Mikhail Borovoi

Abelian Galois cohomology of reductive groups

Introduction The algebraic fundamental group of a reductive group Abelian Galois cohomology The abelianization maps Computation of abelian Galois cohomology Galois cohomology over local fields and number fields References.

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

Hardy-Littlewood varieties and semisimple groups

SummaryWe are interested in counting integer and rational points in affine algebraic varieties, also under congruence conditions. We introduce the notions of a strongly Hardy-Littlewood variety and a relatively Hardy-Littlewood variety, in terms of counting rational points satisfying congruence conditions. The definition of a strongly Hardy-Littlewood variety is given in such a way that varieties for which the Hardy-Littlewood circle method is applicable are strongly Hardy-Littlewood.We prove that certain affine homogeneous spaces of semisimple groups are strongly Hardy-Littlewood varieties. Moreover, we prove that many homogeneous spaces are relatively Hardy-Littlewood, but not strongly Hardy-Littlewood. This yields a new class of varieties for with the asymptotic density of integer points can be computed in terms of a product of local densities.

Extended Picard complexes and linear algebraic groups

Abstract For a smooth geometrically integral variety X over a field k of characteristic 0, we introduce and investigate the extended Picard complex UPic(X). It is a certain complex of Galois modules of length 2, whose zeroth cohomology is and whose first cohomology is Pic(), where is a fixed algebraic closure of k and is obtained from X by extension of scalars to . When X is a k-torsor of a connected linear k-group G, we compute UPic(X) = UPic(G) (in the derived category) in terms of the algebraic fundamental group π1(G). As an application we compute the elementary obstruction for such X.

The algebraic fundamental group of a reductive group scheme over an arbitrary base scheme

We define the algebraic fundamental group π1(G) of a reductive group scheme G over an arbitrary non-empty base scheme and show that the resulting functor G↦ π1(G) is exact.

Homogeneous spaces of Hilbert type

Let k be a global field. Let G be a connected linear algebraic k-group, assumed reductive when k is a function field. It follows from a result of a preprint by Bary-Soroker, Fehm and Petersen that when H is a smooth connected k-subgroup of G, the quotient space G/H is of Hilbert type. We prove a similar result for certain non-connected k-subgroups H of G. In particular, we prove that if G is a simply connected k-group over a number field k, and H is an abelian k-subgroup of G, not necessarily connected, then G/H is of Hilbert type.

Conjugate complex homogeneous spaces with non-isomorphic fundamental groups

Abstract Let X = G / Γ be the quotient of a connected reductive algebraic C-group G by a finite subgroup Γ. We describe the topological fundamental group of the homogeneous space X, which is nonabelian when Γ is nonabelian. Further, we construct an example of a homogeneous space X and an automorphism σ of C such that the topological fundamental groups of X and of the conjugate variety σX are not isomorphic.