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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.14026 |
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Table of Contents:
- 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.