Hasse principle for Kummer varieties

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.