem 3.6, below.
Therefore, we always have α(X ) = 1. Further, Theorem III.6.4.a) implies
that β(X ) = 3.
3. Splitting the Picard group
3.1. –––– In the case of the diagonal cubic surface X (a 0,...,a 3 )
⊂ P 3É, given
by a 0 x 3 0 + . . . + a 3x 3 3 = 0 for a 0, . . . , a 3 ∈\{0}, the 27 lines on X (a 0,...,a 3 )
may easily be written down explicitly. Indeed, for each pair (i, j) ∈ (/3) 2 ,
the system
√ √
3 a0 x 0 + ζ3
i 3 a1 x 1 = 0
√ √
3 a2 x 2 + ζ j 3
3 a3 x 3 = 0
of equations defines a line on X (a 0,...,a 3 ) . Decomposing the index set
{0, . . . , 3} differently into two subsets of two elements each yields
all the lines. √In particular, √ we √ see that
)
the 27 lines may be defined
over K =É( ζ3 , 3 a1 /a 0 , 3 a2 /a 0 , 3 a3 /a 0 .

Sec. 3] SPLITTING THE PICARD GROUP 307
3.2. Fact. –––– Let p be a prime number and a 0 , . . . , a 3 be integers not divisible
by p. Then,
#X (a 0, ... ,a 3 ) (p) =
⎧
p
⎪⎨
2 + ( 1 + χ 3 (a 0 a 1 a 2 2 a2 3 ) + χ 3(a 2 0 a2 1 a 2a 3 )
+ χ 3 (a 0 a 2 1 a 2a 2 3 ) + χ 3(a 2 0 a 1a 2 2 a 3)
+ χ 3 (a 0 a 2 1
⎪⎩
a2 2 a 3) + χ 3 (a 2 0 a 1a 2 a 2 3 )) p + 1 if p ≡ 1 (mod 3) ,
p 2 + p + 1 if p ≡ 2 (mod 3) .
Here, in the case p ≡ 1 (mod 3), χ 3 :∗ p →denotes a cubic residue character.
Proof. If p ≡ 2 (mod 3) then every residue class modulo p has a
unique cubic root. Therefore, the map X (a 0, ... ,a 3 ) (p) → P 2 (p) given by
(x : y : z : w) ↦→ (x : y : z) is bijective. This shows #X (a 0, ... ,a 3 ) (p) = p 2 +p+1.
Turn to the case p ≡ 1 (mod 3). It is classically known that, on a degree m
diagonal variety, the number ofp-rational points for p ≡ 1 (mod m) may
be determined using Jacobi sums. The formula given follows immediately
from [I/R, Chapter 10, Theorem 2] together with the well-known relation
g(χ 3
É
)g(χ 2 3 ) = p for cubic Gauß sums.
□
3.3. Lemma. –––– Let a 0 , . . . , a 3
É
∈\{0}. Then, for each prime p such that
p ∤ 3a 0 · . . . · a 3 ,
χ
(a Pic(X 0 ,...,a 3 )
)(Frob p ) = tr ( )⊗)
Frob p | Pic(X (a 0,...,a 3 )
⎧
χ 3 (a 0 a 1 a
⎪⎨
2 2 a2 3 ) + χ 3(a 2 0 a2 1 a 2a 3 )
+ χ 3 (a 0 a 2 1
=
a 2a 2 3 ) + χ 3(a 2 0 a 1a 2 2 a 3)
+ χ 3 (a 0 a 2 1
⎪⎩
a2 2 a 3) + χ 3 (a 2 0 a 1a 2
É ⊗
a 2 3 ) + 1 if p ≡ 1 (mod 3) ,
1 if p ≡ 2 (mod 3) .
Proof. As we have good reduction, the trace of Frob p on Pic(X (a 0,...,a 3 )
)
is the same as that of Frob on Pic(X (a 0,...,a 3 )
) ⊗. Further, for the number
p
of points on a non-singular cubic surface over a finite field, the Lefschetz trace
formula can be made completely explicit [Man,кIV,ÌÓÖѺ½]. It shows
#X (a 0, ... ,a 3 ) (p) = p 2 + p · tr ( Frob | Pic(X (a ⊗) 0, ... ,a 3 )
) + 1 .
p
The explicit formulas for the numbers of points given in Fact 3.2 therefore yield
the assertion.
□
√
3.4. Notation. –––– For A an integer, denote the fieldÉ(ζ 3 , 3 A) by K.
Further, let G := Gal(K/É), H := Gal(K/É(ζ 3 )), and χ: H → 〈ζ 3 〉 ⊂∗
be a primitive character. Then, we write ν K := ind G H
(χ) for the induced character
and V K for the corresponding G-representation.

308 ON THE SMALLEST POINT ON A DIAGONAL CUBIC SURFACE [Chap. X
If K is of degree three overÉ(ζ 3 ) then V K is an irreducible rank two representation
of G ∼ = S 3 . Otherwise, K =É(ζ 3 ). Then, V K ∼ =⊕M splits
into the direct sum of a trivial and a non-trivial one-dimensional representation
of H ∼ =/2.
We will freely consider V K as a Gal(É/É)-representation.
3.5. Lemma. –––– Let A be any integer. Then, for a prime p not dividing A,
we have
{
νÉ(ζ 3 , 3√ A) χ3 (A) + χ
(Frob p ) =
3 (A) if p ≡ 1 (mod 3) ,
0 if p ≡ 2 (mod 3) .
Proof. The primitive character is unique up to conjugation by an element of G.
Therefore, the induced character λ is well-defined.
The Kummer pairing allows to make a definite choice for χ as follows. Fix an
embedding σ :É(ζ 3 ) →. Then, put χ(g) := σ ( g( 3 √ A)/
3√ A
)
.
If p ≡ 2 (mod 3) then p remains prime inÉ(ζ 3 ). This means, Frob p acts
non-trivially onÉ(ζ 3 ), i.e. Frob p ∈ G\H . Since H is a normal subgroup in G,
the induced character vanishes on such an element.
For p ≡ 1 (mod 3), we have that (p) splits inÉ(ζ 3 ). Let us write (p) = pp.
The choice of p is equivalent to the choice of a homomorphism ι: 〈ζ 3 〉 →∗ p .
The Frobenius Frob p is determined only up to conjugation, we may choose
Frob p = Frob p ∈ H . Then, directly by the definition of an induced
character, νÉ(ζ 3 , 3√ A) (Frob p ) = χ(Frob p ) + χ(Frob p ). We need to show
that χ(Frob p ) = χ 3 (A) or χ(Frob p ) = χ 3 (A).
For this, by the choice made above, we have
χ(Frob p ) := σ ( Frob p ( 3 √
A)/
3√
A
)
.
After reduction modulo p, we may write
Frob( 3 √
A)/
3√
A = (
3√
A) p / 3 √
A = A
p−1
3 .
√
Therefore, Frob p ( 3 A)/
3√ A = ι −1 (A p−1
3 ) which shows
É
χ(Frob p ) = σ ( ι −1 (A p−1
3 ) ) .
É
That final formula is a definition for a cubic residue character at A. □
3.6. Theorem. –––– Let a 0 , . . ., a 3 ∈\{0}. Then, the Gal(É/É)-representation
Pic(X (a 0,...,a 3 )
) ⊗splits into the direct sum
Pic ( X (a 0,...,a 3
)
) ⊗∼ =⊕V K 1
⊕ V K 2
⊕ V K 3

Sec. 3] SPLITTING THE PICARD GROUP 309
√
where K 1 , K 2 , and K 3 denote the fieldsÉ(ζ 3 , 3 a0 a 1 a 2 2 a2 3 ),É(ζ √
√ 3 , 3 a0 a 2 1 a 2a 2 3 ),
andÉ(ζ 3 , 3 a0 a 2 1 a2 2 a É
3), respectively.
Proof. We will show that the representations on both sides have the same character.
For that, by virtue of the Chebotarev density theorem, it suffices to
consider the values at the Frobenii Frob p for p ∤ 3a 0 · . . . · a 3 .
For the representation on the left hand
É
side, χ
(a Pic(X 0 ,...,a 3 )
)(Frob p ) has been computed
in Lemma 3.3. For the representation on the right hand side, Lemma 3.5
shows that exactly the same formula is true.
□
3.7. Corollary. –––– Let a 0 , . . . , a 3 ∈\{0} be integers,
P := Pic ( X (a 0, ... ,a 3
)
) ⊗,
considered as a Gal(É/É)-representation,
√
and denote by χ P be the associated
character. Put K 1 :=É(ζ 3 , 3 a0 a 1 a 2 2 a2 3 ), K √
√ 2 :=É(ζ 3 , 3 a0 a 2 1 a 2a 2 3
), and
K 3 :=É(ζ 3 , 3 a0 a 2 1 a2 2 a 3). Then, for the Artin conductor N χP of χ P , we have
where
N 2 χ P
= D(K 1 ) D(K 2 ) D(K 3 )/(−27) ,
D(K ) :=
{ Disc(K/É) if [K :É(ζ 3 )] = 3 ,
−27 if K =É(ζ 3 ) .
Proof. We have to show N 2 = D(K )/(−3). Assume first that [K :É(ζ
ν K 3 )] = 3.
Then, the conductor-discriminant formula [Ne, Chapter VII, Section (11.9)]
shows Disc(K/É) = NN M N 2 and −3 = Disc(É(ζ
ν K 3 )/É) = NN M which
together yield the assertion. In the opposite case, we have V K =⊕M
and N ν K = NN M = −3.
□
3.8. Lemma. –––– Let a and b be integers different from zero. Then,
Proof. We have, at first,
|Disc (É(ζ 3 , 3 √
ab2 )/É) | ≤ 3 9 a 4 b 4 .
| Disc (É(ζ 3 , 3 √
ab2 )/É) | ≤ | Disc
(É(ζ 3 )/É) |3 · Disc (É( 3 √
ab2 )/É) 2
= 27 · Disc (É( 3 √
ab2 )/É) 2.
Further, by [Mc, Chapter 2, Exercise 41], we know
|Disc (É( 3 √
ab2 )/É) | ≤ 3 3 a 2 b 2 .
This shows
|Disc (É(ζ 3 , 3 √
ab2 )/É) | ≤ 3 9 a 4 b 4 .
□

É
310 ON THE SMALLEST POINT ON A DIAGONAL CUBIC SURFACE [Chap. X
3.9. Corollary. –––– Let a 0 , . . . , a 3 ∈\{0} be integers,
P := Pic ( X (a 0, ... ,a 3
)
) ⊗,
considered as a Gal(É/É)-representation, and χ P be the associated character.
Then, for the Artin conductor N χP of χ P , we have the estimate
| N χP | ≤ 3 12 (a 0 · . . . · a 3 ) 6 .
Proof. Lemma 3.8 shows | D(K i )| ≤ 3 9 (a 0 · . . . · a 3 ) 4 for i = 1, 2, and 3.
The assertion follows immediately from this.
□
4. A technical lemma
4.1. Sublemma. –––– a) (Good reduction)
If p ∤ 3a 0 · . . . · a 3 then the sequence ( #X (a 0,...,a 3
n∈Æ
) (/p n)/p2n) n∈Æis constant.
b) (Bad reduction)
i) If p divides a 0· . . . ·a 3 but not 3 then the sequence ( #X (a 0,...,a 3 ) (/p n)/p2n) becomes stationary as soon as p n does not divide any of the coefficients a 0 , . . . , a 3 .
ii) If p = 3 then the sequence ( #X (a 0,...,a 3 ) (/p n)/p2n) n∈Æbecomes stationary as
soon as 3 n does not divide any of the numbers 3a 0 , . . . , 3a 3 .
4.2. Lemma. –––– There are two positive constants C 1 and C 2 such that, for
all a 0 , . . ., a 3 ∈\{0},
Proof. Cf. Lemma VI.5.34.
C 1 < ∏
p prime
p∤3a 0·····a 3
τ p
(
X
(a 0 , ... ,a 3 ) (Ép) ) < C 2 .
For a prime p of good reduction, Sublemma 4.1 shows
τ p
(
X
(a 0 , ... ,a 3 ) (Ép) ) = det ( 1 − p −1 Frob p | Pic(XÉ) ) · #X (a 0, ... ,a 3 ) (p)
p 2 .
Further, for the number of points on a non-singular cubic surface over a finite
field, the Lefschetz trace formula can be made completely explicit [Man,
кIV,ÌÓÖѺ½]. It shows
#X (a 0, ... ,a 3 ) (p) = p 2 + p · tr ( Frob p | Pic(XÉ) ) + 1 .

Sec. 5] A NEGATIVE RESULT 311
Denoting the eigenvalues of the Frobenius on Pic(XÉ) by λ 1 , . . . , λ 7 , we find
τ p
(
X
(a 0 , ... ,a 3 ) (Ép) )
= (1 − λ 1 p −1 )(1 − λ 2 p −1 ) · . . . · (1 − λ 7 p −1 )
· [1 + (λ 1 + · · · + λ 7 )p −1 + p −2 ]
= (1 − σ 1 p −1 + σ 2 p −2 ∓ . . . − σ 7 p −7 )(1 + σ 1 p −1 + p −2 )
= 1 + (1 − σ 2 1 + σ 2 )p −2 − (σ 1 − σ 1 σ 2 + σ 3 )p −3 ±
± . . . − (σ 5 − σ 1 σ 6 + σ 7 )p −7 + (σ 6 − σ 1 σ 7 )p −8 − σ 7 p −9
where σ i denote the elementary symmetric functions in λ 1 , . . . , λ 7 .
We know |λ i | = 1 for all i. Estimating very roughly, we have |σ j | ≤ ( 7 j ) ≤ 7j
and see
1 − 99p −2 − 7·99p −3 − . . . − 7 7·99p −9 ≤ τ p
(
X
(a 0 , ... ,a 3 ) (Ép) ) ≤
≤ 1 + 99p −2 + 7·99p −3 + . . . + 7 7 · 99p −9 .
I.e., 1 − 99p −2 1 < τ (
1−7/p p X
(a 0 , ... ,a 3 ) (Ép) ) < 1 + 99p −2 1
1−7/p
product over all 1 − 99p −2 1
1−7/p
The left hand side is positive for p > 13.
(respectively 1 + 99p−2
1
1−7/p
. The infinite
) is convergent.
For the small primes remaining,
we need a better lower bound. For this, note that a cubic
surface over
(
a finite fieldp always has at least onep-rational point.
This yields τ p X
(a 0 , ... ,a 3 ) (Ép) ) ≥ (1 − 1/p) 7 /p 2 > 0.
□
4.3. Remark. –––– The correctional factors det ( )
1 − p −1 Frob p | Pic(XÉ) I p
are all positive. Indeed, for a pair of complex conjugate eigenvalues, we have
(1 − λp −1 )(1 − λp −1 ) = |1 − λp −1 | 2 > 0 and an eigenvalue of 1 or (−1)
contributes a factor 1 ± p −1 > 0. Consequently, we always have
(
1 − 1 p) 7
< det
(
1 − p −1 Frob p | Pic(XÉ) I p
) <
(
1 + 1 p) 7.
5. A negative result
5.1. –––– For an integer q ≠ 0, let X (q) ⊂ P 3Ébe the cubic surface given by
qx 3 + 4y 3 + 2z 3 + w 3 = 0 and let
m(X (q) ) := min { H(x : y : z : w) | (x : y : z : w) ∈ X (q) (É)}
be the smallest height of aÉ-rational point on X (q) . We compare m(X (q) ) with
the Tamagawa type number τ (q) := τ (X (q) ).

312 ON THE SMALLEST POINT ON A DIAGONAL CUBIC SURFACE [Chap. X
5.2. Lemma. –––– There is a constant C with the following property. For each
pair (a, b)of natural numbers satisfying gcd(a, b) = 1, thereare two prime numbers
p 1 , p 2 ≡ a (mod b), p 1 ≠ p 2 , such that p 1 , p 2 < C ·b 5.5 .
Proof. Linnik’s Theorem in the version of R. Heath-Brown [H-B92b] shows
there is one prime with the property stated. To get a second one, the following
simple trick helps.
One of the numbers a and a + b is relatively prime to 4b. Assume without
restriction that gcd(a, 4b) = 1. Then, we find p 1 ≡ a (mod 4b) and
p 2 ≡ a + 2b (mod 4b) such that p 1 , p 2 < ˜C ·(4b) 5.5 .
□
5.3. Theorem. –––– Assume the Generalized Riemann Hypothesis. Then, there is
no constant C such that
m(X (q) ) < C
τ (q)
for all q ∈\{0}.
Proof. We will construct a sequence (q i ) i∈Æof prime numbers such that
q i ≡ 1 (mod 72) and m(X (qi) ) τ (q i)
→ ∞ for i → ∞. The proof will consist
of several steps.
Firststep. The prime q i divides exactly one of the four coefficients in the equation
defining X (qi) . In this case, it is known by the work of J.-L. Colliot-Thélène and
his coworkers [CT/K/S, Proof of Proposition 2] that precisely
τ H
(
X (É)
It is therefore sufficient to verify that
Br) = 1 3 τ H(
X (É)
) .
m(X (qi) ) · α(X (qi) )·β(X (qi) )·lim (s −1) t L(s, χ )· (
(qi)
s→1 Pic(X ∏ τ
É) ν X
(q i ) (Éν) ) → ∞.
ν∈Val(É)
Second step. If q is a prime different from 2 then we have rkPic(X (q) ) = 1.
[Man,кIII,ÍÔÖÒÒº½¾] shows that X (q) is a minimal cubic surface
overÉ. By [Man,кIV,ÌÓÖѺ½], we have Pic(X (q) ) =.
Third step. We have α(X (qi) ) = 1 and β(X (qi) ) = 3.
α(X (qi) ) = 1 follows immediately from rk Pic(X (qi) ) = 1. β(X (qi) ) can
be computed by the method indicated in Remark 2.4. We work with a
group G ⊂ W (E 6 ) of order 18 which decomposes the 27 lines into three
orbits of nine lines each. An easy calculation shows
Hom ( (NF ∩ F 0 )/NF 0 ,É/) ∼=/3.
Fourth step. For the height of the smallest point, we have m(X (q) ) ≥ 3 √
q
7 .

Sec. 5] A NEGATIVE RESULT 313
There are no rational solutions of the equation 4y 3 + 2z 3 + w 3 = 0 as
this is impossible, 2-adically. √ |x| ≥ 1 yields |4y 3 + 2z 3 + w 3 | ≥ q and
max{|y|, |z|, |w|} ≥ q 3 . 7
Fifth step. For |q| ≥ 7, one has τ ∞
(
X (q) (Ê) ) = 1
3√
|q|
I where I is independent
of q.
By Lemma 6.15, we have
τ ∞ (X (q) ) = 1
4 3 √
|q|
Z
CX (1, ... ,1) (Ê)
|x 0 |≤ 3√ |q|,|x 1 |≤ 3√ 4,|x 2 |≤ 3√ 2,|x 3 |≤1
ω CX (1, ... ,1) (Ê)
Leray .
Since |x 1 | ≤ 3 √
4, |x2 | ≤ 3 √
2, |x3 | ≤ 1, and
|x 0 | = 3 √
| x 3 1 + x3 2 + x3 3 | ≤ 3 √| x 1 | 3 + | x 2 | 3 + | x 3 3 | ,
the condition |x 0 | ≤ 3 √
|q| is actually empty.
(
Sixth step. There is a positive constant C such that ∏ τ p X (q) (Ép) ) > C for
p prime
every prime q ≡ 1 (mod 72).
By Lemma 4.2, we have C 1 > 0 such that
∏
p prime
p≠2,3,q
τ p
(
X (q) (Ép) ) > C 1 .
It, therefore, remains to give lower bounds for the factors τ 2
(
X (q) (É2) ) ,
τ 3
(
X (q) (É3) ) , and τ q
(
X (q) (Éq) ) .
As 2 ∤q, by virtue of Sublemma 4.1 we have, τ 2
(
X (q) (É2) ) = 1 2 7 · #X (q) (/8)
64
.
Further,
#X (q) (/8) ≥ 1
since q ≡ 1 (mod 8) implies (1 : 0 : 0 : (−1)) ∈ X (q) (/8).
Similarly, τ 3
(
X (q) (É3) ) = ( 2 3 )7 · #X (q) (/9)
81
. Again, q ≡ 1 (mod 9) makes sure
that (1 : 0 : 0 : (−1)) ∈ X (q) (/9) and #X (q) (/9) ≥ 1.
For the prime q, we argue a bit differently. First,
det ( 1 − q −1 Frob p | Pic(X É) (q) ) I q ≥ (1 − 1/q) 7 ≥ (72/73) 7 .
Furthermore, the reduction of X (q) modulo q is the cone over the elliptic
curve given by 4y 3 + 2z 3 + w 3 = 0. Therefore, on X (q) there are at
least (q − 2 √ q + 1)(q − 1) smooth points defined overq. As Hensel’s lemma

314 ON THE SMALLEST POINT ON A DIAGONAL CUBIC SURFACE [Chap. X
may be applied to them, we get
#X (q) (/q n)
lim
≥ (q − 2√ q + 1)(q − 1)
(
> 1 − 2 )(
√ 1 − 1 )
n→∞ q 2n q 2 q q
≥ 72 (
1 − √ 2 ).
73 73
Seventh step. There is a sequence (q i ) i∈Æof primes such that q i ≡ 1 (mod 72)
and [lim (s − 1)L(s, χ (qi)
s→1 Pic(X
)] → ∞ for i → ∞.
É)
Since rk Pic(X (qi) ) = 1, the representation Pic ( É)
X (q i) ⊗contains exactly one
trivial summand. Hence,
L ( )
s, χ (qi) Pic(X = ζ(s) · L(s, χP (qi))
É)
for a representation P (q i) not containing trivial components. Our goal is, therefore,
to show L(1, χ P (qi)) → ∞ for i → ∞.
DenotebyP i thei-thprimenumber p suchthat p ≡ 1 (mod 3). For eachi ∈Æ,
choose a cubic residue r i modulo P i . Then, define q i to be the second smallest
prime number such that
q i ≡ r j (mod P j )
for all j ∈ {1, . . . , i} and q i ≡ 1 (mod 72).
This system of simultaneous congruences is solvable by the Chinese Remainder
Theorem. Its solutions form an arithmetic progression with common difference
72P 1 · . . . · P i . By Chebyshev’s inequalities, we know
72P 1 · . . . · P i ≤ 72e θ(P i) < 72e (2 log2)P i
.
According to our definition, q i is the second smallest prime in this arithmetic
progression. From this, we clearly have that q i > 72P 1 · . . . · P i → ∞
for i → ∞. On the other hand, Lemma 5.2 shows
for certain constants C 1 and C 2 .
q i ≤ C 1 · (72e (2 log2)P i
) 5.5 = C 2 e (11 log2)P i
Corollary 3.9 gives us an estimate for the Artin conductor of the character χ P (qi).
We see N χP ≤ 3 12 (a
(qi) 0 · . . . · a 3 ) 6 = 3 12 8 6 qi 6 ≤ C 3 e (66 log2)P i
for another constant
C 3 . Consequently,
logN χP (qi )
≤ (66 log2)P i + logC 3 .
We observe that (log N χP (qi) )1/2 ≤ P i for i sufficiently large. We assume from
now on that this inequality is fulfilled.

Sec. 5] A NEGATIVE RESULT 315
Recall that P (q i) is actually the direct sum of representations which are induced
from one-dimensional characters (Theorem 3.6). By consequence, it is known
that the Artin L-function L( · , χ P (qi)) is entire. Since we also assume the
Generalized Riemann Hypothesis, we may apply the estimate of W. Duke [Duk,
Proposition 5]. It shows
Here,
logL(1, χ P (qi)) =
∑
p

316 ON THE SMALLEST POINT ON A DIAGONAL CUBIC SURFACE [Chap. X
| D(K 3 )| = |Disc (É(ζ 3 , 3 √
qi )/É) |
= Disc (É( 3 √
qi )/É)2
· |N ( Disc (É(ζ 3 , 3 √
qi )/É( 3 √
qi ) )) |
≥ Disc (É( 3 √
qi )/É) 2.
According to [Mc, Chapter 2, Exercise 41], we know
| Disc (É( 3 √
qi )/É) | ≥ 3q
2
i . □
5.4. Remark. –––– Note that the estimate for L(1, χ P (qi)) is the only point where
we use the Generalized Riemann Hypothesis.
6. The fundamental finiteness property
6.1. –––– In this section, we return to the case of a general diagonal cubic
surface X (a 0, ... ,a 3 )
⊂ P 3Égiven by a 0 x 3 0 + . . . + a 3x 3 3 = 0. Our goal is to
establish the estimate for τ (a 0, ... ,a 3 ) := τ (X (a 0, ... ,a 3 ) ) formulated as Theorem 1.7.
For this, in the subsections below, we will give an individual estimate for each
of the factors occurring in the definition of τ (X (a 0, ... ,a 3 ) ).
i. An estimate for the L-factor. —
É
)
6.2. Proposition. –––– For each ε > 0, there exist positive constants c 1 and c 2
such that
c 1 · |a 0 · . . . · a 3 | −ε < lim (s − 1) t L ( s, χ
s→1
É
)
(a Pic(X 0 , ... ,a 3 ) < c2 · |a 0 · . . . · a 3 | ε
for all (a 0 , . . . , a 3 ) ∈ (\{0}) 4 . Here, t = rk Pic(X ).
Proof. The Galois representation Pic(X (a 0, ... ,a 3 )
) ⊗contains the trivial representation
t times as a direct summand. Therefore,
L ( )
s, χ (a Pic(X 0 , ... ,a 3 ) = ζ(s)t · L(s, χ P )
É
)
where ζ denotes the Riemann zeta function and P is a representation which
does not contain trivial components. All we need to show is
c 1 · |a 0 · . . . · a 3 | −ε < L(1, χ P ) < c 2 · |a 0 · . . . · a 3 | ε .
L( · , χ P ) is the product of at most three factors of the form L( · , λ) where λ is
the non-trivial Dirichlet character ofÉ(ζ 3 )/Éand at most three factors which
are Artin-L-functions L( · , ν K ) for K a purely cubic field extension ofÉ(ζ 3 )

Sec. 6] THE FUNDAMENTAL FINITENESS PROPERTY 317
as above, sayK =É( ζ3 , 3 √
a0 a 1 a 2 2 a2 3)
. AsL(1, λ)does not depend on a0 , . . . , a 3 ,
at all, it will suffice to show
for each ε > 0.
c 1 (ε) · |a 0 · . . . · a 3 | −ε < L(1, ν K ) < c 2 (ε) · |a 0 · . . . · a 3 | ε
V K is the only irreducible two-dimensional representation of Gal(K/É) ∼ = S 3 .
For that reason, by virtue of [Ne, Chapter VII, Corollary (10.5)], we have
ζ K (s) = ζÉ(s) · L(s, λ) · L(s, ν K ) 2
= ζÉ(ζ 3 )(s) · L(s, ν K ) 2
for a complex variable s. It, therefore, suffices in our particular situation to
estimate the residue res s=1 ζ K (s) of the Dedekind zeta function of K.
An estimate from above has been given by C. L. Siegel. In view of the analytic
class number formula, his [Si69, Satz 1] gives
res ζ K (s) < C (log Disc(K/É)) 5
s=1
≤ C [log(3 9 a 4 0a 4 1a 4 2a 4 3)] 5
= C [4 log |a 0 · . . . · a 3 | + 9 log 3] 5
for a certain constant C. The final term is less than c 2 (ε) · |a 0 · . . . · a 3 | ε for
every ε > 0.
On the other hand, H. M. Stark [St, formula (1)] shows
res ζ K (s) > C (ε)·Disc(K/É) −ε/4
s=1
for every ε > 0 which implies res
s=1 ζ K (s) > c 1 (ε)·|a 0 · . . . · a 3 | −ε . □
ii. An estimate for the factors at the finite places. —
6.3. Notation. –––– We will adopt the notation from Chapter VIII, 3.1. More
precisely,
i) For a prime number p and an integer x ≠ 0, we put x (p) := p ν p(x) .
Note x (p) = 1/‖x‖ p for the normalized p-adic valuation.
ii) By putting ν(x) := min
ξ∈p
to/p r.
x=(ξ mod p r )
ν(ξ), we carry the p-adic valuation fromp over
Note that any 0 ≠ x ∈/p rhas the form x = ε·p ν(x) where ε ∈ (/p r) ∗ is
a unit. Clearly, ε is unique only in the case ν(x) = 0.

318 ON THE SMALLEST POINT ON A DIAGONAL CUBIC SURFACE [Chap. X
6.4. Definition. –––– For (a 0 , . . . , a 3 ) ∈4 , r ∈Æ, and ν 0 , . . . , ν 3 ≤ r, put
N (r)
ν 0 , ... ,ν 3 ;a 0 , ... ,a 3
:= { (x 0 , . . . ,x 3 ) ∈ (/p r) 4 |
ν(x 0 ) = ν 0 , . . . ,ν(x 3 ) = ν 3 ; a 0 x 3 0 + . . . + a 3 x 3 3 = 0 ∈/p r}.
For the particular case ν 0 = . . . = ν 3 = 0, we will write
I.e.,
Z (r)
a 0 , ... ,a 3
:= N (r)
0, ... ,0;a 0 , ... ,a 3
.
Z (r)
a 0 , ... ,a 3
= { (x 0 , . . . , x 3 ) ∈ [(/p r) ∗ ] 4 | a 0 x 3 0 + . . . + a 3 x 3 3 = 0 ∈/p r}.
We will use the notation z (r)
a 0 , ... ,a 3
:= #Z (r)
a 0 , ... ,a 3
.
6.5. Sublemma. –––– If p k |a 0 , . . . , a 3 and r > k then we have
z (r)
a 0 , ... ,a 3
= p 4k · z (r−k)
a 0 /p k , ... ,a 3 /p k .
Proof. Since a 0 x 3 0 + . . . + a 3 x 3 3 = p k (a 0 /p k · x 3 0 + . . . + a 3 /p k · x 3 3 ), there is a
surjection
ι: Z a (r)
0 , ... ,a 3
−→ Z (r−k) ,
a 0 /p k , ... ,a 3 /p k
given by (x 0 , . . . , x 3 ) ↦→ ( (x 0 mod p r−k ), . . . , (x 3 mod p r−k ) ) . The kernel of
the homomorphism of modules underlying ι is (p r−k/pr) 4 . □
6.6. Lemma. –––– Assume gcd p
(a 0 , . . . , a 4 ) = p k . Then, there is an estimate
z (r)
a 0 , ... ,a 4
≤ 3p 3r+k .
Proof. Suppose first that k = 0. This means, one of the coefficients is prime
to p. Without restriction, assume p ∤a 0 .
For any (x 1 , x 2 , x 3 ) ∈ (/p r) 3 , there appears an equation of the form a 0 x 3 0 = c.
It cannot have more than three solutions in (/p r) ∗ . Indeed, for p odd, this
follows directly from the fact that (/p r) ∗ is a cyclic group. On the other
hand, in the case p = 2, we have (/2 r) ∗ ∼ =/2 r−2×/2. Again, there
are only up to three solutions possible.
The general case may now easily be deduced from Sublemma 6.5. Indeed, if
k < r then
z (r)
a 0 , ... ,a 3
= p 4k · z (r−k)
a 0 /p k , ... ,a 3 /p k ≤ p 4k · 3p 3(r−k) = 3p 3r+k .
On the other hand, if k ≥ r then the assertion is completely trivial since
z (r)
a 0 , ... ,a 3
= #Z (r)
a 0 , ... ,a 3
< p 4r ≤ p 3r+k < 3p 3r+k .
□

Sec. 6] THE FUNDAMENTAL FINITENESS PROPERTY 319
6.7. Remark. –––– The proof shows that in the case p ≡ 2 (mod 3) one could
reduce the coefficient to 1. Unfortunately, this observation does not lead to a
substantial improvement of our final result.
6.8. Lemma. –––– Let r ∈Æand ν 0 , . . . , ν 3 ≤ r. Then,
#N ν (r)
0 , ... ,ν 3 ;a 0 , ... ,a 3
= z(r) · ϕ(p r−ν 0
) · . . . · ϕ(p r−ν 3
)
p 3ν 0a 0 , ... ,p 3ν 3a 3
.
ϕ(p r ) 4
Proof. As p 3ν 0 a0 x 3 0 + . . . + p 3ν 3 a3 x 3 3 = a 0 (p ν 0 x0 ) 3 + . . . + a 3 (p ν 3 x3 ) 3 , we have
a surjection
π : Z (r) −→ N (r)
p 3ν 0a 0 , ... ,p 3ν 3a 3
ν 0 , ... ,ν 3 ;a 0 , ... ,a 3
,
given by (x 0 , . . . , x 3 ) ↦→ (p ν 0 x0 , . . . , p ν 3 x3 ).
For i = 0, . . . , 3, consider the mapping ι:/p r→/pr, x ↦→ p ν i
x.
If ν i = r then ι is the zero map. All ϕ(p r ) = (p − 1)p r−1 units are mapped
to zero. Otherwise, observe that ι is p ν i
: 1 on its image. Further, ν(ι(x)) = ν i
if and only if x is a unit. By consequence, π is (K (ν 0)
· . . . · K (ν3) ) : 1 when
we put K (ν) := p ν for ν < r and K (r) := (p − 1)p r−1 . Summarizing, we could
have written K (ν) := ϕ(p r )/ϕ(p r−ν ). The assertion follows.
□
6.9. Corollary. –––– Let (a 0 , . . . , a 3 ) ∈ (\{0}) 4 . Then, for the local factor
τ p
(
X
(a 0 , ... ,a 3 ) (Ép) ) , one has
(
τ p X
(a 0 , ... ,a 3 ) (Ép) ) = det ( )
1 − p −1 Frob p | Pic(XÉ) I p
r
r→∞
∑
ν 0 ,...,ν 3 =0
· lim
z (r) · ϕ(p r−ν p 3ν 0a 0 , ... ,p 3ν 0
3a ) · . . . · ϕ(p
r−ν 3)
3
.
p 3r · ϕ(p r ) 4
Proof. By Remark VI.5.19, we have
τ p
(
X
(a 0 , ... ,a 3 ) (Ép) ) = det ( 1 − p −1 Frob p | Pic(XÉ) I p )
Lemma 6.8 yields the assertion.
r
#N ν
r→∞
∑
(r)
0 ,...,ν 3 ;a 0 ,...,a 3
ν 0 ,...,ν 3 =0
· lim
p 3r .
□
6.10. Proposition. –––– Let (a 0 , . . . , a 3 ) ∈ (\{0}) 4 . Then, for each ε such
that 0 < ε < 1 , one has
3
(
τ p X
(a 0 , ... ,a 3 ) (Ép) ) (
≤ 1+ 1 ) (
1
)( 1
) 3·(
(p)
) 1−ε
7·3 ( a
p 1 − 1 1 − 1 0 a(p) 1 a(p) 3
2
a (p) ) ε
3 .
p 1−3ε p ε

320 ON THE SMALLEST POINT ON A DIAGONAL CUBIC SURFACE [Chap. X
Proof. We use the formula from Corollary 6.9. By Remark 4.3, the first factor
is at most (1 + 1/p) 7 . Further, by Lemma 6.6,
z (r)
p 3ν 0a 0 , ... ,p 3ν 3a 3
/p 3r ≤ 3 gcd p
(p 3ν 0
a 0 , . . . , p 3ν 3
a 3 )
(
(p)) Writing k i := ν p (a i ) = ν p a i , we see
= 3 gcd ( p 3ν 0
a (p)
0 , . . . , p3ν 3
a (p)
3
z (r)
p 3ν 0a 0 , ... ,p 3ν 3a 3
/p 3r ≤ 3 gcd(p 3ν 0+k 0
, . . . , p 3ν 3+k 3
)
= 3p min{3ν 0+k 0 , ... ,3ν 3 +k 3 } .
We estimate the minimum by a weighted arithmetic mean with weights 1−ε
1−ε
, and ε,
, 1−ε
3 3
min{3ν 0 + k 0 , . . . , 3ν 3 + k 3 }
This shows
≤ 1 − ε
3
· (3ν 0 + k 0 ) + 1 − ε · (3ν 1 + k 1 )
3
+ 1 − ε · (3ν 2 + k 2 ) + ε(3ν 3 + k 3 )
3
= (1 − ε)(ν 0 + ν 1 + ν 2 ) + 3εν 3
)
.
+ 1 − ε (k 0 + k 1 + k 2 ) + εk 3 .
3
z (r) /p 3r ≤ 3p (1−ε)(ν 0+ν 1 +ν 2 )+3εν 3 + 1−ε
p 3ν 0a 0 , ... ,p 3ν 3 (k 0+k 1 +k 2 )+εk 3
3a 3
(a
= 3p (1−ε)(ν 0+ν 1 +ν 2 )+3εν3 (p)
) 1−ε
· ( 0 a(p) 1 a(p) 3
2
a (p) ) ε.
3
We may therefore write
τ p
(
X
(a 0 , ... ,a 3 ) (Ép) ) ≤
r
r→∞
∑
ν 0 , ... ,ν 3 =0
· lim
(
1 + 1 ) 7 ( · 3 a
(p)
) 1−ε (
0
p
a(p) 1 a(p) 3
2
a (p) ) ε
3
p (1−ε)(ν 0+ν 1 +ν 2 )+3εν 3 · ϕ(p
r−ν 0) · . . . · ϕ(p
r−ν 3)
ϕ(p r ) 4 .
Here, the term under the limit is precisely the product of three copies of the
finite sum r
p
∑
(1−ε)ν · ϕ(p r−ν r−1
) 1
=
ν=0
ϕ(p r )
∑
ν=0
(p ε ) + p 1
ν p − 1 (p ε ) r
and one copy of the finite sum
r
p
∑
3εν · ϕ(p r−ν )
ν=0
ϕ(p r )
=
r−1
∑
ν=0
1
(p 1−3ε ) + p
ν
1
p − 1 (p 1−3ε ) . r
3 ,

Sec. 6] THE FUNDAMENTAL FINITENESS PROPERTY 321
For r → ∞, geometric series do appear while the additional summands tend
to zero.
□
6.11. Remark. –––– Unfortunately, the constants
) 7 (
C p (ε) := · 3
(
1 + 1 p
1
)( 1
) 3
1 − 1 1 − 1 p 1−3ε p ε
diverges. On the other hand, we
is bounded for p → ∞, say C p (ε) ≤ C (ε) .
have the property that the product ∏ p C (ε)
p
have at least that C (ε)
p
6.12. Lemma. –––– Let C > 1 be any constant. Then, for each ε > 0, one has
∏
p prime
p|x
for a suitable constant c (depending on ε).
C ≤ c · x ε
Proof. This follows directly from [Nat, Theorem 7.2] together with [Nat,
Section 7.1, Exercise 7].
□
6.13. Proposition. –––– For each ε such that 0 < ε < 1 , there exists a constant c
3
such that
(
∏τ p X
(a 0 , ... ,a 3 ) (Ép) ) ≤ c · |a 0 · . . . · a 3 | 3 1 − ε 8 · ∏
p prime
for all (a 0 , . . . , a 3 ) ∈ (\{0}) 4 .
p prime
min ‖a i‖ 1 3 −ε
p
i=0, ... ,3
Proof. The product over all primes of good reduction is bounded by virtue of
Lemma 4.2.a). It, therefore, remains to show that
∏
p prime
p|3a 0 ... a 3
(
τ p X
(a 0 , ... ,a 3 ) (Ép) ) ≤ c · |a 0 · . . . · a 3 | 3 1 − ε 8 · ∏ min ‖a i‖ 1 3 −ε
i=0, ... ,3
For this, by Proposition 6.10, we have at first
τ p
(
X
(a 0 , ... ,a 3 ) (Ép) ) ≤ C (ε)
p
= C (ε)
p
·
·
p prime
(a(p)
) 1
0 a(p) 1 a(p) 3 − ε4
2
· (a (p)
(a(p)
0 a(p) 1 a(p) 2 a(p) 3
3 ) 3 4 ε
) 1
3
− ε4 · (a (p)
3 )− 1 3 +ε .
Here, the indices 0, . . . , 3 are interchangeable. Hence, it is even allowed to
write
(
τ p X
(a 0 , ... ,a 3 ) (Ép) ) (a
≤ C p
(ε) (p)
) 1
·
0 a(p) 1 a(p) 2 a(p) 3
− ε4
3
· (max )
a (p) −
1
3
+ε
i i
)
= C p
(ε)
1
· 3
− ε4 · min ‖a i ‖ 1 3 −ε
p .
(a(p)
0 a(p) 1 a(p) 2 a(p) 3
Now, we multiply over all prime divisors of a 0 · . . . · a 3 . Thereby, on the
right hand side, we may twice write the product over all primes since the two
i
p .

322 ON THE SMALLEST POINT ON A DIAGONAL CUBIC SURFACE [Chap. X
rightmost factors are equal to one for p ∤ 3a 0 · . . . · a 3 , anyway.
(
τ p X
(a 0 , ... ,a 3 ) (Ép) )
∏
p prime
p|3a 0 ... a 3
≤ ∏ C p
(ε)
p prime
p|3a 0 ... a 3
(
(p)
) 1
· ∏ a
0 a(p) 1 a(p) 2 a(p) 3 − ε 4
3
· ∏
p prime
= ∏ C p (ε) · | a 0 · . . . · a 3 | 1 3 − ε 4 · ∏
p prime
p|3a 0 ... a 3
p prime
p prime
min ‖a i‖ 1 3 −ε
p
i=0, ... ,3
min ‖a 3
i‖ 1 −ε
p
i=0, ... ,3
when we observe that ∏ p a (p) = | a|. Further, we have C p
(ε)
Lemma 6.12,
∏ C (ε) ≤ c · |3a 0 · . . . · a 3 | ε 8 .
p prime
p|3a 0 ... a 3
We finally estimate 3 ε 8 by a constant. The assertion follows.
iii. An estimate for the factor at the infinite place. —
6.14. Corollary. –––– Let a 0 , . . . , a 3 ∈Ê\{0}. Then,
[
ω CX (a 0 , ... ,a 3 ) (Ê) 1
Leray = dx
3| a 0 |x 2 1 ∧ dx 2 ∧ dx 3
].
0
Proof. We apply Lemma VI.5.26 to U =Ê4 and
f (x 0 , . . . , x 3 ) := a 0 x 3 0 + . . . + a 3 x 3 3 .
≤ C (ε) and, by
Note that {(x 0 , . . . , x 3 ) ∈ CX (a 0,...,a 3 ) (Ê) | x 0 = 0} is a zero set according to
the Leray measure as it is for the hypersurface measure.
□
6.15. Lemma. –––– Let a 0 , . . . , a 3 ∈Ê\{0}. Then,
(
τ ∞ X
(a 0 , ... ,a 3 ) (Ê) ) 1
= √
2 3 | a0 · . . . · a 3 |
Z
CX (1, ... ,1) (Ê)
|x 0 |≤ 3√ |a 0 |, ... ,|x 3 |≤ 3√ |a 3 |
ω CX (1, ... ,1) (Ê)
Leray .
(
Proof. According to the definition of τ ∞ X
(a 0 , ... ,a 3 ) (Ê) ) and the corollary above,
we need to show
Z
1 1
dx 1 ∧dx 2 ∧ dx 3
6 |a 0 |
x 2
CX (a 0 , ... ,a 3 ) 0
(Ê)
|x 0 |≤1, ... ,|x 3 |≤1
Z
1
1
= √ dX
6 3 | a0 · . . . · a 3 | X 2 1 ∧ dX 2 ∧ dX 3 .
CX (1, ... ,1) 0
(Ê)
|X 0 |≤ 3√ |a 0 |, ... ,|X 3 |≤ 3√ |a 3 |
□

Sec. 6] THE FUNDAMENTAL FINITENESS PROPERTY 323
For that, consider the linear mapping l : CX (a 0, ... ,a 3
√ √ ) (Ê) → CX (1, ... ,1) (Ê) given
by (x 0 , . . . , x 3 ) ↦→ ( 3 a0 x 0 , . . . , 3 a3 x 3 ). Then,
( 1
) 3√
l ∗ a1 a 2 a 3 1
dX
X0
2 1 ∧ dX 2 ∧ dX 3 = dx
x 2 1 ∧ dx 2 ∧ dx 3 .
0
This immediately yields the assertion when we take into consideration that
orientations are chosen in such a way that both integrals are positive. □
a 2/3
0
6.16. Proposition. –––– For real numbers 0 < b 0 ≤ b 1 ≤ b 2 ≤ b 3 , we have
Z
(
ω CX (1, ... ,1) (Ê)
Leray ≤ 64 + 64
3 log 3 + 1 3√
3ω2
)b 0 + 64b 0 log b 1
3
b 0
CX (1, ... ,1)(Ê)
| x 0 |≤b 0 , ... ,|x 3 |≤b 3
where ω 2 is the two-dimensional hypersurface measure on the l 3 -unit sphere
S 2 := { (x 1 , x 2 , x 3 ) ∈Ê3 | | x 1 | 3 + | x 2 | 3 + | x 3 | 3 = 1 } .
Proof. First step. We cover the domain of integration by 25 sets as follows.
We put R 0 := [−b 0 , b 0 ] 4 ∩ CX (1, ... ,1) (Ê). Further, for each σ ∈ S 4 , we set
R σ := { (x 0 , . . . , x 3 ) ∈Ê4 | | x σ(0) | ≤ · · · ≤ |x σ(3) |, | x i | ≤ b i , b 0 ≤ | x σ(3) | }
Second step. One has R R σ
ω CX (1, ... ,1) (Ê)
Leray
∩ CX (1, ... ,1) (Ê) .
≤ R R id
ω CX (1, ... ,1) (Ê)
Leray for every σ ∈ S 4 .
Consider the map i σ :Ê4 →Ê4 given by (x 0 , . . . , x 3 ) ↦→ (x σ(0) , . . . , x σ(3) ).
Since CX (1, ... ,1) (Ê) is defined by a symmetric cubic form, it is invariant under i σ .
We claim that i σ (R σ ) ⊆ R id .
Indeed, let (x 0 , . . . , x 3 ) ∈ R σ . Then, i σ (x 0 , . . . , x 3 ) = (x σ(0) , . . . , x σ(3) )
has the properties | x σ(0) | ≤ . . . ≤ | x σ(3) | and b 0 ≤ | x σ(3) |. In order to show
i σ (x 0 , . . . , x 3 ) ∈ R id , all we need to verify is | x σ(i) | ≤ b i for i = 0, . . . , 3.
For this, we use that the b i are sorted. We have | x σ(3) | ≤ b σ(3) ≤ b 3 .
Further, | x σ(2) | ≤ b σ(2) and | x σ(2) | ≤ |x σ(3) | ≤ b σ(3) , one of which is at
most equal to b 2 . Similarly, | x σ(1) | ≤ b σ(1) , | x σ(1) | ≤ |x σ(2) | ≤ b σ(2) ,
and | x σ(1) | ≤ |x σ(3) | ≤ b σ(3) , the smallest of which is not larger than b 1 .
Finally, | x σ(0) | ≤ b σ(0) , | x σ(0) | ≤ |x σ(1) | ≤ b σ(1) , | x σ(0) | ≤ |x σ(2) | ≤ b σ(2) ,
and | x σ(0) | ≤ |x σ(3) | ≤ b σ(3) which shows | x σ(0) | ≤ b 0 .
As x 3 0+ . . . +x 3 3 is a symmetric form, the Leray measure on CX (1, ... ,1) (Ê) is invariant
under the canonical operation of S 4 on CX (1, ... ,1) (Ê) ⊂Ê4 . This means,
we have (i σ ) ∗ ω CX (1, ... ,1) (Ê)
Leray = ω CX (1, ... ,1) (Ê)
Leray for each σ ∈ S 4 .

324 ON THE SMALLEST POINT ON A DIAGONAL CUBIC SURFACE [Chap. X
Altogether,
Z
ω CX (1, ... ,1) (Ê)
Leray
R σ
≤
Z
i −1 σ (R id )
Z
ω CX (1, ... ,1) (Ê)
Leray =
(i σ ) ∗ ω CX (1, ... ,1) (Ê)
Leray =
R id
Z
R id
ω CX (1, ... ,1) (Ê)
Leray .
Third step. We have R R 0
ω CX (1, ... ,1) (Ê)
Leray ≤
3√ 1 3ω2 b
3 0 .
By virtue of Corollary 6.14, we have
Z
Leray = 1 3
R 0
ω CX (1, ... ,1) (Ê)
= 1 3
Z
1
x 2 3
R 0
ZZZ
π(R 0 )
dx 0 ∧ dx 1 ∧ dx 2
1
(x 3 0 + x3 1 + x3 2 )2/3 dx 0 dx 1 dx 2
where π : CX (1, ... ,1) (Ê) →Ê3 , (x 0 , x 1 , x 2 , x 3 ) ↦→ (x 0 , x 1 , x 2 ), denotes the projection
to the first three coordinates.
We enlarge the domain of integration to
R ′ := { (x 1 , x 2 , x 3 ) ∈Ê3 | | x 0 | 3 + | x 1 | 3 + | x 2 | 3 ≤ 3b 3 0 } .
Then, by homogeneity, we see
ZZZ
R ′ 1
(x 3 0 + x3 1 + x3 2 )2/3 dx 0 dx 1 dx 2 = ω 2 ·
Fourth step. We have R
Observe
R id
ω CX (1, ... ,1) (Ê)
3√
3b0
Z
0
1
r 2 · r2 dr = ω 2 · 3√
3b0 .
Leray ≤ ( 8 3 + 8 9 log 3)b 0 + 8 3 b 0 log b 1
| x 3 | = ∣
√x 3 3 0 + x3 1 + x3 2 ∣ ≤
√| 3 x 0 | 3 + | x 1 | 3 + | x 2 | 3 .
For (x 0 , . . . , x 3 ) ∈ R id , this implies | x 3 | ≤ 3 √
3 | x2 | and | x 2 | ≥ b 0 / 3 √
3. We find
Z
Leray = 1 3
R id
ω CX (1, ... ,1) (Ê)
Z
1
x 2 3
R id
≤ 1 Z
1
3 x 2 2
R id
< 1 3
≤ 1 3
Z b 0
dx 0 ∧ dx 1 ∧ dx 2
dx 0 ∧ dx 1 ∧ dx 2
Z
−b 0 |x 1 |∈[| x 0 |,b 1 ]
Z b 0 Z
−b 0 |x 1 |∈[|x 0 |,b 1 ]
Z
|x 2 |≥b 0 / 3√ 3
|x 2 |≥|x 1 |
1
x 2 2
dx 2 dx 1 dx 0
2
√
max{b 0 / 3 3, | x1 |} dx 1 dx 0
b 0
.

Sec. 6] THE FUNDAMENTAL FINITENESS PROPERTY 325
≤ 2 3
⎡
⎢
⎣
Z b 0
Z
−b 0 |x 1 |∈[|x 0 |,b 0 / 3√ 3]
3√
3
dx 1 dx 0 +
b 0
Z b 0
Z
−b 0 |x 1 |∈[b 0 / 3√ 3,b 1 ]
⎤
1
| x 1 | dx ⎥
1 dx 0 ⎦
≤ 2 3 · 4b2 0
3√
3 ·
3√
3
+ 2 Z b 0
2 log
b 0 3
−b 0
3√
3b1
b 0
dx 0
= 8 3 b 0 + 8 3 b 0 log
=
3√
3b1
b 0
( 8
3 + 8 9 log 3 )
b 0 + 8 3 b 0 log b 1
b 0
.
□
6.17. Corollary. –––– For every ε > 0, there exists a constant C such that
τ ∞
(
X
(a 0 , ... ,a 3 ) (Ê) ) ≤ C · | a 0 · . . . · a 3 | − 1 3 +ε · min
i=0, ... ,3 ‖a i‖ 1 3
∞
for each (a 0 , . . . , a 3 ) ∈ (\{0}) 4 .
Proof. We assume without restriction that | a 0 | ≤ . . . ≤ | a 3 |. Then,
Lemma 6.15 and Proposition 6.16 together show that, for certain explicit positive
constants C 1 and C 2 ,
(
τ ∞ X
(a 0 , ... ,a 3 ) (Ê) ) (
√ )
≤ | a 0 · . . . · a 3 | − 3 1 · C 1 | a 0 | 3 1 + C2 | a 0 | 3 1 log
3
| a 1 |
| a 0 |
= | a 0 · . . . · a 3 | − 1 3 · | a0 | 1 3
(
C 1 + 1 3 C 2 log | a 1|
| a 0 |
≤ | a 0 · . . . · a 3 | − 3 1 · min ‖a i‖ 1 3
∞
i=0, ... ,3
(
· C 1 + 1 )
3 C 2 log | a 0 · . . . · a 3 | .
There is a constant C such that C 1 + 1 C 3 2 log | a 0 · . . . · a 3 | ≤ C | a 0 · . . . · a 3 | ε
for every (a 0 , . . . , a 3 ) ∈ (\{0}) 4 .
□
)
iv. The Tamagawa number. —
6.18. Proposition. –––– For every ε > 0, there exists a constant C > 0 such that
1
τ ≥ C · H( ) 1
1 1
a 0
: . . . : 3
a 3 (a 0, ... ,a 3 )
|a 0 · . . . · a 3 | ε
for each (a 0 , . . . , a 3 ) ∈ (\{0}) 4 .

326 ON THE SMALLEST POINT ON A DIAGONAL CUBIC SURFACE [Chap. X
Proof. We may assume that ε is small, say ε < 2 . Then, immediately from the
3
definition of τ (a 0, ... ,a 3 ) , we have
τ (a 0, ... ,a 3 )
= α(X (a 0, ... ,a 3 ) )·β(X (a 0, ... ,a 3 ) ) · lim
s→1
(s − 1) t L ( s, χ Pic(X
(a 0 , ... ,a 3 )
≤ α(X (a 0, ... ,a 3 ) )·β(X (a 0, ... ,a 3 ) ) · lim
s→1
(s − 1) t L ( s, χ Pic(X
(a 0 , ... ,a 3 )
É
)
= α(X (a 0, ... ,a 3 ) )·β(X (a 0, ... ,a 3 ) ) · lim
s→1
(s − 1) t L ( s, χ Pic(X
(a 0 , ... ,a 3 )
)
)
É
(
·τ H X
(a 0 , ... ,a 3 ) (É) Br)
)
)
(
·τ H X
(a 0 , ... ,a 3 ) (É) )
)
)
·∏
ν∈Val(É)
τ ν
(
X
(a 0 , ... ,a 3 ) (Éν) ) .
Let us collect estimates for the factors. First, by Proposition 6.2, we have
lim
s→1
(s − 1) t L ( s, χ Pic(X
(a 0 , ... ,a 3 )
) < c1 · | a 0 · . . . · a 3 | ε 16
for a certain constant c 1 . Further, Proposition 6.13 yields
(
∏ τ p X
(a 0 , ... ,a 3 ) (Ép) ) ≤ c 2 · | a 0 · . . . · a 3 | 3 1 − ε
16 · ∏
p prime
Finally, Corollary 6.17 shows
p prime
min ‖a 3
i‖ 1 − ε 2
i=0, ... ,3
τ ∞
(
X
(a 0 , ... ,a 3 ) (Ê) ) ≤ C · | a 0 · . . . · a 3 | − 1 3 + ε 2 · min
i=0, ... ,3 ‖a i‖ 1 3
∞ .
p .
We assert that the three inequalities together imply the following estimate for
Peyre’s constant τ (a 0, ... ,a 3) = τ (X (a 0, ... ,a 3 ) ),
τ (a 0, ... ,a 3 ) ≤ C 3 · | a 0 · . . . · a 3 | ε 2 · ∏
p prime
min ‖a i‖ 1 3 p
i=0, ... ,3
[ ]
· min ‖a i‖ 1 −
ε
3
∞ · ∏ min ‖a 2
i‖ p .
i=0, ... ,3 i=0, ... ,3
p prime
Indeed, this is trivial in the case τ (a 0, ... ,a 3 )
= 0. Otherwise, X (a 0, ... ,a 3 ) has an
adelic point and we may estimate the factors α and β by constants as shown in
Section 2. By consequence,
1
τ (a 0,...,a 3 )
≥ 1 C 3
·
[
∏
p prime
] −
1
min ‖a 3
i‖ p ·
i=0,...,3
[
] −
1
min ‖a 3
i‖ ∞
i=0,...,3
[
| a 0 · . . . · a 3 | ε 2 · ∏ min ‖a ] −
ε
2
i‖ p
p prime i=0,...,3

Sec. 6] THE FUNDAMENTAL FINITENESS PROPERTY 327
= 1 C 3
·
= 1 C 3
·
∏
max
p prime i=0,...,3
∥ 1 ∥ 1 ∥∥ 3
a i p
[
| a 0 · . . . · a 3 | ε 2 · ∏
p prime
· max
i=0,...,3
∥ 1 ∥ 1 ∥∥ 3
a i
∞
max
i=0,...,3 a(p) i
] ε
2
H ( ) 1
1 1
a 0
: . . . : 3
a 3
[ ] ε .
| a 0 · . . . · a 3 | ε 2
2 · ∏ max
p prime i=0,...,3 a(p) i
It is obvious that max
i=0,...,3 a(p) i ≤ | a (p)
0
· . . . · a (p)
3 | and
This shows
∏ | a (p)
0
· . . . · a (p)
3 | = | a 0 · . . . · a 3 | .
p prime
1
τ (a 0,...,a 3 ) ≥ 1 C 3
·
H ( ) 1
1 1
a 0
: . . . : 3
a 3
| a 0 · . . . · a 3 | ε 2 · | a 0 · . . . · a 3 | ε 2
= 1 C 3
· H( 1
a 0
: . . . :
1
a 3
) 1
3
| a 0 · . . . · a 3 | ε . □
6.19. Lemma. –––– Let (a 0 : . . . : a 3 ) ∈ P 3 (É) be any point such that
a 0 ≠ 0, . . . , a 3
∈
≠ 0. Then,
H(a 0 : . . . : a 3 ) ≤ H( 1 1
a 0
: . . . :
a 3
) 3 .
Proof. First, observe that (a 0 : . . . : a 3 ) ↦→ ( )
1
1
a 0
: . . . :
a 3
is a welldefined
map. Hence, we may assume without restriction that a 0 , . . . , a 3
and gcd(a 0 , . . . , a 3 ) = 1. This yields H(a 0 : . . . : a 3 ) = max |a i|.
i=0,...,3
On the other hand, ( 1 1
a 0
: . . . :
a 3
) = (a 1 a 2 a 3 : . . . : a 0 a 1 a 2 ). Consequently,
H ( 1
a 0
: . . . :
1
a 3
) ≤ [ max
i=0,...,3 |a i|] 3 = H(a 0 : . . . : a 3 ) 3 .
From this, the asserted inequality emerges when the roles of a i and 1 a i
are interchanged.
□
6.20. Corollary. –––– Let a 0 , . . . , a 3 ∈such that gcd(a 0 , . . . , a 3 ) = 1. Then,
|a 0 · . . . · a 3 | ≤ H ( 1
a 0
: . . . :
1
a 3
) 12.
Proof. Observe that | a 0 · . . . · a 3 | ≤ max | a i| 4 = H(a 0 : . . . : a 3 ) 4 and apply
i=0,...,3
Lemma 6.19.
□

328 ON THE SMALLEST POINT ON A DIAGONAL CUBIC SURFACE [Chap. X
6.21. Theorem. –––– For each ε > 0, there exists a constant C (ε) > 0 such that,
for all (a 0 , . . . , a 3 ) ∈ (\{0}) 4 ,
1
τ ≥ C (ε) · H( ) 1
1 1
(a 0,...,a 3 ) a 0
: . . . : 3
−ε
a 3
.
Proof. We may assume that gcd(a 0 , . . . , a 3 ) = 1. Then, by Proposition 6.18,
1
τ ≥ C (ε) · H( ) 1
1 1
a 0
: . . . : 3
a 3
.
(a 0,...,a 3 )
| a 0 · . . . · a 3 | ε 12
(
Corollary 6.20 yields | a 0 · . . . · a 3 | ε 12 ≤ H 1
)
1 ε.
a 0
: . . . :
a 3
6.22. Corollary (Fundamental finiteness). —– For each T > 0, there are only
finitely many diagonal cubic surfaces X (a 0,...,a 3 ) : a 0 x 3 0 + . . . +a 3 x 3 3 = 0 in P 3Ésuch
that τ (a 0,...,a 3) > T .
Proof. This is an immediate consequence of the comparison to the naive height
established in Theorem 6.21.
□
□

CHAPTER XI
THE DIOPHANTINE EQUATION
x 4 + 2y 4 = z 4 + 4w 4∗
Hash, x. There is no definition for this word—nobody knows what hash is.
AMBROSE BIERCE: The Devil’s Dictionary (1906)
1. Introduction
1.1. –––– An algebraic curve C of genus g > 1 admits at most a finite number
ofÉ-rational points. On the other hand, for genus one curves, #C (É) may be
zero, finite non-zero, or infinite. For genus zero curves, one automatically has
#C (É) = ∞ as soon as C (É) ≠ ∅.
1.2. –––– In higher dimensions, there is a conjecture, due to S. Lang, stating
that if X is a variety of general type over a number field then all but finitely many
of its rational points are contained in the union of closed subvarieties which are
not of general type (cf. Conjecture VI.2.2). On the other hand, abelian varieties
(as well as, e.g., elliptic and bielliptic surfaces) behave like genus one curves.
I.e., #X (É) may be zero, finite non-zero, or infinite. Finally, rational and ruled
varieties comport in the same way as curves of genus zero in this respect.
This list does not yet exhaust the classification of algebraic surfaces, to say
nothing of dimension three or higher. In particular, the following problem is
still open.
1.3. Problem. –––– Does there exist a K3 surface X overÉwhich has a finite
non-zero number ofÉ-rational points? I.e., such that 0 < #X (É) < ∞?
1.4. Remark. –––– This question was posed by Sir P. Swinnerton-Dyer as
Problem/Question 6.a) in the problem session to the workshop [Poo/T]. We are
not able to give an answer to it.
(∗) This chapter collects material from the articles:
The Diophantine equation x 4 + 2y 4 = z 4 + 4w 4 , Math. Comp. 75(2006), 935–940 and
The Diophantine equation x 4 + 2y 4 = z 4 + 4w 4 — a number of improvements, Preprint,
both joint with A.-S. Elsenhans.

330 THE DIOPHANTINE EQUATION x 4 + 2y 4 = z 4 + 4w 4 [Chap. XI
1.5. –––– One possible candidate for a K3 surface with the property
0 < #X (É) < ∞ is given by the following
Problem. Find a third point on the projective surface X ⊂ P 3 defined by
x 4 + 2y 4 = z 4 + 4w 4 .
1.6. Remarks. –––– i) Problem 1.5 is also due to Sir P. Swinnerton-Dyer
[Poo/T, Problem/Question 6.c)]. It was raised, in particular, during his
talk [S-D04, very end of the article] at the Göttingen Mathematisches Institut
on June 2 nd , 2004.
ii) x 4 +2y 4 = z 4 +4w 4 is a homogeneous quartic equation. It, therefore, defines
a K3 surface X in P 3 . As trivial solutions of the equation, we consider those
corresponding to theÉ-rational points (1:0:1:0) and (1:0:(−1):0).
iii) Our main result is the following theorem which contains an answer to
Problem 1.5.
1.7. Theorem. –––– The diagonal quartic surface X in P 3 given by
x 4 + 2y 4 = z 4 + 4w 4
(∗)
admits precisely tenÉ-rational points having integral coordinates within the hypercube
|x|, |y|, |z|, |w| < 10 8 .
These are (±1:0:±1:0) and (±1 484 801:±1 203 120:±1169407:±1157520).
1.8. Remark. –––– This result does clearly not exclude the possibility that
#X (É) is actually finite. It might indicate, however, that a proof for this
property is deeper than one originally hoped for.
2. Congruences
2.1. –––– It seems natural to first try to understand the congruences
x 4 + 2y 4 ≡ z 4 + 4w 4 (mod p) (†)
modulo some prime number p. For p = 2 and 5, one finds that all primitive
solutions insatisfy
a) x and z are odd,
b) y and w are even,
c) y is divisible by 5.

Sec. 2] CONGRUENCES 331
For other primes, it follows from the Weil conjectures, proven by P. Deligne
[Del], that the number of solutions of the congruence (†) is
#C X (p) = 1 + (p − 1)(p 2 + p + 1 + E) = p 3 + E(p − 1) .
Here, E is an error-term which may be estimated by |E| ≤ 21p.
Indeed, consider the projective variety X overÉdefined by (∗). It has good reduction
at every prime p ≠ 2. Therefore, [Del, Théorème (8.1)] may be applied
to the reduction X p . This yields #X p (p) = p 2 + p + 1 + E and |E| ≤ 21p.
We note that dim H 2 (X ,Ê) = 22 for every complex surface X of type K3 [Bv,
p. 98].
2.2. –––– Another question of interest is to count the numbers of solutions
to the congruences x 4 + 2y 4 ≡ c (mod p) and z 4 + 4w 4 ≡ c (mod p) for a
certain c ∈.
This means to count thep-rational points on the affine plane curves C l c and C r c
defined overp by x 4 + 2y 4 = c and z 4 + 4w 4 = c, respectively. If p ∤ c
and p ≠ 2 then these are smooth curves of genus three.
By the work of André Weil [We48, Corollaire 3 du Théorème 13], the numbers
ofp-rational points on their projectivizations are given by
#C l c (p) = p + 1 + E l and #C r c (p) = p + 1 + E r ,
where the error-terms can be bounded by |E l |, |E r | ≤ 6 √ p. There may be up
to fourp-rational points on the infinite line. For our purposes, it suffices to
notice that both congruences admit a number of solutions which is close to p.
The case p|c, p ≠ 2, is slightly different since it corresponds to the case of a
reducible curve. The congruence x 4 + ky 4 ≡ 0 (mod p) admits only the trivial
solution if (−k) is not a biquadratic residue modulo p. Otherwise, it has exactly
1 + (p − 1) gcd(p − 1, 4) solutions.
Finally, if p = 2 then #C l 0 (2) = #C l 1 (2) = #C r 0 (2) = #C r 1 (2) = 2.
2.3. Remark. –––– The number of solutions of the congruence (†) is
#C X (p) = ∑ #Cc l (p) · #Cc r (p).
c∈p
Hence, the formulas just mentioned yield an elementary estimate for that count.
They show once more that the dominating term is p 3 . The estimate for the
error is, however, less sharp than the one obtained via the more sophisticated
methods in 2.1.

332 THE DIOPHANTINE EQUATION x 4 + 2y 4 = z 4 + 4w 4 [Chap. XI
3. Naive methods
3.1. –––– The most naive method to search for solutions of (∗) is probably
the following. Start with the set
{(x, y, z, w) ∈|0≤x, y, z, w ≤ N }
and test the equation for every quadruple.
Obviously this method requires about N 4 steps. It can be accelerated using the
congruence conditions for primitive solutions noticed above.
3.2. –––– A somewhat better method is to start with the set
{x 4 + 2y 4 − 4w 4 | x, y, w ∈, 0 ≤ x, y, w ≤ N }
and to search for fourth powers. This set has about N 3 Elements, and the
algorithm takes about N 3 steps. Again, it can be sped up by the above congruence
conditions for primitive solutions. We used this approach for a trial-run
with N = 10 4 .
An interesting aspect of this algorithm is the optimization by further congruences.
Suppose x and y are fixed, then about one half or three-quarter of the
values for w are no solutions to the congruence modulo a new prime. Following
this way, one can find more congruences for w and the size of the set may
be reduced by a constant factor.
4. An algorithm to efficiently search for solutions
i. The basic idea. —
4.1. –––– We need to compute the intersection of two sets
{x 4 + 2y 4 | x, y ∈, 0 ≤ x, y ≤ N } ∩ {z 4 + 4w 4 | z, w ∈, 0 ≤ z, w ≤ N } .
Both have about N 2 elements.
It is a standard problem in Computer Science to find the intersection of two
sets which both fit into memory. Using the congruence conditions modulo 2
and 5, one can reduce the size of the first set by a factor of 20 and the size of the
second set by a factor of 4.

Sec. 4] AN ALGORITHM TO EFFICIENTLY SEARCH FOR SOLUTIONS 333
ii. Some details. —
4.2. –––– The two sets described above are too big, at least for our computers
and interesting values of N . Therefore, we introduce a prime number p p which
we call the page prime.
Define the sets
and
L c := {x 4 + 2y 4 | x, y ∈, 0 ≤ x, y ≤ N , x 4 + 2y 4 ≡ c (mod p p )}
R c := {z 4 + 4w 4 | z, w ∈, 0 ≤ z, w ≤ N , z 4 + 4w 4 ≡ c (mod p p )} .
This means, the intersection problem is divided into p p pieces and the sets L c
and R c fit into the computer’s memory if p p is big enough. We worked with
N = 2.5 · 10 6 and chose p p = 30 011.
For every value of c, our program computes L c and stores this set in a hash table.
Then, it determines the elements of R c and looks them up in the table. Assuming
uniform hashing, the expected running-time of this algorithm is O(N 2 ).
4.3. Remark –––– An important further aspect of this approach is that the
problem may be attacked in parallel on several machines. The calculations
for one particular value of c are independent of the analogous calculations for
another one. Thus, it is possible, say, to let c run from 0 to (p p − 1)/2 on one
machine and, at the same time, from (p p + 1)/2 to (p p − 1) on another.
iii. Some more details. —
iii.1. The page prime. —
4.4. –––– For each value of c, it is necessary to find the solutions of the congruences
x 4 +2y 4 ≡ c (mod p p ) and z 4 +4w 4 ≡ c (mod p p ) in an efficient manner.
We do this in a rather naive way by letting y (w) run from 0 to p p − 1.
For each value of y (w), we compute x 4 (z 4 ). Then, we extract the fourth root
modulo p p .
Note that the page prime fulfills p p ≡ 3 (mod 4). Hence, the fourth roots of
unity modulo p are just ±1 and, therefore, a fourth root modulo p p , if it exists,
is unique up to sign. This makes the algorithm easier to implement.

334 THE DIOPHANTINE EQUATION x 4 + 2y 4 = z 4 + 4w 4 [Chap. XI
4.5. –––– Actually, we do not execute any modular powering operation or even
computation of fourth roots in the lion’s share of the running time. For more
efficiency, all fourth powers and all fourth roots modulo p p are computed and
stored in an array during an initialization step. Thus, the main speed limitation
to find all solutions to a congruence modulo p p is, in fact, the time it takes to
look up values stored in the machine’s main memory.
iii.2. Hashing. —
4.6. –––– We do not compute L c and R c directly, because this would require
the use of multi precision integers within the inner loop. Instead, we choose
two other primes, the hash prime p h and the control prime p c , which fit into
the 32-bit registers of our computers. All computations are done modulo p h
and p c .
More precisely, for each pair (x, y) considered, the expression
(
(x 4 + 2y 4 ) mod p h
)
defines its position in the hash table. In other words, we hash pairs (x, y)
whereas (x, y) ↦→ ( (x 4 + 2y 4 ) mod p h
)
plays the role of the hash function.
For each pair (x, y), we write two entries into the hash table, namely the value
of ( (x 4 + 2y 4 ) mod p c
)
and the value of y.
In the main computation, we worked with the numbers p h = 25 000 009 for the
hash prime and p c = 400 000 009 for the control prime.
4.7. –––– Note that, when working with a particular value of c, there are
around p p pairs ((x mod p p ), (y mod p p )) which fulfill the required congruence
Therefore, approximately
x 4 + 2y 4 ≡ c (mod p p ) .
p p · (N /2
p p
· N /10
p p
) =
N 2
20p p
values will be written into the table. For our choices,
N 2
20p p
≈ 10 412 849 which
means that the hash table will get approximately 41.7% filled.
Like for many other rules, there is an exception to this one. If c = 0 then approximately
1 + (p p − 1) gcd(p p − 1, 4) pairs ((x mod p p ), (y mod p p )) may
satisfy the congruence
x 4 + 2y 4 ≡ 0 (mod p p ) .
As p p ≡ 3 (mod 4) this is not more than 2p p − 1 and the hash table will be
filled not more than about 83.3%.

Sec. 5] GENERAL FORMULATION OF THE METHOD 335
4.8. –––– To resolve collisions within the hash table, we use an open addressing
method. We are not particularly afraid of clustering and choose linear probing.
We feel free to use open addressing as, thanks to the Weil conjectures, we
have a priori estimates available for the load factor.
4.9. –––– The program makes frequent use of fourth powers modulo p h and p c .
Again, we compute these data in the initialization part of our program and store
them in arrays, once and for all.
5. General formulation of the method
5.1. –––– The method described in the previous section is actually a systematic
method to search for solutions of a Diophantine equation. It works efficiently
when the equation is of the form
f (x 1 , . . . , x n ) = g(y 1 , . . . , y m ) .
We find all solutions which are contained within the (n + m)-dimensional cube
{(x 1 , . . . , x n , y 1 , . . . , y m ) ∈n+m | | x i |, |y i | ≤ B} .
The expected running-time of the algorithm is O(B max{n,m} ).
5.2. –––– The basic idea may be formulated as follows.
Algorithm H.
i) Evaluate f on all points of the n-dimensional cube
Store the values within a set L.
ii) Evaluate g on all points of the cube
{(x 1 , . . . , x n ) ∈n | |x i | ≤ B} .
{(y 1 , . . . , y m ) ∈m | |y i | ≤ B}
of dimension m. For each value start a search in order to find out whether
it occurs in L. When a coincidence is detected, reconstruct the corresponding
values of x 1 , . . . , x n and output the solution.
5.3. Remarks. –––– a) In fact, we are interested in the very particular Diophantine
equation
x 4 + 2y 4 = z 4 + 4w 4
which was suggested by Sir Peter Swinnerton-Dyer.

336 THE DIOPHANTINE EQUATION x 4 + 2y 4 = z 4 + 4w 4 [Chap. XI
b.i) In the form stated above, the main disadvantage of Algorithm H is that it
requires an enormous amount of memory. Actually, the set L is too big to be
stored in the main memory even of our biggest computers, already when the
value of B is only moderately large.
For that reason, we introduced the idea of paging. We choose a page prime p p and
work with the sets L r := {s ∈ L | s ≡ r (mod p p )} for r = 0, . . . , p p − 1, separately.
At the cost of some more time spent on initializations, this yields
a reduction of the memory space required by a factor of 1 p p
.
ii) The sets L r were implemented in the form of a hash table with open addressing.
iii) It is possible to achieve a further reduction of the running-time and the memory
required by making use of some obvious congruence conditions modulo 2
and 5.
5.4. –––– The goal of the remainder of this chapter is to describe an improved
implementation of Algorithm H which we used in order to find
all solutions of x 4 + 2y 4 = z 4 + 4w 4 contained within the hypercube
{(x, y, z, w) ∈4 | |x|, |y|, |z|, |w| ≤ 10 8 }.
6. Improvements I – More congruences
6.1. –––– The most obvious way to further reduce the size of the sets L r and
to increase the speed of Algorithm H is to find further congruence conditions
for solutions and evaluate f and g only on points satisfying these conditions.
As the equation, we are interested in, is homogeneous, it is sufficient to restrict
consideration to primitive solutions.
6.2. –––– It should be noticed, however, that this idea is subject to strict limitations.
If we were using the most naive O(B n+m )-algorithm then, for more or less
every l ∈Æ, the congruence f (x 1 , . . . , x n ) ≡ g(y 1 , . . . , y m ) (mod l) caused a
reduction of the number of (n + m)-tuples to be checked. For Algorithm H,
however, the situation is by far less fortunate.
One may gain something only if there are residue classes (r mod l) which are
represented by f , but not by g, or vice versa. Values, the residue class of which
is not represented by g, do not need to be stored into L r . Values, the residue
class of which is not represented by f , do not need to be searched for.
Unfortunately, if l is prime and not very small then the Weil conjectures ensure
that all residue classes modulo l are represented by both f and g. In this case, the
idea fails completely. The same is, however, not true for prime powers l = p k .
Hensel’s Lemma does not work when all partial derivatives ∂f
∂x i
(x 1 , . . . , x n ),

Sec. 6] IMPROVEMENTS I – MORE CONGRUENCES 337
respectively ∂g
∂y i
(y 1 , . . . , y m ), are divisible by p. This makes it possible that
certain residue classes (r mod p k ) are not representable although (r mod p) is.
i. The prime 5. Congruences modulo 625. —
6.3. –––– In the algorithm described in the previous chapter, we made use
of the fact that y is always divisible by 5. However, at this point, one can
do a lot better. When one takes into consideration that a 4 ≡ 1 (mod 5) for
every a ∈not divisible by 5, a systematic inspection shows that there are
actually two cases.
Either, 5|w. Then, 5∤x and 5∤z. Or, otherwise, 5|x. Then, 5∤z and 5∤w.
Note that, in the latter case, one indeed has z 4 + 4w 4 ≡ 1 + 4 ≡ 0 (mod 5).
6.4. –––– The Case 5|w. We call this case “N” and use the letter N at a
prominent position in the naming of the relevant files of source code. N stands
for “normal”. To consider this case as the ordinary one is justified by the fact
that all primitive solutions known actually belong to it. Note, however, that
we have no theoretical reason to believe that this case should in whatever sense
be better than the other one.
In case N, we rearrange the equation to f N (x, z) = g N (y, w) where
f N (x, z) := x 4 − z 4 and g N (y, w) := 4w 4 − 2y 4 .
As y andw are both divisible by 5, we get g N (y, w) = 4w 4 −2y 4 ≡ 0 (mod 625).
Consequently, f N (x, z) ≡ 0 (mod 625).
This yields an enormous reduction of the set L r . To see this, recall 5∤x and 5∤z.
That means, for x, there are precisely ϕ(625) possibilities in/625. Further,
for each such value, the congruence z 4 ≡ x 4 (mod 625) may not have
more than four solutions. All in all, there are 4 · ϕ(625) = 2 000 possible pairs
(x, z) ∈ (/625) 2.
Further, these pairs are very easy to find, computationally. The fourth
roots of unity modulo 625 are ±1 and ±182. For each x ∈/625∗ , put
z := (±x mod 625) and z := (±182x mod 625).
We store the values of f N into the set L r . Only 2 000 out of 625 2 values (0.512%)
need to be computed and stored. Then, each value of g N is looked up in L r .
Here, as y and w are both divisible by 5, only one value out of 25 (4%) needs to
be computed and searched for.
6.5. –––– The Case 5|x. We call this case “S” and use the letter S at a prominent
position in the naming of the relevant files of source code. S stands for
“Sonderfall” which means “exceptional case”. It is not known whether there
exists a solution belonging to case S.

338 THE DIOPHANTINE EQUATION x 4 + 2y 4 = z 4 + 4w 4 [Chap. XI
Here, we simply interchange both sides of the equation. Define
f S (z, w) := z 4 + 4w 4 and g S (x, y) := x 4 + 2y 4 .
As x and y are divisible by 5, we get x 4 + 2y 4 ≡ 0 (mod 625) and, therefore,
z 4 + 4w 4 ≡ 0 (mod 625).
Again, this congruence allows only 4 · ϕ(625) = 2 000 solutions
(z, w) ∈ (/625) 2
and these pairs are easily computable, too. The fourth roots of (−4)
in/625are ±181 and ±183. For each x ∈/625∗ , one has to consider
z := (±181x mod 625) and z := (±183x mod 625).
We store the values of f S into the set L r . Then, we search through L r for the
values of g S . As above, only 2 000 out of 625 2 values need to be computed
and stored and one value out of 25 needs to be computed and searched for.
ii. The prime 2. —
6.6. –––– Any primitive solution is of the form that x and z are odd while y
and w are even.
6.7. –––– In case S, there is no way to do better than that as both f S and g S
represent (r mod 2 k ) for k ≥ 4 if and only if r ≡ 1 (mod 16).
IncaseN, thesituationissomewhatbetter. g N (y, w) = 4w 4 −2y 4 isalwaysdivisible
by 32 while f N (x, z) = x 4 −z 4 ≡ 0 (mod 32), as may be seen by inspecting
the fourth roots of unity modulo 32, implies the condition x ≡ ±z (mod 8).
This may be used to halve the size of L r .
iii. The prime 3. —
6.8. –––– Looking for further congruence conditions, a primitive solution
must necessarily satisfy, we did not find any reason to distinguish more cases.
But there are a few more congruences which we used in order to reduce the size
of the sets L r .
To explain them, let us first note two theorems on binary quadratic forms.
They may both be easily deduced from [H/W, Theorems 246 and 247].
6.9. Theorem. –––– The quadratic formsq 1 (a, b) := a 2 +b 2 , q 2 (a, b) := a 2 −2b 2 ,
and q 3 (a, b) := a 2 + 2b 2 admit the property below.
Suppose n 0 := q i (a 0 , b 0 ) is divisible by a prime p which is not represented by q i .
Then, p|a 0 and p|b 0 .

Sec. 6] IMPROVEMENTS I – MORE CONGRUENCES 339
6.10. Theorem. –––– A prime number p is representedbyq 1 , q 2 , or q 3 , respectively,
if and only if (0 mod p) is represented in a non-trivial way. In particular,
i) p is represented by q 1 if and only if p = 2 or p ≡ 1 (mod 4).
ii) p is represented by q 2 if and only if p = 2 or ( 2
p) = 1. The latter means
p ≡ 1, 7 (mod 8).
iii) p is represented by q 3 if and only if p = 2 or ( −2
p
to p ≡ 1, 3 (mod 8).
6.11. Remark –––– There is the obvious asymptotic estimate
Further,
♯{q i (a, b) | a, b ∈, q i (a, b) ∈È, q i (a, b) ≤ n} ∼
) = 1. The latter is equivalent
n
2 logn .
n
♯{q i (a, b) | a, b ∈, |q i (a, b)| ≤ n} ∼ C i √
logn
where C 1 , C 2 , and C 3 are constants which can be expressed explicitly by Euler
products. (For q 1 , this is worked out in [Brü, Satz (1.8.2)]. For the other
forms, J. Brüdern’s argument works in the same way without essential changes.)
6.12. Congruences modulo 81. ––––
In case N,
g N (y, w) = (2w 2 ) 2 − 2(y 2 ) 2 = q 2 (2w 2 , y 2 )
where q 2 does not represent the prime 3. Therefore, if 3|g N (y, w) then 3|2w 2
and 3|y 2 which implies y and w are both divisible by 3. By consequence, if
3|g N (y, w) then, automatically, 81|g N (y, w).
If 3| f N (x, z) but 81∤ f N (x, z) then f N (x, z) does not need to be stored into L r .
Further, if 3|x and 3|z then f N (x, z) does not need to be stored, either, as it
cannot lead to a primitive solution. This reduces the size of the set L r by a
factor of 1 + 4 · 1
9 3 (1 − 1 ) = 131 ≈ 53.9%.
3 81 243
In case S, the situation is the other way round.
f S (z, w) = (z 2 ) 2 + (2w 2 ) 2 = q 1 (z 2 , 2w 2 )
and q 1 does not represent the prime 3. Therefore, if 3| f S (z, w) then 3|z 2 and
3|2w 2 which implies that z and w are both divisible by 3 and 81| f S (z, w).
We use this in order to reduce the time spent on reading. If 3|g S (x, y) but
81∤g S (x, y) or if 3|x and 3|y then g S (x, y) does not need to be searched for.
Although modular operations are not at all fast, the reduction of the number of
attempts to read by 53.9% is highly noticeable.

340 THE DIOPHANTINE EQUATION x 4 + 2y 4 = z 4 + 4w 4 [Chap. XI
iv. Some more hypothetical improvements. —
6.13. –––– i) In the argument for case N given above, p = 3 might be replaced
by any other prime p ≡ 3, 5 (mod 8).
In case S, the same argument as above works for every prime p ≡ 3 (mod 8).
For primes p ≡ 5 (mod 8), the strategy could be reversed. q 3 is a binary
quadratic form which represents (0 mod p) only in the trivial manner. Therefore,
if p|g S (x, y) then p|x and p|y. It is unnecessary to store f S (z, w) if p|z
and p|w or if p| f S (z, w) but p 4 ∤ f S (z, w).
i ′ ) Each argument mentioned may be extended to some primes p ≡ 1 (mod 8).
For example, in case N, what is actually needed is that 2 is not a fourth power
modulo p. This is true, e.g., for p = 17, 41, and 97, but not for p = 73 and 89.
ii) f N and f S do not represent the residue classes of 6, 7, 10, and 11 modulo 17.
g N and (−g S ) do not represent 1, 3, and 9 modulo 13. This could be used to
reduce the load for writing as well as reading.
6.14. Remarks. –––– a) We did not implement these improvements as it seems
the gains would be marginal or the cost of additional computations would
even dominate the effect. It is, however, foreseeable that these congruences will
eventually become valuable when the speed of the CPU’s available will continue
to grow faster than the speed of memory. Observe that alone the congruences
noticed in a) could reduce the amount of data to be stored into L to a size
asymptotically less than εB 2 for any ε > 0.
b) For every prime p different from 2, 5, 13, and 17, the quartic forms f N , g N ,
f S , and g S represent all residue classes modulo p. This means, ii) may not be
carried over to any further primes.
This can be seen as follows. Let b be equal to f N , f S , g N , or g S . (0 mod p) is
represented by b, trivially. Otherwise, b(x, y) = r defines an affine curve
C r of genus three with at most four points on the infinite line. The Weil
conjectures [We48, Corollaire 3 du Théorème 13] imply that [(p+1−6 √ p)−4]
is a lower bound for the number ofp-rational points on C r . This is a positive
number as soon as p ≥ 43. In this case, every residue class (r mod p) is
represented, at least, once.
For the remaining primes up to p = 41, an experiment shows that all residue
classes modulo p are represented by f N , f S , g N , as well as g S .

Sec. 7] IMPROVEMENTS II – ADAPTION TO OUR HARDWARE 341
7. Improvements II – Adaption to our hardware
i. A 64 bit based implementation of the algorithm. —
7.1. –––– We migrated the implementation of Algorithm H from a 32 bit
processor to a 64 bit processor. This means, the new hardware supports addition
and multiplication of 64 bit integers. Even more, every operation on (unsigned)
integers is automatically modulo 2 64 .
From this, various optimizations of the implementation described in Section 4
are almost compelling. The basic idea is that 64 bits should be enough to
define hash value and control value by selection of bits instead of using (notoriously
slow) modular operations. Hash value and control value are two integers
significantly less than 2 32 which should be independent on each other.
Note, however, that the congruence conditions modulo 2 imposed imply that
x 4 ≡ z 4 ≡ 1 (mod 16) and 2y 4 ≡ 4w 4 ≡ 0 (mod 16). This means, the four
least significant bits of f and g may not be used as they are always the same.
7.2. –––– The description of the algorithm below is based on case S, case N
being completely analogous.
Algorithm H64.
I. Initialization. Fix B := 10 8 . Initialize a hash table of 2 27 = 134 217 728
integers, each being 32 bit long. Fix the page prime p p := 200 003.
Further, define two functions, the hash function h and the control function c,
which map 64 bit integers to 27 bit integers and 31 bit integers, respectively, by
selecting certain bits. Do not use any of the bits twice to ensure h and c are
independent on each other and do not use the four least significant bits.
II) Loop. Let r run from 0 to p p − 1 and execute steps A. and B. for each r.
A. Writing. Build up the hash table, which is meant to encode the set L r ,
as follows.
a) Find all pairs (z, w) of non-negative integers less than or equal to B which
satisfy z 4 + 4w 4 ≡ r (mod p p ) and all the congruence-conditions for primitive
solutions, listed above. (Make systematic use of the Chinese remainder theorem.)
b) Execute steps i) and ii) below for each such pair.
i) Evaluate f S (z, w) := (z 4 + 4w 4 mod 2 64 ).
ii) Use the hash value h( f S (z, w)) and linear probing to find a free place in the
hash table and store the control value c( f S (z, w)) there.

342 THE DIOPHANTINE EQUATION x 4 + 2y 4 = z 4 + 4w 4 [Chap. XI
B. Reading. Search within the hash table, as follows.
a) Find all pairs (x, y) of non-negative integers less than or equal to B which satisfy
x 4 + 2y 4 ≡ r (mod p p ) and all the congruence conditions for primitive solutions,
listed above. (Make systematic use of the Chinese remainder theorem.)
b) Execute steps i) and ii) below for each such pair.
i) Evaluate g S (x, y) := (x 4 + 2y 4 mod 2 64 ) on all points found in step a).
ii) Search for the control value c(g S (x, y)) in the hash table, starting at the
hash value h(g S (x, y)) and using linear probing, until a free position is found.
Report all hits and the corresponding values of x and y.
7.3. Remarks (Some details of the implementation). —– i) The fourth powers
and fourth roots modulo p p are computed during the initialization part of the
program and stored into arrays because arithmetic modulo p p is slower than
memory access.
ii) The control value is limited to 31 bits as it is implemented as a signed integer.
We use the value (−1) as a marker for an unoccupied place in the hash table.
iii) In contrast to our previous programs, we do not precompute large tables of
fourth powers modulo 2 64 because an access to these tables is slower than the
execution of two multiplications in a row (at least on our computer).
iv) It is the impact of the congruences modulo 625, 8, and 81, described above,
that the set of pairs (y, w) [(x, y)] to be read is significantly bigger than the
set of pairs (x, z) [(z, w)] to be written. They differ actually by a factor of
625 2
· 243
2000·25 112
6252
· 2 ≈ 33.901 in case N and · 112
2 000·25 243
≈ 3.601 in case S.
As a consequence of this, only a small part of the running-time is spent on writing.
The lion’s share is spent on unsuccessful searches within L.
7.4. Remarks (Post-processing). —– i) Most of the hits found in the hash table
actually do not correspond to solutions of the Diophantine equation. Hits indicate
only a similarity of bit-patterns. Thus, for each pair of x and y reported, one
needs to check whether a suitable pair of z and w does exist. We do this by recomputing
z 4 +4w 4 for all z and w which fulfill the given congruence conditions
modulo p p and powers of the small primes.
Although this method is entirely primitive, only about 3% of the total runningtime
is actually spent on post-processing. One reason for this is that postprocessing
is not called very often, on average only once on about five pages.
For those pages, the writing part of the algorithm needs to be recapitulated.
This is, however, not time-critical as only a small part of the running-time is
spent on writing, anyway.
ii) An interesting alternative for post-processing would be to apply the theory of
binary quadratic forms. The obvious strategy is to factorize x 4 +2y 4 completely

Sec. 7] IMPROVEMENTS II – ADAPTION TO OUR HARDWARE 343
into prime powers and to deduce from the decomposition all pairs (a, b) such
that a 2 + b 2 = x 4 + 2y 4 . Then, one may check whether for one of them both
a and b are perfect squares.
2
7.5. Remark. –––– The migration to a more bit-based implementation led to
an increase of the speed of our programs by a factor of approximately 1.35.
ii. Adaption to the memory architecture of our computer. —
7.6. –––– The factor of 1.35 is less than what we actually hoped for. For that
reason, we made various tests in order to find out what the limiting bottleneck
of our program is. It turned out that the major slowdown is the access of the
processor to main memory.
Our programs are, in fact, doing only two things, integer arithmetic and memory
access. The integer execution units of modern processors are highly optimized
circuits and several of them work in parallel inside one processor. They
work a lot faster than main memory does. In order to reach a further improvement,
it will therefore be necessary to take the architecture of memory into
closer consideration.
7.7. The Situation. –––– Computer designers try to bridge the gap between the
fast processor and the slow memory by building a memory hierarchy which
consists of several cache levels.
The cache is a very small and fast memory inside the processor. The first
cache level, called L1 cache, of our processor consists of a data cache and an
instruction cache. Both are 64 kbyte in size. The cache manager stores the most
recently used data into the cache in order to make sure a second access to them
will be fast.
If the cache manager does not find necessary data within the L1 cache then the
processor is forced to wait. In order to deliver data, the cache management first
checks the L2 cache which is 1024 kbyte large. It consists of 16384 lines of
64 byte, each.
7.8. Our Program. –––– Our program fits into the instruction cache, completely.
Therefore, no problem should arise from this.
When we consider the data cache, however, the situation is entirely different.
The cache manager stores the 1024 most recently used memory lines, each being
64 byte long, within the L1 data cache.
This strategy is definitely good for many applications. It guarantees main
memory may be scanned at a high speed. On the other hand, for our application,
it fails completely. The reason is that access to our 500 Mbyte hash table is

344 THE DIOPHANTINE EQUATION x 4 + 2y 4 = z 4 + 4w 4 [Chap. XI
completely random. An access directly to the L1 cache happens in by far less
than 0.1% of the cases. In all other cases, the processor has to wait.
Even worse, it is clear that in most cases we do not even access the L2 cache.
This means, the cache manager needs to access main memory in order to transfer
the corresponding memory line of 64 byte into the L1 cache. After this, the
processor may use the data. In the case that there is no free line available
within the L1 cache, the cache manager must restore old data back to main
memory, first. This process takes us 60 nanoseconds, at least, which seems to
be short, but the processor could execute more than 100 integer instructions
during the same time.
The philosophy for further optimization must, therefore, be to adapt the programs
as much as possible to our hardware, first of all to the sizes of the L1 and
L2 caches.
7.9. Programmer’s position. –––– Unfortunately, the whole memory hierarchy
is invisible from the point of view of a higher programming language, such asC,
since such languages are designed for being machine-independent. Further, the
hardware executes the cache management in an automatic manner. This means,
even by programming in assembly, one cannot control the cache completely
although some new assembly instructions such as prefetch allow certain direct
manipulations.
7.10. A way out. –––– A practical way, nonetheless to gain some influence on
thememoryhierarchy, istorearrangethealgorithm inanapparentlynonsensical
manner, thereby making memory access less chaotic. One may then hope that
the automatic management of the cache, when confronted with the modified
algorithm, is able to react more properly. This should allow the program to
run faster.
iii. Our first trial. —
7.11. –––– Our first idea for this was to work with two arrays instead of one.
Algorithm M.
i) Store the values of f into an array and the values of g into a another one.
Write successively calculated values into successive positions. It is clear that this
part of the algorithm is not troublesome as it involves a linear memory access
which is perfectly supported by the memory management.
ii) Then, use Quicksort in order to sort both arrays. In addition to being
fast, Quicksort is known to have a good memory locality when large arrays
are sorted.

Sec. 7] IMPROVEMENTS II – ADAPTION TO OUR HARDWARE 345
iii) In a final step, search for matches by going linearly through both arrays as
in Mergesort.
7.12. Remark –––– Unfortunately, the idea behind Algorithm M is too simple
to give it any chance of being superior to the previous algorithms. However, it is
a worthwhile experiment. Indeed, our implementation of Algorithm M causes
at least 30 times more memory transfer compared with the previous programs
but, actually, it is only three times slower. This indicates that our approach
is reasonable.
iv. Hashing with partial presorting. —
7.13. –––– Our final algorithm is a combination of sorting and hashing. An important
aspect of it is that the sorting step has to be considerably faster than the
Quicksort algorithm. For that reason, we adopted some ideas from linear-time
sorting algorithms such as Radix sort or Bucket sort.
7.14. –––– The algorithm works as follows. Again, the description is based on
case S, case N being analogous.
Algorithm H64B.
I. Initialization. Fix B := 10 8 . Initialize a hash table H of 2 27 = 134 217 728
integers, each being 32 bit long. Fix the page prime p p := 200 003.
In addition, initialize 1024 auxiliary arrays A i each of which may contain
2 17 = 131 072 long (64 bit) integers.
Further, define two functions, the hash function h and the control function c,
which map 64 bit integers to 27 bit integers and 31 bit integers, respectively, by
selecting certain bits. Do not use any of the bits twice to ensure h and c are
independent on each other and do not use the four least significant bits.
Finally, let h (10) denote the function mapping 64 bit integers to integers
within [0, 1023] which is given by the ten most significant bits of h. In other
words, for every x, h (10) (x) is the same as h(x) shifted to the right by 17 bits.
II) Outer Loop. Let r run from 0 to p p − 1 and execute A. and B. for each r.
A. Writing. Build up the hash table, which is meant to encode the set L r ,
as follows.
a) Preparation. Find all pairs (z, w) of non-negative integers less than or equal
to B which satisfy z 4 + 4w 4 ≡ r (mod p p ) and all the congruence-conditions
for primitive solutions, listed above. (Make systematic use of the Chinese
remainder theorem.)
b) Inner Loop. Execute steps i) – iii) below for each such pair.

346 THE DIOPHANTINE EQUATION x 4 + 2y 4 = z 4 + 4w 4 [Chap. XI
i) Evaluate f S (z, w) := (z 4 + 4w 4 mod 2 64 ).
ii) Do not store f S (z, w) into the hash table, immediately. Put
first.
i := h (10) ( f S (z, w)),
iii) Add f S (z, w) to the auxiliary array A i . Maintain A i as an unordered list.
I.e., always write to the lowest unoccupied address.
If there is no space left in A i then output an error message and abort the algorithm.
c) Storing. Let i run from 0 to 1023. For each i let j run through the addresses
occupied in A i .
For fixed i and j, extract from the 64 bit integer A i [j] the 27 bit hash value
h(A i [j]) and the 31 bit control value c(A i [j]).
Usethehashvalueh(A i [j])andlinear probingtofindafreeplaceinthehashtable
and store the control value c(A i [j]) there.
d) Clearing up. Clear the auxiliary arrays A i for all i ∈ [0, 1023] to make them
available for reuse.
B. Reading. Search within the hash table, as follows.
a) Preparation. Find all pairs (x, y) of non-negative integers less than or equal
to B which satisfy x 4 + 2y 4 ≡ r (mod p p ) and all the congruence conditions
for primitive solutions, listed above. (Make systematic use of the Chinese
remainder theorem.)
b) Inner Loop. Execute steps i) – iii) below for each such pair.
i) Evaluate g S (x, y) := (x 4 + 2y 4 mod 2 64 ).
ii) Do not look up g S (x, y) in the hash table, immediately. Put
first.
i := h (10) (g S (x, y)),
iii) Add g S (x, y) to the auxiliary array A i . Maintain A i as an unordered list.
I.e., always write to the lowest unoccupied address.
If there is no space left in A i then call d[i] and add g S (x, y) to A i , afterwards.
c) Searching. Clearing all buffers. Let i run from 0 to 1023. For each i, call d[i].
When this is finished, go to the next iteration of the outer loop.

Sec. 7] IMPROVEMENTS II – ADAPTION TO OUR HARDWARE 347
Subroutine d[i]) Clearing a buffer. Let j run through the addresses occupied
in A i . For fixed j, search for the control value c(A i [j]) within the hash table H ,
starting at the hash value h(A i [j]) and using linear probing, until a free place
is found. Report all hits and the corresponding values of x and y.
Having done this, declare A i to be empty.
7.15. Remark –––– The auxiliary arrays A i play the role of a buffer. Thus, one
could say that we introduced some buffering into the management of the hash table
H . However, this description misses the point.
What is more important is that the values of f S to be stored into L r are partially
sorted according to the 10 most significant bits of h( f S (z, w)) by putting them
into the auxiliary arrays A i . When the hash table is then built up, the records
arrive almost in order. The same is true for reading.
What we actually did is, therefore, to introduce some partial presorting into the
management of the hash table.
7.16. Remark –––– It is our experience that each auxiliary array carries more
or less the same load. In particular, in step II.A.b.iii), when the buffers are filled
up for writing, a buffer overflow should never occur. For this reason, we feel
free to treat this possibility as a fatal error.
v. Running time. —
7.17. –––– Algorithm H64B uses about three times more memory than our
previous algorithms but our implementation runs almost three times as fast.
It was this factor which made it possible to attack the bound B = 10 8 in a
reasonable amount of time.
The final version of our programs took almost exactly 100 days of CPU time
on an AMD Opteron 248 processor. This time is composed almost equally
of 50 days for case N and 50 days for case S. The main computation was
executed in parallel on two machines in February and March, 2005.
7.18. Why is this algorithm faster? –––– To answer this question, one has to
look at the impact of the cache. For the old program, the cache memory was
mostly useless. For the new program, the situation is completely different.
When the auxiliary arrays are filled in step II.A.b.iii) and II.B.b.iii), access to
these arrays is linear. There are only 1024 of them which is exactly the number
of lines in the L1 cache. When an access does not hit into that innermost cache
then the corresponding memory line is moved to it and the next seven accesses
to the same auxiliary array are accesses to that line. Altogether, seven of eight
memory accesses hit into the L1 cache.

348 THE DIOPHANTINE EQUATION x 4 + 2y 4 = z 4 + 4w 4 [Chap. XI
When an auxiliary array is emptied in step II.A.c) or II.B.d[i]), the situation
is similar. There are a high number of accesses to a very short segment of the
hash table. This segment fits completely into the L2 cache. It has to be moved
into that cache, once. Then, it can be used many times. Again, access to the
auxiliary array is linear and a hit into the L1 cache occurs in seven of eight cases.
All in all, for Algorithm H64B, most memory accesses are hits into the cache.
This means, at the cost of some more data transfer altogether, we achieved that
main memory may be mostly used at the speed of the cache.
8. The solution found
8.1. –––– Ironically, the premature version of our algorithm as described in
Section 4 already solved Problem 1.5.
The improvements made it possible to shift the search bound to 10 8 which
was far beyond our original expectations. However, they did not produce any
new solution.
Thus, let us conclude this chapter by some comments on the simpler approach.
8.2. –––– Test versions of the program were written in Delphi.
The definitive version was written in C. It took about 130 hours of CPU time
on a 3.00 GHz Pentium 4 processor with 512 kbyte cache memory. The main
computation was executed in parallel on two machines during the very first
days of December, 2004.
8.3. –––– Instead of looking for solutions of x 4 + 2y 4 = z 4 + 4w 4 , the algorithm
searches, in fact, for solutions to the corresponding simultaneous congruences
modulo p p and p c which, in addition, fulfill that ( (x 4 + 2y 4 ) mod p h
)
and ( (z 4 + 4w 4 ) mod p h
)
are “almost equal”.
To this modified problem, we found approximately 3800 solutions such that
(y, w) ≠ (0, 0). These congruence solutions were checked by an exact computation
using O. Forster’s [Fo] Pascal-style multi-precision interpreter language
ARIBAS.
8.4. –––– Among the congruence solutions, exact equality occurred only once.
This solution is as follows.
==> 1484801**4 + 2 * 1203120**4.
-: 90509_10498_47564_80468_99201
==> 1169407**4 + 4 * 1157520**4.
-: 90509_10498_47564_80468_99201

CHAPTER XII
NEW SUMS OF THREE CUBES ∗
There are three cubes (levels) in total, each of which are connected by a
single door. Players have the goal of moving the character from room to
room, cube to cube in an attempt to find the final exit door of all three cubes.
If this door is found the character will appear to leave the cube, walk across
the table surface and vanish. The game then begins again.
JULIAN OLIVER: levelHead, A 3D Spatial Memory Game (2007)
1. Introduction
It is a long standing problem as to whether every rational integer
n ≢ 4, 5 (mod 9) can be written as a sum of three integral cubes. According
to the web page [Be-list] of Daniel Bernstein, the first attacks by computer
were carried out as early as 1955.
Nevertheless, for example for n = 3, there is still no solution known apart from
the obvious ones: (1, 1, 1), (4, 4, −5), (4, −5, 4), and (−5, 4, 4). For n = 30, the
first solution was found by N. Elkies and his coworkers in 2000 [El]. It is interesting
to note that, in 1992, D. R. Heath-Brown [H-B92a] had made a prediction
on the density of the solutions for n = 30 without knowing any solution explicitly.
Over the years, a number of algorithms have been developed in order to attack
the general problem. An excellent overview concerning the various approaches
invented up to around 2000 has been given in [B/P/T/Y], published recently.
The historically first algorithm which has a complexity of O(B 1+ε ) for a search
bound of B is the method of R. Heath-Brown [H/L/R]. The best algorithm
presently known is Elkies’ method described in [El].
(∗) This chapter is a revised version of the article: New sums of three cubes, to appear in:
Math. Comp., joint with A.-S. Elsenhans.

350 NEW SUMS OF THREE CUBES [Chap. XII
2. Elkies’ method
This
√
algorithm is geometric
√
in nature. The idea is to cover the curve
Y = 3 1 − X 3
, X ∈ [0, 1/ 3 2], by very small parallelograms which we call flagstones.
The algorithm finds all rational points of the particular form (x/z, y/z)
which are contained in one of the flagstones for z ∈Æup to a given bound B.
(This means we are searching for triples such that x 3 + y 3 − z 3 will be small.)
For each flagstone, this is equivalent to the detection of all points of the standard
lattice3 that are contained in a certain pyramid. The problem is that, viewed
in the standard coordinates, this pyramid has an enormous height in comparison
with the two other dimensions. Thus, it has an extremely sharp apex. Searching
naively for lattice points in such a pyramid would be highly inefficient.
The idea to overcome this difficulty is to work in coordinates more
adapted to this pyramid. The drawback in this case is that the basis
{ (1, 0, 0), (0, 1, 0), (0, 0, 1) } of the standard lattice is then far from being
reduced in whatever sense. One needs to apply lattice basis reduction. Having
done that, searching for lattice points within the pyramid is essentially
equivalent to a search for small points of the lattice. For this, one may use the
well-known algorithm of Fincke-Pohst [F/P].
The size of the flagstones is somewhat arbitrary. Smaller flagstones mean that
more time is required for lattice basis reductions. Larger ones lead to more time
being spent on the algorithm of Fincke-Pohst. The optimum size depends on
details of the implementation.
3. Implementation
Our implementation of Elkies’ method is written in C and C++. We took care
that only initialization parts of the code were written inC++ or made use of the
multi precision floats of GMP.
The time-critical parts were written in plain C making some use of the instruction
asm. It turned out that, for most of the computations, 128-bit fixed-point
arithmetic was sufficiently precise. We realized the 128-bit fixed-point numbers
as arrays consisting of two long ints. The arithmetic of the fixed-point
numbers was implemented in such a way that all loops (of length two) were
manually unrolled.
For lattice basis reduction, we implemented a version of LLL for three-dimensional
lattices. It turned out that adjacent flagstones led to similar reduced bases.
That is why enormous savings could be achieved by doing LLL incrementally.

Sec. 4] RESULTS 351
We start theLLL computation for the next flagstone with a reduced basis of the
previous one and not with a naive basis.
Another substantial improvement was realized in the Fincke-Pohst part.
Here, one has to compute many adjacent values of the same cubic polynomial
in three variables. An implementation of a difference scheme reduced most
of these computations to a few additions of values obtained before.
Some details. — We searched systematically for solutions of x 3 + y 3 + z 3 = n
where the positive integer n < 1000 is neither a cube nor twice a cube and
| x|, |y|, |z| < 10 14 . The length of the flagstones was chosen dynamically. It was
around 8.4 · 10 −12 near x = 0 and around 6.6 · 10 −14 near x = 1/ 3 √
2. The area
of the flagstones was essentially constant at a value near 1.7 · 10 −40 . This led to
a total number of a bit more than 10 13 flagstones to deal with.
We chose the widths of the flagstones such that all points in a horizontal
distance of < 10 −30 from the curve are contained in one of the flagstones.
This should make sure that all solutions of heights between 10 11 and 10 14 are
certainly found. Indeed, if we arrange variables such that | x| ≤ |y| ≤ |z|
then the point (| x/z|, |y/z|) is at a horizontal distance from the curve of,
√
in
ds
first order approximation, 1/3
|
ds s=(1−X 3 ) · n/|z| 3 . Since X := | x/z| < 1/ 3 2,
the derivative is always less than 0.53.
The whole search took around ten months of CPU time. Only 14% of that
time was spent on lattice basis reductions. The lion’s share was spent searching
for small lattice points. I.e., on our implementation of the algorithm of Fincke-
Pohst.
4. Results
In comparison with the computations of [B/P/T/Y] and the lists, dating back
to 2001 and published in [Be-list], 3519 new solutions have been found.
Among them, there are the following. For each of the nine numbers on the
left, no solution had been given before, neither in D. Bernstein’s lists, nor
in [B/P/T/Y].
156 = 26 577 110 807 569 3 − 18 161 093 358 005 3 − 23 381 515 025 762 3
318 = 1 970 320 861 387 3 + 1 750 553 226 136 3 − 2 352 152 467 181 3
= 30 828 727 881 037 3 + 27 378 037 791 169 3 − 36 796 384 363 814 3
366 = 241 832 223 257 3 + 167 734 571 306 3 − 266 193 616 507 3
420 = 8 859 060 149 051 3 − 2 680 209 928 162 3 − 8 776 520 527 687 3

352 NEW SUMS OF THREE CUBES [Chap. XII
564 = 53 872 419 107 3 − 1 300 749 634 3 − 53 872 166 335 3
758 = 662 325 744 409 3 + 109 962 567 936 3 − 663 334 553 003 3
= 83 471 297 139 078 3 + 77 308 024 343 011 3 − 101 433 242 878 565 3
789 = 18 918 117 957 926 3 + 4 836 228 687 485 3 − 19 022 888 796 058 3
894 = 19 868 127 639 556 3 + 2 322 626 411 251 3 − 19 878 702 430 997 3
948 = 103 458 528 103 519 3 + 6 604 706 697 037 3 − 103 467 499 687 004 3
For 13 values of n, for which exactly one solution was known, we found a
second one. Among those, there is n = 30. The second solution for n = 30 is
30 = 3 982 933 876 681 3 − 636 600 549 515 3 − 3 977 505 554 546 3 .
A second and a third solution for n = 75 are
75 = 2 576 191 140 760 3 + 1 217 343 443 218 3 − 2 663 786 047 493 3
= 59 897 299 698 355 3 − 47 258 398 396 091 3 − 47 819 328 945 509 3 .
On the other hand, the highest numbers of solutions found are 93 for n = 792
and 85 for n = 720.
This fits well with some speculations made in [H-B92a]. D. R. Heath-Brown
predicts ∼A(n) logB essentially different solutions of height

APPENDIX
Diophantus, with this name which is frequent in Greece, was a true Greek,
disciple of Greek science, if one who towers high above his contemporaries.
He was Greek in what he accomplished, as well as in what he was not able
to accomplish. But we must not forget that Greek science, as it conquered
the East from Alexandria . . . , brought new ideas back home from these
campaigns, that Greek mathematics as such has ceased to pick up whatever
it found worth picking up here and there.
MORITZ CANTOR (1907, translated by N. Schappacher)
1. A script in GAP
This is the script in GAP which was used to compute H 1( G, Pic(XÉ) )
and rkPic(X ) for all groups which may operate on the 27 lines upon a smooth
cubic surface. The mathematical background is explained in Section 8, Subsection
iv of Chapter II.
TABLE 1. A GAP script
# A script in GAP to compute H^1(G, Pic) = (NS \cap S_0) / NS_0 and the Picard rank for all
# Galois groups of a cubic surface.
# Here, S is the free abelian group on the 27 lines, S_0 \subset S the subgroup of all
# principal divisors, and N: S --> S the norm map.
# We compute the intersection matrix of the 27 lines.
# g must be a faithful representation of W(E_6) in S_27.
SchnittMatrixBerechnen := function (g)
local i, j, mat, len, tmp;
mat := NullMat(27, 27);
for i in [1..27] do
for j in [1..27] do
if (i = j) then
mat[i][j] := -1;
else
tmp := [i, j]; Sort(tmp);
len := Size(Orbit(g, tmp, OnSets));
# Operation of W(E_6) on unordered pairs of lines.
# A pair of skew lines has orbit length 27*8.
# A pair of intersecting lines has orbit length 27*5.
if (len = 27*5) then
mat[i][j] := 1;
fi;
fi;
od;
od;
return mat;

354 APPENDIX
end;
# Compute the 27x27-matrix representing the norm map N: S --> S.
NBerechnen := function (orb, u)
local N, akt, j, k, l;
# The matrix N.
N := NullMat(27, 27);
for j in [1..Size(orb)] do
akt := orb[j];
for k in [1..Size(akt)] do
for l in [1..Size(akt)] do
N[akt[k]][akt[l]] := Size(u) / Size(akt);
od;
od;
od;
return N;
end;
# Build up a matrix containing a minimal system of generators of NS in its columns.
IBerechnen := function (orb, u)
local my_I, j, k, akt;
# The matrix I.
my_I := NullMat(27, Size(orb));
for j in [1..Size(orb)] do
akt := orb[j];
for k in [1..Size(akt)] do
# The line akt[k] lies in orbit akt numbered j.
my_I[akt[k]][j] := Size(u) / Size(akt);
od;
od;
return my_I;
end;
# A separate routine in order to make a lattice base from the column vectors of a matrix.
NormMat := function (m)
return TransposedMat(HermiteNormalFormIntegerMat(BaseIntMat(TransposedMat(m))));
end;
# The discriminant of a lattice. The lattice needs not be maximal.
# The lattice base is supposed to be given in the column vectors of m.
# It is important that m is a lattice base. The function will return 0 for a linearly
# dependent system.
Disc := function (m)
return Determinant(TransposedMat(m) * m);
end;
# We compute H^1 of the Picard group and the arithmetic Picard rank using Manin’s formulas.
h1_pic := function (schnitt_matrix, u)
local orb, I_mat, AI, N, K, linke_seite, rechte_seite, disc_links,
disc_rechts, index, rang, mul, scal_links, schnitt;
# Everything depends only on the combinatorial structure of the orbits.
orb := Orbits(u, [1..27]);
I_mat := IBerechnen(orb, u); # I_mat represents NS.
AI := schnitt_matrix * I_mat;
N := NBerechnen(orb, u); # N represents the norm map.
# Normalization: We choose all integral vectors in the kernel.

APPENDIX 355
# Computation of NS \cap S_0.
K := NullspaceIntMat(TransposedMat(AI));
# K collects the relations among the generators of NS modulo S_0.
linke_seite := NormMat(I_mat * TransposedMat(K));
# linke_seite represents the lattice NS \cap S_0.
K := NullspaceIntMat(TransposedMat(schnitt_matrix));
# The rows of K are generators of S_0.
rechte_seite := NormMat(N * TransposedMat(K));
# The columns of N * TransposedMat(K) are generators of NS_0.
# The discriminats.
disc_links := Disc(linke_seite);
disc_rechts := Disc(rechte_seite);
index := RootInt(disc_rechts / disc_links);
rang := Size(orb) - Size(TransposedMat(rechte_seite));
# The order of H^1(G, Pic).
# The arithmetic Picard rank.
AppendTo("h1pic.txt"," #H^1 = ", index);
if index > 3 then
# We have to compute the primary decomposition.
# This code suffices since index may be only 1, 2, 3, 4, or 9.
if index = 4 then mul := 2; else mul := 3; fi;
scal_links := mul * linke_seite;
schnitt := TransposedMat(BaseIntersectionIntMats(TransposedMat(rechte_seite),
TransposedMat(scal_links)));
if Disc(schnitt) = Disc(scal_links) then
AppendTo("h1pic.txt"," [ ", mul, ", ", mul, " ]");
else
AppendTo("h1pic.txt"," More complicated than the direct sum of Z/", mul, "Z");
fi;
fi;
AppendTo("h1pic.txt", ", Rk(Pic) = ", rang, ",");
end;
# Adds the orbit structure of the lines to the list.
bahnstruktur := function (u)
local ol;
ol := ShallowCopy(OrbitLengths(u, [1..27]));
Sort(ol);
AppendTo("h1pic.txt", " Orbits ", ol);
end;
# Our group taken from the library.
we6 := TransitiveGroup(27, 1161);
schnitt_matrix := SchnittMatrixBerechnen(we6);
# The subgroups.
ugv := ConjugacyClassesSubgroups(we6);;
# We compute H^1(G, Pic) for all Galois groups.
for i in [1..Size(ugv)] do
u := Representative(ugv[i]);
AppendTo("h1pic.txt", i, " #U = ", Size(u), " ");
AppendTo("h1pic.txt", AbelianInvariants(u), ",");
if IsTransitive(u) and (Size(u) > 1) then
AppendTo("h1pic.txt", " transitive");
else
h1_pic(schnitt_matrix, u);
bahnstruktur(u);
fi;
AppendTo("h1pic.txt", "\n");
od;

356 APPENDIX
2. The list
This is the list produced by theGAP script. For each subgroup of W (E 6 ), we give
its abelian quotient, the corresponding values of H 1( G, Pic(XÉ) ) and rkPic(X ),
and the orbit structure of the 27 lines.
We make use of this list in several ways. For our presentation of the Brauer-
Manin obstruction in the case of diagonal cubic surfaces given in Section III.6,
the list is of fundamental importance. On the other hand, we also use it in the
arguments given in III.5.33 and Remarks III.5.37.
It is further applied in Examples VI.5.7 and VI.5.8. Here, the goal is simply to
show that one has Picard rank 2 in the situations considered.
Finally, we give reference to the list in Remarks VI.5.13.ii) and X.2.4 in order
to prove that the factor β is bounded from above by 9.
TABLE 2. H 1 (G, Pic) and rk Pic(X ) for smooth cubic surfaces
1 #U = 1 [ ], #H^1 = 1, Rk(Pic) = 7, Orbits [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ]
2 #U = 2 [ 2 ], #H^1 = 1, Rk(Pic) = 6, Orbits [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 2, 2, 2, 2, 2, 2 ]
3 #U = 2 [ 2 ], #H^1 = 4 [ 2, 2 ], Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2, 2 ]
4 #U = 2 [ 2 ], #H^1 = 1, Rk(Pic) = 5, Orbits [ 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2,
2, 2, 2 ]
5 #U = 2 [ 2 ], #H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2,
2, 2 ]
6 #U = 3 [ 3 ], #H^1 = 9 [ 3, 3 ], Rk(Pic) = 1, Orbits [ 3, 3, 3, 3, 3, 3, 3, 3, 3 ]
7 #U = 3 [ 3 ], #H^1 = 1, Rk(Pic) = 5, Orbits [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3,
3, 3 ]
8 #U = 3 [ 3 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 3, 3, 3, 3, 3, 3, 3, 3, 3 ]
9 #U = 4 [ 4 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 4, 4, 4, 4, 4, 4 ]
10 #U = 4 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 5, Orbits [ 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2,
2, 2, 4 ]
11 #U = 4 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 1, 1, 2, 2, 2, 4, 4, 4, 4 ]
12 #U = 4 [ 2, 2 ], #H^1 = 4 [ 2, 2 ], Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 2, 2, 4, 4, 4, 4 ]
13 #U = 4 [ 4 ], #H^1 = 4 [ 2, 2 ], Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 4, 4, 4, 4 ]
14 #U = 4 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 2, 2, 4, 4, 4, 4 ]
15 #U = 4 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 2, 2, 2, 2, 2, 2, 4, 4, 4 ]
16 #U = 4 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 2, 2, 2, 2, 4, 4, 4 ]
17 #U = 4 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 4, 4, 4, 4, 4 ]
18 #U = 4 [ 4 ], #H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 1, 1, 2, 4, 4, 4, 4, 4 ]
19 #U = 4 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 2, 2, 4, 4, 4, 4 ]
20 #U = 4 [ 4 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 4, 4, 4, 4, 4 ]
21 #U = 4 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 4, 4, 4, 4 ]
22 #U = 4 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4 ]
23 #U = 5 [ 5 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 5, 5, 5, 5, 5 ]
24 #U = 6 [ 2 ], #H^1 = 1, Rk(Pic) = 5, Orbits [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3,
3, 3 ]
25 #U = 6 [ 2 ], #H^1 = 4 [ 2, 2 ], Rk(Pic) = 1, Orbits [ 3, 3, 3, 6, 6, 6 ]
26 #U = 6 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 6, 6, 6 ]
27 #U = 6 [ 2, 3 ], #H^1 = 4 [ 2, 2 ], Rk(Pic) = 1, Orbits [ 3, 6, 6, 6, 6 ]
28 #U = 6 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 6 ]
29 #U = 6 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 3, 6, 6, 6 ]
30 #U = 6 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 2, 6, 6, 6 ]
31 #U = 6 [ 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 2, 6, 6, 6 ]
32 #U = 6 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 3, 3, 3, 6, 6 ]

APPENDIX 357
33 #U = 6 [ 2 ], #H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 6 ]
34 #U = 6 [ 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 3, 3, 3, 3, 3, 3, 3, 6 ]
35 #U = 6 [ 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 2, 3, 3, 6, 6 ]
36 #U = 6 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 2, 3, 3, 6, 6 ]
37 #U = 6 [ 2 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 3, 3, 3, 6, 6, 6 ]
38 #U = 6 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 3, 6, 6, 6 ]
39 #U = 6 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 6, 6, 6, 6 ]
40 #U = 8 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 8, 8, 8 ]
41 #U = 8 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 2, 2, 8, 8 ]
42 #U = 8 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 3, Orbits [ 1, 1, 1, 4, 4, 4, 4, 4, 4 ]
43 #U = 8 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 2, 2, 8, 8 ]
44 #U = 8 [ 2, 2 ], #H^1 = 4 [ 2, 2 ], Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 4, 4, 4, 4 ]
45 #U = 8 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 4, 4, 8, 8 ]
46 #U = 8 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 2, 2, 2, 2, 2, 2, 4, 4, 4 ]
47 #U = 8 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 2, 2, 4, 4, 8 ]
48 #U = 8 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 3, Orbits [ 1, 1, 1, 4, 4, 4, 4, 4, 4 ]
49 #U = 8 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 1, 1, 2, 4, 4, 4, 4, 4 ]
50 #U = 8 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 8, 8 ]
51 #U = 8 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 8, 8 ]
52 #U = 8 [ 2, 4 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 8, 8 ]
53 #U = 8 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 4, 4, 4, 4, 4 ]
54 #U = 8 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 4, 4, 4, 4, 8 ]
55 #U = 8 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 8, 8 ]
56 #U = 8 [ 2, 4 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 8, 8 ]
57 #U = 8 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 4, 8, 8 ]
58 #U = 8 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 4, 4, 4, 8 ]
59 #U = 8 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 4, 4, 4, 8 ]
60 #U = 8 [ 2, 4 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 4, 4, 8 ]
61 #U = 8 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 4, 4, 4, 4, 4 ]
62 #U = 8 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 8, 8 ]
63 #U = 8 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 4, 8, 8 ]
64 #U = 8 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 4, 4, 8 ]
65 #U = 8 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 4, 4, 8 ]
66 #U = 8 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 8, 8 ]
67 #U = 8 [ 8 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 8, 8 ]
68 #U = 8 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 4, 4, 4, 8 ]
69 #U = 8 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 4, 4, 8 ]
70 #U = 9 [ 3, 3 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 3, 3, 3, 3, 3, 3, 9 ]
71 #U = 9 [ 3, 3 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 9, 9, 9 ]
72 #U = 9 [ 3, 3 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 3, 3, 3, 9, 9 ]
73 #U = 9 [ 9 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 9, 9 ]
74 #U = 10 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 2, 5, 5, 5, 10 ]
75 #U = 10 [ 2, 5 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 2, 5, 5, 5, 10 ]
76 #U = 10 [ 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 5, 5, 5, 5, 5 ]
77 #U = 12 [ 3 ], #H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 1, 1, 4, 4, 4, 4, 6 ]
78 #U = 12 [ 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 6, 6, 12 ]
79 #U = 12 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 2, 2, 2, 3, 3, 3, 3, 6 ]
80 #U = 12 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 2, 6, 6, 6 ]
81 #U = 12 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 3, 3, 6, 12 ]
82 #U = 12 [ 2, 2 ], #H^1 = 4 [ 2, 2 ], Rk(Pic) = 1, Orbits [ 3, 6, 6, 6, 6 ]
83 #U = 12 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 4, 6, 12 ]
84 #U = 12 [ 3, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 4, 6, 12 ]
85 #U = 12 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 4, 6, 12 ]
86 #U = 12 [ 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 4, 6, 12 ]
87 #U = 12 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 3, 3, 4, 6, 6 ]
88 #U = 12 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 3, 3, 4, 6, 6 ]
89 #U = 12 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 2, 3, 3, 6, 6 ]
90 #U = 12 [ 3, 4 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 12, 12 ]
91 #U = 12 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 6, 6, 12 ]
92 #U = 12 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 6, 12 ]
93 #U = 12 [ 2, 2, 3 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 6, 6, 12 ]
94 #U = 12 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 3, 6, 6, 6 ]
95 #U = 12 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 6, 6, 12 ]
96 #U = 12 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 4, 6, 12 ]
97 #U = 12 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 3, 3, 4, 6, 6 ]

358 APPENDIX
98 #U = 12 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 6, 6, 6, 6 ]
99 #U = 16 [ 2, 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 2, 2, 16 ]
100 #U = 16 [ 2, 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 3, Orbits [ 1, 1, 1, 4, 4, 4, 4, 4, 4 ]
101 #U = 16 [ 2, 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 16 ]
102 #U = 16 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 8, 8, 8 ]
103 #U = 16 [ 2, 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 8, 8 ]
104 #U = 16 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 4, 4, 8, 8 ]
105 #U = 16 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ]
106 #U = 16 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 4, 4, 8, 8 ]
107 #U = 16 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ]
108 #U = 16 [ 4, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ]
109 #U = 16 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 8, 8 ]
110 #U = 16 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 2, 4, 8, 8 ]
111 #U = 16 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ]
112 #U = 16 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 8, 8 ]
113 #U = 16 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ]
114 #U = 16 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 8, 8 ]
115 #U = 16 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ]
116 #U = 16 [ 2, 2, 4 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 8, 8 ]
117 #U = 16 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 16 ]
118 #U = 16 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ]
119 #U = 16 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 8, 8 ]
120 #U = 16 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 16 ]
121 #U = 16 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 4, 4, 8, 8 ]
122 #U = 16 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ]
123 #U = 16 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 4, 4, 8 ]
124 #U = 16 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 8, 8 ]
125 #U = 16 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 2, 4, 4, 4, 8 ]
126 #U = 16 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 8, 8 ]
127 #U = 16 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 8, 8 ]
128 #U = 18 [ 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 3, 3, 3, 3, 3, 3, 9 ]
129 #U = 18 [ 2, 3 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 3, 3, 3, 9, 9 ]
130 #U = 18 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 3, 3, 3, 3, 3, 3, 9 ]
131 #U = 18 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 3, 3, 18 ]
132 #U = 18 [ 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 3, 3, 18 ]
133 #U = 18 [ 2, 3 ], #H^1 = 4 [ 2, 2 ], Rk(Pic) = 1, Orbits [ 6, 6, 6, 9 ]
134 #U = 18 [ 2, 3, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 6, 6, 9 ]
135 #U = 18 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 6, 6, 9 ]
136 #U = 18 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ]
137 #U = 18 [ 2 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 9, 9, 9 ]
138 #U = 18 [ 2 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 3, 6, 9, 9 ]
139 #U = 18 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 18 ]
140 #U = 18 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 6, 6, 9 ]
141 #U = 18 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 6, 6, 6, 9 ]
142 #U = 18 [ 2, 3, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 18 ]
143 #U = 18 [ 2, 3 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 3, 6, 9, 9 ]
144 #U = 18 [ 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 9, 9 ]
145 #U = 18 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ]
146 #U = 20 [ 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 2, 5, 10, 10 ]
147 #U = 20 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 2, 5, 5, 5, 10 ]
148 #U = 20 [ 4 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 5, 5, 5, 10 ]
149 #U = 24 [ 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
150 #U = 24 [ 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 6, 8, 8 ]
151 #U = 24 [ 2, 3 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 6, 8, 8 ]
152 #U = 24 [ 2 ], #H^1 = 1, Rk(Pic) = 4, Orbits [ 1, 1, 1, 1, 1, 4, 4, 4, 4, 6 ]
153 #U = 24 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
154 #U = 24 [ 3 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 8, 8, 8 ]
155 #U = 24 [ 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
156 #U = 24 [ 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 4, 4, 6, 8 ]
157 #U = 24 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 6, 6, 12 ]
158 #U = 24 [ 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ]
159 #U = 24 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 6, 6, 12 ]
160 #U = 24 [ 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 6, 8, 8 ]
161 #U = 24 [ 2, 3 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ]
162 #U = 24 [ 2, 3 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ]

APPENDIX 359
163 #U = 24 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 6, 6, 12 ]
164 #U = 24 [ 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ]
165 #U = 24 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 4, 4, 6, 8 ]
166 #U = 24 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 6, 8, 8 ]
167 #U = 24 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 4, 6, 12 ]
168 #U = 24 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 3, 3, 4, 6, 6 ]
169 #U = 24 [ 8 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
170 #U = 24 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 4, 6, 12 ]
171 #U = 24 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 4, 6, 12 ]
172 #U = 24 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 12, 12 ]
173 #U = 24 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 4, 6, 12 ]
174 #U = 24 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 4, 6, 12 ]
175 #U = 24 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 4, 6, 12 ]
176 #U = 24 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 6, 6, 12 ]
177 #U = 27 [ 3, 3 ], transitive
178 #U = 27 [ 3, 3, 3 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 9, 9, 9 ]
179 #U = 27 [ 3, 3 ], transitive
180 #U = 32 [ 2, 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 8, 8, 8 ]
181 #U = 32 [ 2, 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 2, 4, 16 ]
182 #U = 32 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 8, 8, 8 ]
183 #U = 32 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ]
184 #U = 32 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ]
185 #U = 32 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ]
186 #U = 32 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ]
187 #U = 32 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 4, 4, 8, 8 ]
188 #U = 32 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ]
189 #U = 32 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ]
190 #U = 32 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ]
191 #U = 32 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ]
192 #U = 32 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ]
193 #U = 32 [ 2, 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 8, 8 ]
194 #U = 32 [ 2, 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ]
195 #U = 36 [ 4 ], #H^1 = 4 [ 2, 2 ], Rk(Pic) = 1, Orbits [ 6, 6, 6, 9 ]
196 #U = 36 [ 2, 2 ], #H^1 = 4 [ 2, 2 ], Rk(Pic) = 1, Orbits [ 6, 6, 6, 9 ]
197 #U = 36 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 3, 3, 3, 3, 3, 3, 9 ]
198 #U = 36 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 3, 3, 18 ]
199 #U = 36 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 6, 6, 9 ]
200 #U = 36 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 6, 6, 9 ]
201 #U = 36 [ 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 6, 6, 6, 9 ]
202 #U = 36 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 6, 6, 6, 9 ]
203 #U = 36 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 6, 9, 12 ]
204 #U = 36 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 18 ]
205 #U = 36 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 18 ]
206 #U = 36 [ 2, 2, 3 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 6, 9, 12 ]
207 #U = 36 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 18 ]
208 #U = 36 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 6, 6, 9 ]
209 #U = 36 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 18 ]
210 #U = 36 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 18 ]
211 #U = 36 [ 2, 2 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 3, 6, 9, 9 ]
212 #U = 36 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 6, 6, 9 ]
213 #U = 36 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ]
214 #U = 40 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 2, 5, 10, 10 ]
215 #U = 48 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 6, 16 ]
216 #U = 48 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
217 #U = 48 [ 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 6, 16 ]
218 #U = 48 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 2, 2, 6, 8, 8 ]
219 #U = 48 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 6, 16 ]
220 #U = 48 [ 3, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 6, 16 ]
221 #U = 48 [ 2, 2, 3 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 4, 6, 8, 8 ]
222 #U = 48 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 4, 6, 8, 8 ]
223 #U = 48 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ]
224 #U = 48 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 2, 2, 4, 4, 6, 8 ]
225 #U = 48 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 2, 6, 8, 8 ]
226 #U = 48 [ 2, 2, 3 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ]
227 #U = 48 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ]

360 APPENDIX
228 #U = 48 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ]
229 #U = 48 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 6, 6, 12 ]
230 #U = 48 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 6, 16 ]
231 #U = 48 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 4, 6, 8, 8 ]
232 #U = 48 [ 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
233 #U = 48 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ]
234 #U = 48 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ]
235 #U = 48 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
236 #U = 48 [ 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
237 #U = 48 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ]
238 #U = 48 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 4, 6, 12 ]
239 #U = 54 [ 2 ], transitive
240 #U = 54 [ 2 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 9, 9, 9 ]
241 #U = 54 [ 2, 3 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 9, 9, 9 ]
242 #U = 54 [ 2, 3, 3 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 9, 9, 9 ]
243 #U = 54 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ]
244 #U = 54 [ 2, 3, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ]
245 #U = 54 [ 2, 3 ], transitive
246 #U = 54 [ 2, 3 ], transitive
247 #U = 60 [ ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 6, 6, 15 ]
248 #U = 60 [ ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 5, 5, 5, 10 ]
249 #U = 64 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 8, 8, 8 ]
250 #U = 64 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ]
251 #U = 64 [ 2, 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 4, 4, 16 ]
252 #U = 64 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ]
253 #U = 64 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ]
254 #U = 64 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ]
255 #U = 64 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ]
256 #U = 72 [ 2, 2 ], #H^1 = 4 [ 2, 2 ], Rk(Pic) = 1, Orbits [ 6, 6, 6, 9 ]
257 #U = 72 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 6, 9, 12 ]
258 #U = 72 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 6, 9, 12 ]
259 #U = 72 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 6, 6, 6, 9 ]
260 #U = 72 [ 3, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
261 #U = 72 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 6, 9, 12 ]
262 #U = 72 [ 2, 4 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 6, 9, 12 ]
263 #U = 72 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 3, 3, 6, 6, 9 ]
264 #U = 72 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 6, 18 ]
265 #U = 80 [ 5 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 10, 16 ]
266 #U = 81 [ 3, 3 ], transitive
267 #U = 96 [ 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
268 #U = 96 [ 3 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 8, 8, 8 ]
269 #U = 96 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
270 #U = 96 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 2, Orbits [ 1, 4, 6, 8, 8 ]
271 #U = 96 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 2, 6, 16 ]
272 #U = 96 [ 4 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
273 #U = 96 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
274 #U = 96 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 6, 16 ]
275 #U = 96 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 6, 16 ]
276 #U = 96 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 6, 16 ]
277 #U = 96 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 6, 16 ]
278 #U = 96 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 6, 16 ]
279 #U = 96 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 3, 12, 12 ]
280 #U = 108 [ 2, 2 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 9, 9, 9 ]
281 #U = 108 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ]
282 #U = 108 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ]
283 #U = 108 [ 2, 2, 3 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 9, 9, 9 ]
284 #U = 108 [ 2, 2 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 9, 9, 9 ]
285 #U = 108 [ 4 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ]
286 #U = 108 [ 4 ], transitive
287 #U = 108 [ 3, 4 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ]
288 #U = 108 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ]
289 #U = 108 [ 2, 2 ], transitive
290 #U = 108 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ]
291 #U = 120 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 6, 6, 15 ]
292 #U = 120 [ 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 12, 15 ]

APPENDIX 361
293 #U = 120 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 2, 5, 10, 10 ]
294 #U = 120 [ 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 5, 5, 5, 10 ]
295 #U = 120 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 2, 5, 10, 10 ]
296 #U = 120 [ 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 12, 15 ]
297 #U = 128 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ]
298 #U = 144 [ 2, 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 6, 9, 12 ]
299 #U = 144 [ 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
300 #U = 160 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 10, 16 ]
301 #U = 162 [ 2, 3 ], transitive
302 #U = 162 [ 2, 3 ], transitive
303 #U = 162 [ 2 ], transitive
304 #U = 192 [ 2 ], #H^1 = 1, Rk(Pic) = 3, Orbits [ 1, 1, 1, 8, 8, 8 ]
305 #U = 192 [ 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
306 #U = 192 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ]
307 #U = 192 [ 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
308 #U = 192 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ]
309 #U = 192 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
310 #U = 192 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
311 #U = 192 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 4, 6, 16 ]
312 #U = 216 [ 2, 2, 2 ], #H^1 = 3, Rk(Pic) = 1, Orbits [ 9, 9, 9 ]
313 #U = 216 [ 2, 2 ], transitive
314 #U = 216 [ 2, 4 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ]
315 #U = 216 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ]
316 #U = 216 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ]
317 #U = 216 [ 2, 2 ], transitive
318 #U = 216 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ]
319 #U = 216 [ 8 ], transitive
320 #U = 216 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ]
321 #U = 216 [ 2, 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ]
322 #U = 240 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 2, 5, 10, 10 ]
323 #U = 240 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 12, 15 ]
324 #U = 288 [ 3, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
325 #U = 320 [ 4 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 10, 16 ]
326 #U = 324 [ 3 ], transitive
327 #U = 324 [ 2, 2 ], transitive
328 #U = 360 [ ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 6, 6, 15 ]
329 #U = 384 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
330 #U = 384 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 2, 8, 16 ]
331 #U = 432 [ 2, 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 9, 18 ]
332 #U = 432 [ 2, 2 ], transitive
333 #U = 576 [ 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
334 #U = 576 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
335 #U = 576 [ 2, 3 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
336 #U = 648 [ 2 ], transitive
337 #U = 648 [ 2 ], transitive
338 #U = 648 [ 2, 3 ], transitive
339 #U = 648 [ 3 ], transitive
340 #U = 720 [ 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 12, 15 ]
341 #U = 720 [ 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 12, 15 ]
342 #U = 720 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 6, 6, 15 ]
343 #U = 960 [ ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 10, 16 ]
344 #U = 1152 [ 2, 2 ], #H^1 = 1, Rk(Pic) = 1, Orbits [ 3, 24 ]
345 #U = 1296 [ 2 ], transitive
346 #U = 1296 [ 2, 2 ], transitive
347 #U = 1440 [ 2, 2 ], #H^1 = 2, Rk(Pic) = 1, Orbits [ 12, 15 ]
348 #U = 1920 [ 2 ], #H^1 = 1, Rk(Pic) = 2, Orbits [ 1, 10, 16 ]
349 #U = 25920 [ ], transitive
350 #U = 51840 [ 2 ], transitive

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INDEX
A
Abbes, A. . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
abelian surface . . . . . . . . . . . . . . . . . . 218, 255
abelian variety . . . . . . . . . . . . . . . . . . 218, 329
accumulating subvariety xxi, 212, 254f, 258
accumulating surface. . . . . . . . . . . . .255, 258
adelic intersection product . . . . . . . . xx, 183
adelic intersection theory. . . .xx, 183 – 188
adelic metric.175, 182, 187, 193f, 199, 207,
222f
adelic Picard group . xx, 173, 177, 187, 193
adelic point . . xv – xix, xxii, 99ff, 105, 118,
128, 234, 239, 286, 302 f, 326
adelic topology . . . . . . . . . . . . . . . . . . . . . . 101
adelicly metrized invertible sheaf . . . . 175ff
adjunction formula . . . . . . . . . . . . . . . . . . 232
affine cone . . . . . . . . . . . . . . . . . . . . . . 232, 283
algebraic surface . . . . . . . . . . . . . . . . . . . 71, 85
algebraic surfaces
classification of . . . . . . . . . . . . . . . . . . . . 329
Algorithm
Elkies’ method . . . . . . . . . . xxiii, 294, 349
FFT point counting. . . . . . .248, 251, 260
for numerical integration.251, 291f, 298
obvious . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
of Fincke-Pohst. . . . . . . . . . . . . . . . . . .350 f
of Tate. . . . . . . . . . . . . . . . . . . . . . . . . . . .142
to compute approximate value for Peyre’s
constant . . . . . . . . . . . . . . . 251, 290, 293
to detect conics on quartic threefold 257
to detect lines on cubic threefold . . 255,
260
to generate triangular mesh. . . . . . . . .291
to search for solutions of Diophantine
equation.xxi, xxiii, 259, 284, 294, 296,
332 – 348
naive . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
to solve system of equations over finite
field. . . . . . . . . . . . . . . . . . . . . . . .255, 257
to test for conic through two points.256
to test lines for irrationality . . . . . . . 254f
to test whetherconic is contained in quartic
threefold. . . . . . . . . . . . . . . . . . . . .257
to verify Galois group acting on 27 lines
xxii, 288f, 293, 296f, 299
2-adic Hensel lift . . . . . . . . . . . . . . . . . . 293
algorithms, for computation of volume 227
alteration. . . . . . . . . . . . . . . . . . . . . . . . . . . .164
AMD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
Amir-Moez, A. R.. . . . . . . . . . . . . . . . . . . . . .3
Amitsur, S. A. . . . . . . . . . . . . . . . . . . . . . . . . 37
apex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
Arakelov, S. Yu. . . . . . . . . . . . . . . . . . 156, 160
Arakelov degree . . . . . . . . . . . . . xx, 157, 162
Arakelov geometry . . . . . . . . . . . . . . 190, 192
archimedean valuation . . . . . . . . . . . . . . . 152
ARIBAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
arithmetic Chern class. . . . . . . . . . . . . . . .162
arithmetic Chow group. . . . . . . . .161 – 164
arithmetic cycle . . . . . . . . xx, 160f, 165, 173

384 INDEX
linearly equivalent to zero. . . . . . . . . .161
arithmetic degree . . . . . . . . . . 161, 168, 185f
arithmetic Hilbert-Samuel formula . . . . . xx
arithmetic intersection product . . 163, 165,
168, 185, 208
arithmetic intersection theory . . . . . xx, 155
arithmetic Picard group . . . . . . . . . . . . . . 156
arithmetic variety . . . . . . . . . xix, 156, 158 ff,
162 – 167, 169f, 172, 185
singular . . . . . . . . . . . . . . . . . . . . . 165 – 168
Artin, M. . . . . . . . . . . . . . . . . . . . . . . . 4, 34, 38
Artin conductor. . . . . . . . . . . . . . . .309f, 314
Artin L-function. . . . . .228, 303, 314ff, 326
assembly. . . . . . . . . . . . . . . . . . . . . . . .344, 350
Auslander-Goldman, Theorem of xvii, 71,
84
automorphisms of P n−1 . . . . . . . . . . . 29, 155
Azumaya algebra . . 61 – 70, 78, 80, 82f, 85,
104, 109
B
bad
prime . . . . . . . . . . . . . . . . . . . . . . . 179f, 251
reduction . . . . . . . . . . . . . . . . . . . . .136, 297
barycenter . . . . . . . . . . . . . . . . . . . . . . . . . . 292
Batyrev, V. V. . . . . . . . . . . . . . . . . . . . . . . . .217
Batyrev and Manin, Conjecture ofxxi, 214,
216 – 219, 255
Bernstein, D. . . . . . . . . . . . . . . 259, 349, 351f
Bezout’s theorem . . . . . . . . . . . . . . . . . . . . 253
bielliptic surface. . . . . . . . . . . . . . . . .255, 329
Bierce, A. . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
Bilu, Yu.. . . . . . . . . . . . . . . . . . . .xxi, 190, 192
equidistribution theorem ofxxi, 190, 192
biquadratic reciprocity low . . . . . . . . . . . . xv
Bismut, J.-M. . . . . . . . . . . . . . . . . . . . . . . . . 170
Bogomolov, F. . . . . . . . . . . . . . . . . . . . . . . . 219
Conjecture of . . . . . . . . . . . . . . . . . . . . . 219
Bost, J.-B. . . . . . . . . . . . . . . . . . . . . . . . . xx, 167
Bouche, T.. . . . . . . . . . . . . . . . . . . . . . . . . . .170
Bourbaki, N.. . . . . . . . . . . . . . . . . . . . .3 f, 176
Brauer, R. . . . . . . . . . . . . . . . . . . . . . . . . . . 4, 22
Brauer group . . . . . . .xv ff, 22, 66, 71 f, 74 ff,
84 – 97, 286
cohomological. . . . .xvii, 70 – 76, 84 – 97
of cubic surface. . . . . . . . . . . . . . . . . . . . .96
Brauer-Manin obstruction . . . . . . . . . . . xv ff,
xix, xxii, 97, 103, 105 f, 110, 113, 122,
128f, 144f, 212, 237, 239, 244, 246,
281, 286f, 293, 301ff
is the only obstruction . . . . . . . . . . . . . 106
to rational points . . . . . . . . . . . . . . . . . . 106
to the Hasse principle . . . xviii, 106, 113,
122ff, 130, 212, 239, 286
to weak approximation. .xxii, 106, 122f,
127, 130, 283, 286
Brauer-Severi variety. . . . .xvi f, 3 f, 22, 28 f,
32 – 39, 46, 48, 51 ff, 102
that splits . . . . . . . . . . . . . . . . . . . . xvi, 28ff
Bremner, A.. . . . . . . . . . . . . . . . . . . . . .xv, 103
Brown-Gersten-Quillen spectral sequence
164
Bucket sort. . . . . . . . . . . . . . . . . . . . . . . . . .345
buffer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346f
Burgos Gil, J. I. . . . . . . . . . . . . . . . . . 156, 165
C
C . . . . . . . . . . . . . . . . . . . . . . 296, 344, 348, 350
C++ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
Cantor, M. . . . . . . . . . . . . . . . . . . . . . . . . . . 353
Cartan, E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Cassels, J. W. S. . . . . . . . . . .xv, 103, 265, 283
Čech cocycle . . . . . . . . . . . . . . . . . . . . . . . . . 69
Čech cohomology. . . . . . . . . . . . . . . . . . .68ff
center. . . . . . . . . . .3, 7, 22, 28, 45, 48, 52, 62
central simple algebra . . . . . . . . . . . . . . . xvi f,
3 ff, 10, 22 f, 35 – 39, 45 – 54, 58, 61ff,
74, 76, 78, 82 f
index of. . . . . . . . . . . . . . . . . . . . . . . . . . . .22
that splits . . . . . . . . . . . . . . xvi, 23f, 45, 81
Chambert-Loir, A. . . . . . . . . . . . . . . . xxi, 192
equidistribution theorem of . . . . xxi, 192
Châtelet, F. . . . . . . . . . . . . . . . . . . . . 4f, 34, 38
Chebotarev density theorem .290, 309, 315
Chebyshev’s inequalities. . . . . . . . . . . . . .314
Chinese remainder theorem. . . .341f, 345f
circle method . . . . . xiii, xvi, 238ff, 244, 282
class number formula . . . . . . . . . . . . . . . . 317
classification of algebraic surfaces . . . . . 329
clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

INDEX 385
cocycle . . . . . . . . . . . 6f, 24f, 30 – 34, 37 f, 47
cohomologous to another . . . . . 6, 24, 30
trivial . . . . . . . . . . . . . . . . . . . . . . . . . . . 6, 33
cocycle relations . . . . . . . . . . . . . 12, 108, 180
local in étale topology. . . . . . . . . . . . . . .65
cohomological Brauer group . xvii, 70 – 76,
84 – 97
Colliot-Thélène, J.-L..xv f, xix, 4, 103, 106,
129f, 239, 286f, 302, 312
Conjecture of . . . . . . . . . . . . . . . . . . . . . 106
Colliot-Thélène and Sansuc, Conjecture of
287
Colliot-Thélène, Kanevsky, and Sansuc,
Theorem of. . . . . . . . . . . . . . . . . . . .129 f
collision. . . . . . . . . . . . . . . . . . . . . . . . . . . . .335
computer graphics . . . . . . . . . . . . . . . . . . . 291
conductor-discriminant formula . . . . . . 309
congruencexiii, xix, 38, 144 f, 238, 269, 287,
314, 330 – 334, 336 – 342, 345f, 348
conic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256ff
Conjecture
of Batyrev and Manin . . . . . . . . . xxi, 214,
216 – 219, 255
of Bogomolov . . . . . . . . . . . . . . . . . . . . . 219
of Colliot-Thélène . . . . . . . . . . . . . . . . . 106
of Colliot-Thélène and Sansuc . . . . . . 287
of Lang . . . . . . . . . xii, xxi, 214ff, 255, 329
geometric. . . . . . . . . . . . . . . . . . . . . . .214
strong . . . . . . . . . . . . . . . . . . . . . . . . . . 214
weak. . . . . . . . . . . . . . . . . . . . . . .214, 217
of Manin . . . . . . xiv, xxi, 214, 223f, 239 f,
243f, 246, 256, 261, 263, 265, 282, 284,
293, 296, 301
of Manin-Peyre . . . . . . . . . . . 237, 239, 261
of Mordell . . . . . . . . . . . . . . . . . . . . . . . . 215
control function. . . . . . . . . . . . . . . . .341, 345
convolution. . . . . . . . . . . . . . . . . . . .205, 247 f
Corn, P. K. . . . . . . . . . . . . . . . . . . . . . . xvii, 96
cubic reciprocity low . . . . . . . . . . . . . xv, 145
cubic surface. . . . . . . . . . . . . . .xv – xix, xxi f,
94 – 97, 103, 106, 113f, 122, 125 – 130,
143, 224 – 228, 234, 239 f, 252f, 256,
281 – 328, 353
diagonal . xxii, 103, 128ff, 143, 301 – 328
general . . . . . . . . xxii, 106, 240, 281 – 300
minimal . . . . . . . . . . . . . . . . . . . . . . . . . . 312
cubic threefold . . . . xxi, 215, 240, 243 – 263
diagonal . . . . . . . . . . . . . . . . . xxi f, 213, 224
curvature . . . . . . . . . . . . . . . . . . . . . . 199, 201f
current . . . . . . . . . . . . . . . . . . . . . . . 198, 201
form . . . . . . . . . . . .167, 170, 177, 179, 183
cusp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
D
de Jong, A. J. . . . . . . . . . . . . . . . . . . . . . . . . 164
decomposition type . . . . . . . . . . . . . . . . . . 289
decoupled equation. . . . . . . . .xxiii, 259, 335
Dedekind zeta function . . . . . . . . . . 250, 317
Deligne, P. . . . . . . . . . . . . . 165, 235, 247, 269
Deligne-Beilinson cohomology. . . . . . . .165
∂∂-lemma. . . . . . . . . . . . . . . . . . . . . . . . . . .162
Derenthal, U.. . . . . . . . . . . . . . . . . . . . . . . .228
descent. . .xvi, 8, 10 – 15, 19, 22, 25, 31, 33,
37 f, 47, 49, 51, 58 ff, 64f, 181
Dickson, L. E. . . . . . . . . . . . . . . . . . . . . . . . 288
’s list. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .288
Dieudonné, J.. . . . . . . . . . . . . . . . . . . . . . . . .17
difference scheme . . . . . . . . . . . . . . . . . . . . 351
Diophantine equation . . . . . . . . . . . xi – xiv,
xvi, xxi ff, 149, 259, 266, 284, 294, 296,
329, 332 – 348
decoupled . . . . . . . . . . . . . . . xxiii, 259, 335
Diophantus. . . . . . . . . . . . . . . . . . . . . . .xi, 353
Dirichlet character . . . . . . . . . . . . . . . . . . . 316
Dirichlet’s prime number theorem . . . . 270
distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
division algebra . . . . . . . . . . . . . . . . . . 78 – 81
Duke, W.. . . . . . . . . . . . . . . . . . . . . . . . . . . .315
dynamical system . . . . . . . . . . . . . . . . . . . . 195
E
Eckardt point. . . . . . . . . . . . . . . . . . . . . . . .253
Edidin, D. . . . . . . . . . . . . . . . . . . . . . . . . . . . .85
effective cone . . . . . . . . . . . . . . . . . . 224 – 227
Eisenstein polynomial. . . . . . . . . . . . . . . .115
Ekedahl, T. . . . . . . . . . . . . . . . . . . . . . . . . . . 297
elementary symmetric functions. .153, 311
Elkies, N. . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
’ method . . . . . . . . . . . . . . . xxiii, 294, 349f

386 INDEX
elliptic cone . . . . . . . . . . . . . . . . . . . . . . . . . 252
elliptic curve . . . . . . . . . . . . . . . . . . . . . . . . 137
elliptic surface . . . . . . . . . . . . . . . . . . . . . . . 329
Elsenhans, A.-S. . . .243, 265, 281, 301, 329,
349
Emerson, R. W.. . . . . . . . . . . . . . . . . . . . . .301
Enriques surface . . . . . . . . . . . . . . . . . . . . . . 89
equidistribution . . . xvi, xx, 142, 190f, 208,
246, 266, 284, 301
theorem of Bilu . . . . . . . . . . . xxi, 190, 192
theorem of Chambert-Loir . . . . . xxi, 192
theorem of Szpiro, Ullmo, and Zhangxx,
190
étale cohomology . . . . . . . . . . . . 75, 222, 235
étale morphism . . . . . . . . . . . . . . . . . . . . .229f
étale neighbourhood. . . . . . . . . . . .69, 87, 90
étale topology . . . . . . . . . . . . . . 65, 68, 71, 73
Euler product. . . . . . . . .249 ff, 290, 296, 298
Euler sequence. . . . . . . . . . . . . . . . . . . . . . .233
even . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
expanding map . . . . . . . . . . . . . . . . . . . . . . 196
exponential sequence . . . . . . . . . . . . . 88, 222
F
factor α . . . . xiv, xxi, 224 – 228, 234, 238f,
245f, 283, 301, 306, 326
factor β . . .xvi, 228, 244, 283, 301, 305, 326
Faltings, G. . . . . . . . . . . . . . . . . .149, 156, 215
Fano variety . . . . . . . . . . . . . . . . . xii, xiv, xxi,
213f, 217, 220f, 223, 236 f, 239, 243f,
246, 265, 282, 285, 301
Faraday, M.. . . . . . . . . . . . . . . . . . . . . . . . . . .99
Fermat cubic . . . . . . . . . . . . . . . . . . . . . . . . 213
FFT convolution . . . . . . . . . . . . . . . . . . . . 248
FFT point counting . . . . . . . . . . . . . 251, 260
Fincke-Pohst, algorithm of . . . . . . . . . . 350f
first arithmetic Chern class . . . . . . . . . . . 162
first cohomology set . . . . . . . . . . . . . . . . . . . 6
flagstone . . . . . . . . . . . . . . . . . . . . . . . . . . . 350f
Forster, O. . . . . . . . . . . . . . . . . . . . . . . . . . . 348
fpqc-topology . . . . . . . . . . . . . . . . . . . . . . . . 65
Franke, J. . . . . . . . . . . . . . . . . . . . . . . . xiv, 223
Frobenius eigenvalues. . . . . . . . . . . .236, 311
Fubini-Study metric. . . . . . . . . . . .156ff, 193
functor of points . . . . . . . . . . . 5, 41, 53, 55 ff
functor represented by a scheme . . . . . . . 41
fundamental finiteness . xx, xxii, 149 f, 160,
167, 194, 267, 270, 279, 304, 316, 328
G
G-group . . . . . . . . . . . . . . . . . . . . . . . . 5 – 8, 34
G-morphism . . . . . . . . . . . . . . . . . . . . . . . . . . 5
G-set . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 – 8, 94
Gabber, O. . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
GAP xvii, 95f, 126, 225f, 285, 289, 305, 353,
356
Gauß-Legendre formula . . . . . . . . . . . . . 251f
Gauß sum . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
Generalized Riemann Hypothesis284, 312,
315f
genus one curve . 119, 129, 135, 138ff, 258,
329
Gilbert, W. S.. . . . . . . . . . . . . . . . . . . . . . . .243
Gillet, H.. . . . .xx, 156, 160, 163ff, 167, 170
global class field theory . . . . . . . . . xv, 86, 91
global evaluation map. .see also Manin map
gluing data . . . . . . . . . . . . . . 65, 68f, 108, 180
GMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
Godeaux surface . . . . . . . . . . . . . . . . . . . . . 215
good
prime . . . . . . . . . . . . . . . . . . . . . . . . 179, 299
reduction . 269f, 274, 297, 299, 307, 310,
321, 331
Gröbner base . . . . . . 127, 288, 293, 297, 299
Graßmann
functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . .54
Green’s
current . . . . . . . . . . . . . . . . . 160f, 163, 165
form of log type . . . . . . . . . . . . . . 163, 165
Grothendieck, A. . . xvii, 4f, 17, 22, 39, 41,
53, 71, 85
Guy, M. J. T. . . . . . . . . . . . . . . . . . . . . . xv, 103
H
Hankel, H. . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
Hardy, G. H. . . . . . . . . . . . . . . . . . . . . . . . . xiii
hash function . . . . . . . . . . . . . . .334, 341, 345
hash table . . . . . . . . . . . . . . 259, 335, 345, 347
hashing. . . . . . . . . . . . . . . . . . . .xxiii, 334, 345

INDEX 387
uniform . . . . . . . . . . . . . . . . . . . . . . . . . . 333
Hasse, H. . . . . . . . . . . . . . . . . . . . . . xiii, 4, 265
’s bound . . . . . . . . . . . . . . . . . . . . . . 119, 139
Hasse principle. . .xiii – xvi, xix, 102f, 106,
113, 265f, 281, 284f, 287
Brauer-Manin obstruction to . xviii, 106,
113, 122ff, 130, 212, 239, 286
counterexample to . . . . . xviii, 103, 122ff
obstruction to . . . . . . . . . . . . . . . . . . . . .296
Hassett, B.. . . . . . . . . . . . . . . . . . . . . . . . . . . .85
Hawking, S. . . . . . . . . . . . . . . . . . . . . . . . . . 149
Heath-Brown, D. R. . . . xvi, 145, 239f, 312,
349, 352
’s congruences . . . . . . . . . . . . . . . . . . . . . 145
height
absolute 152, 154, 168, 172, 188, 193, 222
adelic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
anticanonical . . xiii, 217, 222, 237, 265 f,
282, 284, 301
Bost-Gillet-Soulé . . . . . . . . . . . . . . xx, 167 f
canonical. .190, 192, 217, 222, 237, 265 f,
282, 284, 301
canonical . . . . . . . . . . . . . . . . . . . . . . . . . xiii
defined by adelic metric . . . . . . . . . . . . 222
defined by an invertible sheaf. . .xix, 154
l 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .158
logarithmic . . . . . . . . . . . . . . . . . . . 154, 158
Néron-Tate . . . . . . . . . . . . . . . . . . . . . . 190 f
naive . . . . xiii, xix f, 149 – 152, 158, 173,
190ff, 244ff, 266, 279, 282, 301, 328
normalized. . . . . . . . . . . . . . . . . . . . . . . .172
of smallest point . . . . . . . . xxii, 281, 295 f
with respect to hermitian line bundlexix,
158
Hensel, K. . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
’s lemma . . . .xiii, 66ff, 75, 104, 115, 118,
135, 141, 230, 269, 290, 313, 336
hermitian line bundle .xx, 156 – 159, 162 f,
167f, 183, 223
continuous. . . . . . . . . . . . . . . . . . . . . . . .157
smooth . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
hermitian metric . . xx, 156, 159f, 174, 200,
202, 245
almost semiample . . 179, 188, 194, 199ff,
204
bounded . . . . . . . . . . . . . . . . . . . . . . . . . . 174
continuous . . . . 156, 173f, 200f, 222, 246
positive . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
smooth . . . . . . . . . . . . . . . . . 162, 173f, 199
heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
Hirzebruch, F. . . . . . . . . . . . . . . . . . . . . . . . 247
Hochschild-Serre spectral sequence xvi, 90,
111
hypersurface measure. . . . . . . . . . .231, 322f
I
index
of central simple algebra . . . . . . . . . . . . 22
induced
character. . . . . . . . . . . . . . . . . . . . . . . . .307f
representation. . . . . . . . . . . . . . . . . . . .307f
inertia group . . . . . . . . . . . . . . . . . . . . . . . . 283
inflation . . . . . . . . . . . . . . . . 7 f, 112, 127, 134
integrable metric. . . . . . . . . . . . . . . . . . . . .177
integrable topology. . . . . . . . . . . . .177f, 183
integrably metrized invertible sheaf. . . .xx,
177, 188
intersection of two sets . . . . . . . . . . . . . . .332
invariant map. . . . . . . . . . . . . . . . . . . . . . . . .86
irrationality test for lines . . . . . . . . . . . . 254f
isomorphism test
J
for cubic surfaces . . . . . . . . . . . . . . . . . . 296
Jacobi sum . . . . . . . . . . . . . . . . . 116, 248, 307
Julia set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
K
K3 surface . . . . xxii f, 89, 92, 94, 218ff, 255,
329f
Kanevsky, D.. . . . . .xix, 103, 129f, 239, 286
Kelvin, W. Thomson 1st Baron . . . . . . . 211
Kersten, I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Khayyám, O. . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Kleiman, S.. . . . . . . . . . . . . . . . . . . . . . . . . .186
Kodaira classification . . . . . . . . . . . . 213, 255
Kodaira dimension. . . . . . . . . . . . . . . . . . .255
Kramer, J. . . . . . . . . . . . . . . . . . . . . . . 156, 165
Kresch, A.. . . . . . . . . . . . . . . . . . . . . . . . . . . .85

388 INDEX
Kühn, U. . . . . . . . . . . . . . . . . . . . . . . . 156, 165
Kummer pairing . . . . . . . . . . . . . . . . . . . . . 308
Kummer sequence . . . . . . . . 72, 87f, 91, 222
Kummer surface . . . . . . . . . . . . . . . . . . . . . 218
L
L1 cache . . . . . . . . . . . . . . . . . . . . . .343f, 347f
L2 cache . . . . . . . . . . . . . . . . . . . . . . . 343f, 348
l 3 -unit sphere . . . . . . . . . . . . . . . . . . . . . . . . 323
Lang, S. . . . . . . . xii, xxi, 214 – 217, 255, 329
’s conjecture. . . . .xii, xxi, 214ff, 255, 329
geometric. . . . . . . . . . . . . . . . . . . . . . .214
strong . . . . . . . . . . . . . . . . . . . . . . . . . . 214
weak. . . . . . . . . . . . . . . . . . . . . . .214, 217
Langian exceptional set. . . . . . . . . . . . . .214f
Larsen, K. . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
lattice basis reduction. . . . . . . . . . .124, 350f
Lebeau, G. . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Lebesgue measure . . . . . . . . . . . . . . . 169, 224
Lefschetz hyperplane theorem . . . . . . . . . 89
Lefschetz theorem on (1, 1)-classes. . . . . .88
Lefschetz trace formula. . . . . .235, 307, 310
Legendre, A.-M.. . . . . . . . . . . . .xiii, 265, 283
’s Theorem . . . . . . . . . . . . . . . xiii, 265, 283
Lemma
of Grothendieck . . . . . . . . . . . . . . . . . . . .39
of Grothendieck and Dieudonné. . . . .17
of Hensel . . . xiii, 66 ff, 75, 104, 115, 118,
135, 141, 230, 269, 290, 313, 336
of Wedderburn and Brauer . . . . . . . . . . 22
of Yoneda. . . . . . . . . . . . . . . . . . . . . . . . .107
of Zorn. . . . . . . . . . . . . . . . . . . . . . . .10, 107
Leray measure . . 231f, 238, 245f, 304, 322f
Lichtenbaum, S. . . . . . . . . . . . . . . . . . . . . . . 92
Theorem of . . . . . . . . . . . . . . . . . . . . . . . . 92
Lichtenbaum duality. . . . . . . . . . . . . .92, 144
Lind, C.-E. . . . . . . . . . . . . . . . . . . . . . xiv f, 103
line . . . xii, xviii, 115, 117 ff, 121, 125f, 128,
131, 136, 212f, 215f, 219, 225ff, 244,
252 – 258, 260, 285, 287f, 299
non-obvious. . . . . . . . . .xxi, 216, 255, 260
obvious . . . . . . . . . . . . . xxi, 215 f, 253, 255
sporadic. . . . . . . . . . . . . .xxi, 216, 255, 260
–s, 27 on cubic surface . . see also 27 lines
on cubic surface
linear probing . . . . . . . . . . . . . . . . . . . . . . . 335
linear subspace. . . . . . . . . . . . . . .33 f, 38, 244
Linnik’s Theorem. . . . . . . . . . . . . . . . . . . .312
Littlewood, J. E. . . . . . . . . . . . . . . . . . . . . . xiii
LLL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350f
local evaluation map . . . . . . 103f, 129f, 135
local measure . . . . . . . . . . . . . . . . . . . . . . . . 230
log-factor. . . . . . . . . . . . . . . . . . . . . . . .xiv, 223
Lovasz, L. . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
M
Maclagan-Wedderburn, J. H.. . . . . . . . .3, 22
Magma .. . . . . . . . . . . . . . . . . . . . . . . . . 294, 297
Manin, Yu. I. . . . xiv f, xix, 93, 96, 103, 113,
217, 223, 281
’s conjecture . . .xiv, xxi, 214, 223f, 239f,
243f, 246, 256, 261, 263, 265, 282, 284,
293, 296, 301
’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Manin map . . . . . . . . . . . xvii, 103f, 110, 134
Manin-Peyre, Conjecture of. .237, 239, 261
maple.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .118
Maxwell, J. C. . . . . . . . . . . . . . . . . . . . . . . . . 99
McKinnon, D.. . . . . . . . . . . . . . . . . . . . . . .218
memory architecture. . . . . . . . . . . . . . . . .343
Mergesort . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
Merkurjev, A. S. . . . . . . . . . . . . . . . . . . . . . . . 4
Merkurjev-Suslin
Theorem of . . . . . . . . . . . . . . . . . . . . . . . . . 4
mesh generation
2.5-dimensional. . . . . . . . . . . . . . . . . . . .292
metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
bounded . . . . . . . . . . . . . . . . . . . . . . . . . . 174
continuous. . . . . . . . . . . . . . . . . . . . . . . .174
induced by a model . . . . . . . . . . . . . . . . 174
minimum metric. . . . . . .xx, 156ff, 173, 246
Minkowski, H. . . . . . . . . . . . . . . . . . . . . . . xiii
’s lattice point theorem. . . . . . . . . . . . .171
model. . . . . . . . . . .xix, 101f, 104, 108 f, 129,
135f, 139, 142 f, 173ff, 177, 179, 181ff,
187f, 193, 220, 223, 229, 234, 245, 252,
267, 283, 300, 303
modular operation . . . . . . . . . . . . . . . . . . . 341

INDEX 389
Molien, T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Monte-Carlo method. . . . . . . . . . . . . . . . .227
Mordell, L. J. . . . . . . . xiv, xix, 103, 113, 215
’s conjecture. . . . . . . . . . . . . . . . . . . . . . .215
’s examples of cubic surfaces . . . xiv, 103,
113 – 128
Moriwaki, A.. . . . . . . . . . . . . . . . . . . . . . . .165
morphism twisted by σ. . . . . . . . . . . . . . . . .8
moving lemma. . . . . . . . . . . . . . . . . . . . . . .164
Mumford, D. . . . . . . . . . . . . . . . . . . . . . . . . . . 4
N
Néron-Severi group. . . . . . . . . . . . . . . .88, 91
nef . . . . . . . . . . . . . . . . . . 177 – 180, 183, 185 f
Noether, E.. . . . . . . . . . . . . . . . . . . . . . . . . . . .4
Noether-Lefschetz Theorem . xvi, 244, 252
non-archimedean valuation . . . . . . . . . . . 152
non-Azumaya locus . . . . . . . . . . . 62, 85, 110
non-obvious line . . . . . . . . xxi, 216, 255, 260
norm
of global section . . . . . . . . . . . . . . . . . . . 169
norm variety . . . . . . . . . . . . . . . . . . . . . . . . 265
normalized valuation. . . . . . . . . . . . . . . . .150
Northcott’s theorem . . . . . . . . . . . . . . . . . 152
numerical integration . . . . . . 251, 291f, 298
O
obvious line . . . . . . . . . . . xxi, 215f, 253, 255
odd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
Oliver, J. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
open addressing. . . . . . . . . . . . . . . . . . . . . .335
Opteron processor . . . . . . . . . . . . . . . . . . . 347
order. . . . . . . . . . . . . . . . . . .76 – 79, 82ff, 187
maximal . . . . . . . . . . . . . . . 77 – 83, 85, 187
P
p-adic measure . . . . . . . . . . . . . . . . . . . . . 229 f
p-adic unsolvability . . . . . . . . . . . . . . . . . . 212
p-adic valuation . . . . . . . . . . . . . . . . . 270, 317
p-adic numbers . . . . . . . . . . . . . . . . . . . . . . xiii
page prime . . . . . . . . . . . . . 333, 336, 341, 345
paging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
parametrization . . . . . . . . . . . . . . . . . . . . . . xii
Pari................................250
partial presorting . . . . . . . . . . . . . . . . . . . . 347
Pentium 4 processor . . . . . . . . . . . . . . . . . 348
periodicity isomorphism, for cohomology
of cyclic group . . . . . . . . . . . . . . . . . . 137
permutation representation. . . . . . . . . . .285
primitive. . . . . . . . . . . . . . . . . . . . . . . . . .288
Peyre, E. . xvi, xix, xxi, 217, 224, 237 – 240,
244, 266f, 282, 284, 296, 298, 301, 303
Peyre’s constant . . . . . . . . . . . . . . . . xxi f, 224,
237 – 240, 243ff, 247, 260, 266ff, 277,
282ff, 295f, 298, 301 f, 311, 325f
Peyre’s Tamagawa type number . . . . see also
Peyre’s constant
Picard functor . . . . . . . . . . . . . . . . . . . . . . . . 87
Picard group. . . . . . . . . . . . . . . . . . . . . . . . . .91
Picard rank
of cubic surface. . . . . . . . . . . . . . . . . . . . .97
Picard scheme . . . . . . . . . . . . . . . . . . . . . . . . 87
Poincaré residue map. . . . . . . . . . . . . . . . .233
Poincaré-Lelong equation. . . . . . . . . . . . .162
post-processing . . . . . . . . . . . . . . . . . . . . . . 342
presorting, partial. . . . . . . . . . . . . . . . . . . .345
prime number theorem
Dirichlet’s. . . . . . . . . . . . . . . . . . . . . . . . .270
principal divisor
norm of. . . . . . . . . . . . . . . . . . . . . . . . . . .137
product formula . . . . . . . . . . . . . . . . . . . . . 150
projective plane. . . . . . . . . . . . . . . . . . . . . .257
projective space of lines. . . . . . . . . . . . . . . .42
Proposition
of Châtelet-Artin. . . . . . . . . . . . . . . .34, 38
of Severi . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Puiseux expansion . . . . . . . . . . . . . . . . . . . 252
pull-back
of Azumaya algebra. . . . . . . . . . . . . . . . .63
on adelic Picard group . . . . . . . . . . . . . 182
on arithmetic Chow group . . . . . . . . 163f
on group cohomology . . . . . . . . . . . . . . . 7
pure cubic field
discriminant of. . . . . . . . . . . . . . . .309, 316
push-forward
on arithmetic Chow group . . . . . . . . 164f
pyramid. . . . . . . . . . . . . . . . . . . . . . . . . . . . .350
Q
quadratic form . . . . . . . . . . . . . . . . 338ff, 342

390 INDEX
quadratic reciprocity low . . . . . . . . . . . . . xiv
quadric. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .257
quartic surface
diagonal . . . . . . . . . . . . . . . . . . . . . . . . . . 218
quartic threefold.xxi f, 240, 243 – 279, 284,
301
diagonal . . . . . . . . . . . . xxi f, 213, 224, 284
Quicksort . . . . . . . . . . . . . . . . . . . . . . . . . .344f
Quillen, D.. . . . . . . . . . . . . . . . . . . . . . . . . .170
R
Radix sort . . . . . . . . . . . . . . . . . . . . . . . . . . .345
random number generator. . .xxii, 284, 292
rational points
Brauer-Manin obstruction to . . . . . . . 106
rational surface . . . . . . . . . . 4, 89, 92ff, 255f
rational variety . . . . . . . . . . . . . . . . . . . . . . 329
Reading . . . . . . . . . . . . . 259, 339f, 342, 346 ff
reciprocity low
biquadratic. . . . . . . . . . . . . . . . . . . . . . . . .xv
cubic . . . . . . . . . . . . . . . . . . . . . . . . . . xv, 145
quadratic . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
reduced trace . . . . . . . . . . . . . . . . . . . . . . . . . 76
representability, of a functor . . . . . . . . . . . 53
restriction . . . . . . . . . . . . . . . . . . . . . 7, 36, 134
Riemann zeta function . . . . . . . . . . . . . . . 316
ruled surface. . . . . . . . . . . . . . . .252, 255, 258
ruled variety. . . . . . . . . . . . . . . . . . . . . . . . .329
S
Sansuc, J.-J. . . . . . . xix, 103, 129f, 239, 286f
Schappacher, N. . . . . . . . . . . . . . . . . . . xi, 353
Schröer, S.. . . . . . . . . . . . . . . . . . . . . . . . . . . .85
Schwarz, H. A. . . . . . . . . . . . . . . . . . . . . . . 292
’s cylindrical surface . . . . . . . . . . . . . . . 292
search bound . . . xxiii, 260, 268, 284, 293 ff,
304, 348f
searching for rational points . . . . . . . see also
searching for solutions of Diophantine
equation
searching for solutions of Diophantine
equation.xxi, xxiii, 259, 284, 294, 296,
332 – 348
naive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
selection of bits . . . . . . . . . . . . . . . . . . . . . . 341
semi-abelian variety
split . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
semipositive. . . . . .177ff, 181, 188, 194, 204
semipositive cone . . .177ff, 181 – 184, 187f
sequence
generic. . . . . . . . . . . . . . . . . . . . . . . . . . . .204
small . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Serre, J.-P. . . . . . . . . . . . . . . . . . . . . . . . . . 4, 189
’s homological characterization of regularity
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
’s vanishing theorem . . . . . . . . . . . . . . . . 17
Theorem of . . . . . . . . . . . . . . . . 24, 30, 189
Severi, F. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4, 32
sheaf of Azumaya algebras . . . . . . . . xvii, 61
Siegel, C. L. . . . . . . . . . . . . . . . . 265, 283, 317
’s estimate. . . . . . . . . . . . . . . . . . . . . . . . .317
σ-linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Sikorsky, I. I. . . . . . . . . . . . . . . . . . . . . . . . . 265
similarity
of Azumaya algebras . . . . . . . . . . . 65 f, 70
of central simple algebras . . . . . . 4, 22, 38
simple group . . . . . . . . . . . . . . . . . . . . . . . . 285
SINGULAR.. . . . . . . . . . . . . . . . . . . . . .288, 293
64 bit processor. . . . . . . . . . . . . . . . . . . . . .341
Skolem-Noether, Theorem of . . . 23, 67, 69
smallest point . . . . . . . . . . . . . . . . . . . 283, 295
solutions of Diophantine equation
algorithm to search for. . .xxi, xxiii, 259,
284, 294, 296, 332 – 348
naive . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
sorting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
Soulé, C. . . . . . xx, 156, 160, 163ff, 167, 170
splitting field . . 23, 28, 32, 34ff, 38, 46f, 49,
58, 60, 75, 288
sporadic line. . . . . . . . . . . .xxi, 216, 255, 260
∗-product of Green’s currents. . . . .163, 165
Stark, H. M.. . . . . . . . . . . . . . . . . . . . . . . . .317
statistical parameters . . . . . . . . . . . 261ff, 295
Steiner surface . . . . . . . . . . . . . . . . . . . . . . . 258
successive differences . . . . . . . . . . . . . . . . . 298
successive minima.xx, 169f, 172, 188, 194f
Sullivan, A.. . . . . . . . . . . . . . . . . . . . . . . . . .243
surface . . . . . . . . . . . . . . . . . . . 85, 94, 224, 252
abelian . . . . . . . . . . . . . . . . . . . . . . . 218, 255

INDEX 391
algebraic. . . . . . . . . . . . . . . . . . . . . . . .71, 85
bielliptic. . . . . . . . . . . . . . . . . . . . . .255, 329
cubic. . .xv – xix, xxi f, 94 – 97, 103, 106,
113f, 122, 125 – 130, 143, 224 – 228,
234, 239f, 252f, 256, 281 – 328, 353
diagonal . . . . . . . . xxii, 103, 128 ff, 143,
301 – 328
general . . . . . . xxii, 106, 240, 281 – 300
elliptic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
Enriques . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Godeaux . . . . . . . . . . . . . . . . . . . . . . . . . .215
in P 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxiii
K3 . . . xxii f, 89, 92, 94, 218ff, 255, 329ff
Kummer . . . . . . . . . . . . . . . . . . . . . . . . . . 218
non-minimal . . . . . . . . . . . . . . . . . . . . . . 220
non-separated . . . . . . . . . . . . . . . . . . . . . . 85
of general type. . . . . . . . . . . . . . .215, 254 f
of Kodaira dimension one . . . . . . . . . . 255
of Kodaira dimension zero . . . . . . . . . 217
rational . . . . . . . . . . . . . . . 4, 89, 92ff, 255 f
ruled. . . . . . . . . . . . . . . . . . . . .252, 255, 258
Steiner. . . . . . . . . . . . . . . . . . . . . . . . . . . .258
surfaces
classification of . . . . . . . . . . . . . . . . . . . . 329
Suslin, A. A. . . . . . . . . . . . . . . . . . . . . . . . . . . .4
suspicious
pair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .257
point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
Swinnerton-Dyer, Sir P. xiv, xvii, xix, xxiii,
96, 103, 124, 133, 263, 297, 329f, 335
’s list. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .133
Sylow subgroup . . . . . . . . . . . . . . . . . 285, 287
Szpiro, L. . . . . . . . . . . . . . . . . . . . xx, 190, 192
Szpiro, Ullmo, and Zhang, equidistribution
theorem of . . . . . . . . . . . . . . . . . . xx, 190
T
Tamagawa measure. . . . . . . . . .234, 246, 303
Tamagawa number see also Peyre’s constant
Tate, J. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
’s algorithm . . . . . . . . . . . . . . . . . . . . . . . 142
Tate cohomology . . . . . . . . . . . . . . . . . . . . . 94
tensor field . . . . . . . . . . . . . . . . . . . . . . . . . . 230
test for conic through two points . . . . . 256
Theorem
Hilbert 90 . . . . . . . . . . . 26, 33, 71, 90, 112
of Auslander-Goldman. . . . . .xvii, 71, 84
of Bezout . . . . . . . . . . . . . . . . . . . . . . . . . 253
of Bilu . . . . . . . . . . . . . . . . . . . xxi, 190, 192
of Chambert-Loir . . . . . . . . . . . . . xxi, 192
of Colliot-Thélène, Kanevsky, and Sansuc
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 129f
of Hasse-Minkowski . . . . . . . . . . . . . . . 281
of Legendre. . . . . . . . . . . . . . .xiii, 265, 283
of Lichtenbaum . . . . . . . . . . . . . . . . . . . . 92
of Linnik . . . . . . . . . . . . . . . . . . . . . . . . . 312
of Merkurjev-Suslin . . . . . . . . . . . . . . . . . . 4
of Noether-Lefschetz . . . . . . . . . . 244, 252
of Northcott . . . . . . . . . . . . . . . . . . . . . . 152
of Serre . . . . . . . . . . . . . . . . . . . . 24, 30, 189
of Skolem-Noether. . . . . . . . . . .23, 67, 69
of successive minima . . . . . . . . . . 170, 188
of Szpiro, Ullmo, and Zhang. . . .xx, 190
of Tietze . . . . . . . . . . . . . . . . . . . . . . . . . . 160
of Tsen. . . . . . . . . . . . . . . . . . . . . . . . . . . . .89
of Zak . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
Weak Lefschetz. . . . . . . . . . . . . . . . . . . .247
Thompson, S. . . . . . . . . . . . . . . . . . . . . . . . 211
threefold . . . . . . . . . . . . . . . . . . . 212, 282, 302
cubic . . . . . . . . . . .xxi, 215, 240, 243 – 263
diagonal. . . . . . . . . . . . . . .xxi f, 213, 224
quartic. . . .xxi f, 240, 243 – 279, 284, 301
diagonal . . . . . . . . . . xxi f, 213, 224, 284
Tietze’s Theorem . . . . . . . . . . . . . . . . . . . . 160
Tor-formula . . . . . . . . . . . . . . . . . . . . . . . . . 163
toric variety . . . . . . . . . . . . . . . . . . . . . . . . . 192
Tregub, S. L.. . . . . . . . . . . . . . . . . . . . . . . . . .37
triangular mesh . . . . . . . . . . . . . . . . . . . . . . 291
Tschinkel, Y. . . . . . . . .xiv, xvi, xix, 223, 239
Tsen’s theorem. . . . . . . . . . . . . . . . . . . . . . . .89
27 lines on cubic surfacexvii, xxii, 95f, 106,
126ff, 131f, 144, 225 – 228, 284 – 288,
293, 297, 299 f, 305f, 312, 353
2.5-dimensional mesh generation. . . . . .292
Tychonov topology . . . . . . . . . . . . . . . . . . 102
U
Ullmo, E. . . . . . . . . . . . . . . . xx, 190, 192, 208

392 INDEX
V
valuation
archimedean . . . . . . . . . . . . . . . . . . . . . . 152
lying above another. . . . . . . . . . . . . . . .151
non-archimedean . . . . . . . . . . . . . . . . . . 152
van der Waerden’s criterion . . . . . . . . . . . 288
variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
Fano . . . . . . . . . . . . . . . . . . . . . . xii, xiv, xxi,
213f, 217, 220f, 223, 236 f, 239, 243f,
246, 265, 282, 285, 301
of general type xii f, 213 ff, 217, 254 f, 329
of intermediate type . . . xii, 26, 31, 213f,
217, 297
Vistoli, A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
von Neumann, J. . . . . . . . . . . . . . . . . . . . . 281
W
weak approximation. . .xvi, xix, xxii, 102f,
106, 128, 240
Brauer-Manin obstruction to . . xxii, 106,
122f, 127, 130, 283, 286
counterexample to . . . . . . . . . . . . . . . . . xvi
Weak Lefschetz Theorem. . . . . . . . . . . . .247
Weil, A. . . . . . . . . . . . . . . . . . . . . . xix, 22, 331
Weil conjectures . . . 235, 247, 269, 274, 331
for curves . . . . . . . . . . . . . . . . . . . . 331, 340
Witt, E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Writing . . . . . . . . . . . . . 259, 340ff, 345, 347f
Y
Yoneda’s lemma. . . . . . . . . . . . . . . . . . . . . .107
Z
Zak’s theorem . . . . . . . . . . . . . . . . . . . . . . . 253
zeroth cohomology set . . . . . . . . . . . . . . . . . 5
Zhang, S. . . . . . . . . . . xx, 172f, 187, 190, 192
Zorn’s lemma . . . . . . . . . . . . . . . . . . . . 10, 107