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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2508.14026 |
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| _version_ | 1866908576699645952 |
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| author | Bartel, Alex Morgan, Adam |
| author_facet | Bartel, Alex Morgan, Adam |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14026 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Galois module structures and the Hasse principle in twist families via the distribution of Selmer groups Bartel, Alex Morgan, Adam Number Theory 11G05, 11G10, 20C10, 14G12, 14G05 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. |
| title | Galois module structures and the Hasse principle in twist families via the distribution of Selmer groups |
| topic | Number Theory 11G05, 11G10, 20C10, 14G12, 14G05 |
| url | https://arxiv.org/abs/2508.14026 |