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arXiv · 2309.02374

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

Adam Morgan
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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.

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Adam Morgan. 2023-09-05. Hasse principle for Kummer varieties in the case of generic 2-torsion. https://arxiv.org/abs/2309.02374

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