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

2025/08/19 by Alex Bartel, Adam Morgan, Bartel, Alex +1
Mathematics · #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2508.14026

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.

Related

• The distribution of ℓ^∞-Selmer groups in degree ℓ twist families I
• The distribution of ℓ^∞-Selmer groups in degree ℓ twist families II
• Hasse principle for Kummer varieties in the case of generic 2-torsion
• Governing fields and statistics for 4-Selmer groups and 8-class groups
• Density of Selmer ranks in families of even Galois representations, Wiles' formula, and global reciprocity