Hasse principle for generalised Kummer varieties

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.

25 pages

References in corpus (3)

• On the fibration method for zero-cycles and rational points
• Failures of weak approximation in families
• The Brauer-Manin obstruction on Kummer varieties and ranks of twists of abelian varieties

Cited by in corpus (7)

• Many cubic surfaces contain rational points
• Effective bounds for Brauer groups of Kummer surfaces over number fields
• Arithmetic of rational points and zero-cycles on products of Kummer varieties and K3 surfaces
• Second descent and rational points on Kummer varieties
• Rational points on elliptic K3 surfaces of quadratic twist type
• Transcendental Brauer-Manin obstructions on singular K3 surfaces
• Ramified descent