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arXiv · 2508.14026

Galois module structures and the Hasse principle in twist families via the distribution of Selmer groups

Alex Bartel·Adam Morgan
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Abstract

We address several seemingly disparate problems in arithmetic geometry: the statistical behaviour of the Galois module structure of Mordell--Weil groups of a fixed elliptic curve over varying quadratic extensions; the frequency of failure of the Hasse principle in quadratic twist families of genus $1$ hyperelliptic curves; and the Hasse principle for Kummer varieties. The common technical ingredient for all of these is a result on the distribution of $2$-Selmer ranks in certain sparse families of quadratic twists of a given abelian variety.

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Alex Bartel, Adam Morgan. 2025-08-19. Galois module structures and the Hasse principle in twist families via the distribution of Selmer groups. https://arxiv.org/abs/2508.14026

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