On the local-global principle for twists of abelian varieties
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866912913034313728 |
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| author | Coppola, Nirvana La Porta, Lorenzo Longo, Matteo |
| author_facet | Coppola, Nirvana La Porta, Lorenzo Longo, Matteo |
| contents | This paper investigates the existence of a local-global principle for certain twists of abelian varieties defined over number fields. Our main focus is to determine when, for $m$ a positive integer, locally $m$-atic twists of an abelian variety $A$ over a number field $K$ are globally $m$-atic. We define and study a "Tate-Shafarevich cohomology set" that governs the obstruction to the local-global principle for $m$-atic twists. We prove that, under some mild assumptions, this set is finite, and give criteria for it to be trivial, i.e. for the local-global principle to be satisfied. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_17299 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the local-global principle for twists of abelian varieties Coppola, Nirvana La Porta, Lorenzo Longo, Matteo Number Theory Primary: 11G05, 11G10, Secondary: 11F80, 12G05, 14K15 This paper investigates the existence of a local-global principle for certain twists of abelian varieties defined over number fields. Our main focus is to determine when, for $m$ a positive integer, locally $m$-atic twists of an abelian variety $A$ over a number field $K$ are globally $m$-atic. We define and study a "Tate-Shafarevich cohomology set" that governs the obstruction to the local-global principle for $m$-atic twists. We prove that, under some mild assumptions, this set is finite, and give criteria for it to be trivial, i.e. for the local-global principle to be satisfied. |
| title | On the local-global principle for twists of abelian varieties |
| topic | Number Theory Primary: 11G05, 11G10, Secondary: 11F80, 12G05, 14K15 |
| url | https://arxiv.org/abs/2602.17299 |