Vol. 10, No. 4, 2016

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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

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.

Kummer varieties, Hasse principle
Mathematical Subject Classification 2010
Primary: 14G05
Secondary: 11J95
Received: 8 May 2015
Revised: 8 February 2016
Accepted: 12 March 2016
Published: 20 June 2016
Yonatan Harpaz
Département de Mathématiques et Applications
École Normale Supérieure
45 rue d’Ulm
Paris 75005
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