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

H1

(simpler

Gal(k/k), Pic(Xk)

)

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 H1

(

Gal(k/k), Pic(X

k

)

)

= /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

x4

+

2y4

=

z4

+

4w4

. (¶)

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

P3

, our algorithm is of

complexity essentially

O(B2)

for a search bound of B.

In the final implementation, we could work with the search bound B =

108.

We dis-

covered the following solution of the Diophantine equation (¶):

1 484

8014

+ 2 · 1 203

1204

= 9 050 910 498 475 648 046 899 201,

1 169

4074

+ 4 · 1 157

5204

= 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

varieties.

In Chapter VI we describe our investigations regarding the particular families

“ax3

=

by3

+

z3

+

v3

+

w3”,

a, b = 1, . . . , 100, and

“ax4

=

by4

+

z4

+

v4

+

w4”,

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