Alexei N. Skorobogatov

Torsors and Rational Points

1. Introduction 2. Torsors: general theory 3. Examples of torsors 4. Abelian torsors 5. Obstructions over number fields 6. Abelian descent and Manin obstruction 7. Conic bundle surfaces 8. Bielliptic surfaces 9. Homogenous spaces.

A finiteness theorem for the Brauer group of abelian varieties and 3 surfaces

Let k be a field finitely generated over the field of rational numbers, and Br(k) the Brauer group of k. For an algebraic variety X over k we consider the cohomological Brauer–Grothendieck group Br(X). We prove that the quotient of Br(X) by the image of Br(k) is finite if X is a K3 surface. When X is an abelian variety over k, and X is the variety over an algebraic closure k of k obtained from X by the extension of the ground field, we prove that the image of Br(X) in Br(X) is finite.

The elementary obstruction and homogeneous spaces

Let k be a field of characteristic zero, and let k be an algebraic closure of k. For a geometrically integral variety X over k, we write k(X) for the function field of X = X ×k k. If X has a smooth k-point, the natural embedding of multiplicative groups k ∗ ↪→ k(X)∗ admits a Galois-equivariant retraction. In the first part of this article, equivalent conditions to the existence of such a retraction are given over local and then over global fields. Those conditions are expressed in terms of the Brauer group of X. In the second part of the article, we restrict attention to varieties that are homogeneous spaces of connected but otherwise arbitrary algebraic groups, with connected geometric stabilizers. For k local or global, and for such a variety X, in many situations but not all, the existence of a Galois-equivariant retraction to k ∗ ↪→ k(X)∗ ensures the existence of a k-rational point on X. For homogeneous spaces of linear algebraic groups, the technique also handles the case where k is the function field of a complex surface. Resume Soient k un corps de caracteristique nulle et k une cloture algebrique de k. Pour une k-variete X geometriquement integre, on note k(X) le corps des fonctions de X = X ×k k. Si X possede un k-point lisse, le plongement naturel de groupes multiplicatifs k ∗ ↪→ k(X)∗ admet une retraction equivariante pour l’action du groupe de Galois de k sur k. Dans la premiere partie de l’article, sur les corps locaux puis sur les corps globaux, on donne des conditions equivalentes a l’existence d’une telle retraction equivariante. Ces conditions s’expriment en terme du groupe de Brauer de la variete X. Dans la seconde partie de l’article, on considere le cas des espaces homogenes de groupes algebriques connexes, non necessairement lineaires, avec groupes d’isotropie DUKE MATHEMATICAL JOURNAL Vol. 141, No. 2, c

Non-abelian descent and the arithmetic of Enriques surfaces

The Brauer-Manin obstruction to the Hasse principle and weak approximation provides a fruitful general approach to rational points on varieties over number fields. A fundamental problem here can be stated as follows: is it possible to describe in purely geometric terms the class of smooth projective varieties for which the Brauer-Manin obstruction is the only obstruction to the Hasse principle and weak approximation? In recent examples where the Brauer-Manin obstruction is not the only one (see [1, 7, 14]), the key role is played by etale Galois coverings with a non-abelian Galois group. This has left open the question whether similar examples exist for varieties with an abelian geometric fundamental group. The case of principal homogeneous spaces of abelian varieties and that of rational surfaces (which are geometrically simply connected), where the Brauer-Manin obstruction is expected to be the only one, might seem to suggest that as long as the geometric fundamental group is abelian, the Brauer-Manin obstruction should still be the only one. The Manin obstruction was linked to the classical abelian descent by ColliotThelene and Sansuc [2]. In [8], the authors introduced the non-abelian descent as a new tool for studying rational points. The present paper enriches the non-abelian theory with a general method for constructing non-abelian torsors, and then applies it to an example which answers the above question in the negative.

Non-abelian Cohomology and Rational Points

Using non-abelian cohomology we introduce new obstructions to the Hasse principle. In particular, we generalize the classical descent formalism to principal homogeneous spaces under noncommutative algebraic groups and give explicit examples of application.

On the Brauer group of diagonal quartic surfaces

We obtain an easy sufficient condition for the Brauer group of a diagonal quartic surface D over Q to be algebraic. We also give an upper bound for the order of the quotient of the Brauer group of D by the image of the Brauer group of Q. The proof is based on the isomorphism of the Fermat quartic surface with a Kummer surface due to Mizukami.

Rational solutions of certain equations involving norms

Let k be an algebraic closure of k. In the case when P(t) has at most one root in k, the open subset of the affine variety (1) given by P(t)y~O is a principal homogeneous space under an algebraic k-torus. In this case it is well known that the Brauer Manin obstruction is the only obstruction to the Hasse principle and weak approximation on any smooth projective model of this variety (Colliot-Th~l~ne and Sansuc [CSanl]). In this paper we prove the same result when P(t) has exactly two roots in k and no other roots in k, and k is the field of rational numbers Q. An immediate change of variables then reduces (1) to the equation ta~

The Brauer group of Kummer surfaces and torsion of elliptic curves

We relate the Brauer group of a Kummer surface to the Brauer group of the corresponding abelian surface. For many pairs of elliptic curves over the rational numbers we prove that the Kummer surface attached to their product has trivial Brauer group.

Exponential sums, the geometry of hyperplane sections, and some diophantine problems

We estimate exponential sums with additive character along an affine variety given by a system of homogeneous equations, with a homogeneous function in the exponent. The proof uses the results of Deligne’s Weil Conjectures II and a generalization of Lefschetz hyperplane theorem to singular varieties. We apply our estimate to obtain an upperbound for the number of integer solutions of a system of homogeneous equations in a box. Another application is devoted to uniform distribution of solutions of a system of homogeneous congruences modulo a prime in the following sense: the portion of solutions in a box is proportional to the volume of the box, provided the box is not very small.

Good reduction of the Brauer-Manin obstruction

For a smooth and projective variety over a number field with torsion free geometric Picard group and finite transcendental Brauer group we show that only the archimedean places, the primes of bad reduction and the primes dividing the order of the transcendental Brauer group can turn up in the description of the Brauer–Manin set. AMS Subject Classification: 14F22, 14G05, 11G35, 11G25