brauer groups, tamagawa measures, and rational points

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