brauer groups, tamagawa measures, and rational points

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