Kummer varieties and their Brauer groups

TL;DR

This paper investigates the Brauer groups of Kummer varieties linked to 2-coverings of abelian varieties, providing conditions for triviality, examples with specific properties, and insights into the Brauer--Manin set over number fields.

Contribution

It offers new criteria for the triviality of the Brauer group of Kummer varieties and constructs explicit examples with particular geometric properties.

Findings

Odd order elements of the Brauer group do not obstruct the Hasse principle.

Provided conditions for the triviality of the Brauer group.

Demonstrated non-emptiness of the Brauer--Manin set for certain Kummer varieties.

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.