Hasse principle for Kummer varieties

Harpaz, Yonatan; Skorobogatov, Alexei

doi:10.2140/ant.2016.10.813

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Hasse principle for Kummer varieties

Yonatan Harpaz and Alexei N. Skorobogatov

Vol. 10 (2016), No. 4, 813–841

DOI: 10.2140/ant.2016.10.813

Abstract

The existence of rational points on the Kummer variety associated to a $2$ -covering of an abelian variety $A$ over a number field can sometimes be established through the variation of the $2$ -Selmer group of quadratic twists of $A$ . In the case when the Galois action on the $2$ -torsion of $A$ has a large image, we prove, under mild additional hypotheses and assuming the finiteness of relevant Shafarevich–Tate groups, that the Hasse principle holds for the associated Kummer varieties. This provides further evidence for the conjecture that the Brauer–Manin obstruction controls rational points on K3 surfaces.

Keywords

Kummer varieties, Hasse principle

Mathematical Subject Classification 2010

Primary: 14G05

Secondary: 11J95

Milestones

Received: 8 May 2015

Revised: 8 February 2016

Accepted: 12 March 2016

Published: 20 June 2016

Authors

Yonatan Harpaz
Département de Mathématiques et Applications École Normale Supérieure 45 rue d’Ulm Paris 75005 France

Alexei N. Skorobogatov
Department of Mathematics Imperial College South Kensington Campus London, SW7 2BZ United Kingdom Institute for Information Transmission Problems Russian Academy of Sciences 19 Bolshoi Karetnyi Moscow, 127994 Russia

Vol. 10, No. 4, 2016