Kummer varieties and their Brauer groups

Skorobogatov, Alexei N.; Zarhin, Yuri G.

doi:10.4310/PAMQ.2017.v13.n2.a5

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Kummer varieties and their Brauer groups

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Zarhin, Yuri G.
Max Planck Institute for Mathematics, Max Planck Society;

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http://dx.doi.org/10.4310/PAMQ.2017.v13.n2.a5
(Publisher version)

https://doi.org/10.48550/arXiv.1612.05993
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Citation

Skorobogatov, A. N., & Zarhin, Y. G. (2017). Kummer varieties and their Brauer groups. Pure and Applied Mathematics Quarterly, 13(2), 337-368. doi:10.4310/PAMQ.2017.v13.n2.a5.

Cite as: https://hdl.handle.net/21.11116/0000-0003-C217-8

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.