introduction 9
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 that
corresponds more or less to the “first case” of [CT/K/S]. Our argument is, however,
shorter and than the original one. The point is that we make use of the
fact that
Gal(k/k), Pic(Xk)
may be only 0, /2 , /3 , ( /2
or ( /3
Further, the group ( /3 )2 appears only once in a very particular case. Thus, in
order to prove H1
Gal(k/k), Pic(X
= /3 , it is almost sufficient to construct
an element of order three.
Part C collects two reports on practical experiments. Chapter V is concerned with
the Diophantine equation
. (¶)
This equation gives an example of a K3 surface X defined over . It is an open
question whether there exists a K3 surface over that has a finite non-zero number
of -rational points.
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 Peter 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 for -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
, our algorithm is of
complexity essentially
for a search bound of B.
In the final implementation, we could work with the search bound B =
We dis-
covered the following solution of the Diophantine equation (¶):
1 484
+ 2 · 1 203
= 9 050 910 498 475 648 046 899 201,
1 169
+ 4 · 1 157
= 9 050 910 498 475 648 046 899 201.
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 108 [EJ2, EJ3].
The reader probably thinks that this particular equation is not of fundamental
importance, and doing so he or she is definitely right. Let us, however, empha-
size that Chapter V discusses an efficient point search algorithm, which works in
much more generality. The two final chapters show it at work in experimental
investigations related to the Manin conjecture for two important families of Fano
In Chapter VI we describe our investigations regarding the particular families
a, b = 1, . . . , 100, and
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, for one the cubic threefolds, the non-obvious lines in Table 1 have
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