Hasse principle for Kummer varieties in the case of generic 2-torsion

TL;DR

This paper extends the applicability of the descent-fibration method to prove the Hasse principle for certain Kummer varieties, overcoming previous limitations related to Galois image size and Shafarevich--Tate groups.

Contribution

It combines second descent techniques with parity results on Selmer ranks to handle cases with maximal Galois image, enabling the method for K3 surfaces from genus 2 Jacobians.

Findings

Proves the Hasse principle for specific Kummer varieties under new conditions.

Overcomes Galois image restrictions in descent methods.

Applies to K3 surfaces from genus 2 curves without rational Weierstrass points.

Abstract

Conditional on finiteness of relevant Shafarevich--Tate groups, Harpaz and Skorobogatov used Swinnerton-Dyer's descent-fibration method to establish the Hasse principle for Kummer varieties associated to a 2-covering of a principally polarised abelian variety under certain largeness assumptions on its mod 2 Galois image. Their method breaks down however when the Galois image is maximal, due to the possible failure of the Shafarevich--Tate group of quadratic twists of A to have square order. In this work we overcome this obstruction by combining second descent ideas in the spirit of Harpaz and Smith with results on the parity of 2-infinity Selmer ranks in quadratic twist families. This allows Swinnerton-Dyer's method to be successfully applied to K3 surfaces arising as quotients of 2-coverings of Jacobians of genus 2 curves with no rational Weierstrass points.