Kummer varieties and their Brauer groups

Abstract

We study Kummer varieties attached to 2-coverings of abelian varieties of arbitrary dimension. Over a number field we show that the subgroup of odd order elements of the Brauer group does not obstruct the Hasse principle. Sufficient conditions for the triviality of the Brauer group are given, which allow us to give an example of a Kummer K3 surface of geometric Picard rank 17 over the rationals with trivial Brauer group. We establish the nonemptyness of the Brauer-Manin set of everywhere locally soluble Kummer varieties attached to 2-coverings of products of hyperelliptic Jacobians with large Galois action on 2-torsion.

Second descent and rational points on Kummer varieties

“…and the image of E L in Pic(W α ) tor is L. In particular, the Kummer lattice is generated over Π 0 by the classes E L . This description of the Kummer lattice was established by Nikulin [18] in the case of Kummer surfaces and extended to general Kummer varieties by Skorobogatov and Zarhin in [29].…”

“…We note that [2] has no non-constant invertible functions the same holds for W α and so we have a canonical isomorphism of Galois modules H 1 (W α , Q/Z(1)) ∼ = Pic(W α ) tor . We note that the injectivity of the residue map [29]. Given an affine-linear map L : Z α −→ μ 2 , we may realize the corresponding element of Pic(W α ) tor as a degree 2 covering of W α .…”

“…While the main theorem of [10] can be considered as establishing Conjecture 1.1 for the Kummer surfaces of type (1) and (2), what it actually states is that under the 2-primary Tate-Shafarevich conjecture (for the relevant abelian varieties) the Kummer surfaces of type (1) and (2) satisfy the Hasse principle. The gap between these two claims can be explained by a recent paper of Skorobogatov and Zarhin [29], which shows, in particular, that there is no 2-primary Brauer-Manin obstruction for Kummer surfaces of type (1) and (2) (see also Remark 1.2).…”

Index of fibrations and Brauer classes that never obstruct the Hasse principle

“…Let Y ′ → Y be the blow up along f −1 (0). Then σ extends to Y ′ ; the quotient X = Y ′ /σ is smooth and is called the Kummer variety Kum(Y ) attached to Y (see [SZ16] for more details).…”

“…Given an abelian variety A of dimesion ≥ 2 and a 2-covering of A, one can construct a Kummer variety attached to this 2-covering. Let C be the class of all such Kummer varieites over a number field k. It was proven by Skorobogatov and Zarhin that Question 1.1 has positive answer for C with n = 2 [SZ16], and later Skorobogatov extended the result to answer Question 1.3 in [CV17, Theorem A.1]. They're approach relied on proving results about Br(X) and its odd torsion part.…”