The Brauer group of Kummer surfaces and torsion of elliptic curves

Abstract

In this paper we are interested in computing the Brauer group of K3 surfaces. To an element of the Brauer–Grothendieck group Br(X) of a smooth projective variety X over a number field k class field theory associates the corresponding Brauer–Manin obstruction, which is a closed condition satisfied by k-points inside the topological space of adelic points of X, see [20, Ch. 5.2]. If such a condition is non-trivial, X is a counterexample to weak approximation, and if no adelic point satisfies this condition, X is a counterexample to the Hasse principle. The computation of Br(X) is thus a first step in the computation of the Brauer–Manin obstruction on X. Let k be an arbitrary field with a separable closure k, Γ = Gal(k/k). Recall that for a variety X over k the subgroup Br0(X) ⊂ Br(X) denotes the image of Br(k) in Br(X), and Br1(X) ⊂ Br(X) denotes the kernel of the natural map Br(X) → Br(X), where X = X ×k k. In [22] we showed that if X is a K3 surface over a field k finitely generated over Q, then Br(X)/Br0(X) is finite. No general approach to the computation of Br(X)/Br0(X) seems to be known; in fact until recently there was not a single K3 surface over a number field for which Br(X)/Br0(X) was known. One of the aims of this paper is to give examples of K3 surfaces X over Q such that Br(X) = Br(Q). We study a particular kind of K3 surfaces, namely Kummer surfaces X = Kum(A) constructed from abelian surfaces A. Let Br(X)n denote the n-torsion subgroup of Br(X). Section 1 is devoted to the geometry of Kummer surfaces. We show that there is a natural isomorphism of Γ-modules Br(X)−→Br(A) (Proposition 1.3). When A is a product of two elliptic curves, the algebraic Brauer group Br1(X) often coincides with Br(k), see Proposition 1.4. Section 2 starts with a general remark on the etale cohomology of abelian varieties which may be of independent interest (Proposition 2.2). It implies that if n is an odd integer, then for any abelian variety A the group Br(A)n/Br1(A)n is canonically isomorphic to the quotient of Het(A, μn) Γ by (NS (A)/n), where NS (A) is the Neron–Severi group (Corollary 2.3). For any n ≥ 1 we prove that Br(X)n/Br1(X)n is a subgroup of Br(A)n/Br1(A)n, and this inclusion is an equality for odd n, see