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

TL;DR

This paper investigates the distribution of Selmer groups in quadratic twist families of elliptic curves and hyperelliptic curves, revealing insights into Galois module structures and the Hasse principle failures.

Contribution

It introduces new results on the distribution of 2-Selmer ranks in sparse quadratic twist families, linking Galois modules and the Hasse principle.

Findings

Distribution of 2-Selmer ranks in quadratic twists

Frequency of Hasse principle failures in genus 1 curves

Galois module structures in Mordell-Weil groups

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.