Kummer varieties and their Brauer groups

Alexei N. Skorobogatov (Department of Mathematics, Imperial College London, United Kingdom; and Institute for the Information Transmission Problems, Russian Academy of Sciences, Moscow, Russia)

Yuri G. Zarhin (Department of Mathematics, Pennsylvania State University, University Park, Pa., U.S.A.)

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 non-emptyness 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.

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Received 8 January 2017

Published 14 September 2018