Hasse principle for generalised Kummer varieties

TL;DR

This paper proves the Hasse principle for certain generalized Kummer varieties linked to abelian varieties over number fields, using Selmer group variations and Galois action assumptions, under mild hypotheses.

Contribution

It establishes the Hasse principle for generalized Kummer varieties with large Galois action on 2-torsion, assuming finiteness of Shafarevich-Tate groups, extending previous methods.

Findings

Hasse principle holds under specified conditions.

Large Galois image on 2-torsion is crucial.

Finiteness of Shafarevich-Tate groups is assumed.

Abstract

The existence of rational points on Kummer varieties associated to 2-coverings of abelian varieties over number fields can sometimes be proved through the variation of the Selmer group in the family of quadratic twists of the underlying abelian variety, using an idea of Swinnerton-Dyer. Following Mazur and Rubin, we consider the case when the Galois action on the 2-torsion has a large image. Under mild additional hypotheses we prove the Hasse principle for the associated Kummer varieties assuming the finiteness of relevant Shafarevich-Tate groups.