Finiteness and cofiniteness of fine Selmer groups over function fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916901707317248 |
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| author | Ghosh, Sohan Ray, Jishnu Suzuki, Takashi |
| author_facet | Ghosh, Sohan Ray, Jishnu Suzuki, Takashi |
| contents | We prove that the dual fine Selmer group of an abelian variety over the unramified $\mathbb{Z}_{p}$-extension of a function field is finitely generated over $\mathbb{Z}_{p}$. This is a function field version of a conjecture of Coates--Sujatha. We further prove that the fine Selmer group is finite (respectively zero) if the separable $p$-primary torsion of the abelian variety is finite (respectively zero). These results are then generalized to certain ramified $p$-adic Lie extensions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_06938 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Finiteness and cofiniteness of fine Selmer groups over function fields Ghosh, Sohan Ray, Jishnu Suzuki, Takashi Number Theory Algebraic Geometry 11R23 (Primary) 11G10, 11R58, 14G17, 14K15 (Secondary) We prove that the dual fine Selmer group of an abelian variety over the unramified $\mathbb{Z}_{p}$-extension of a function field is finitely generated over $\mathbb{Z}_{p}$. This is a function field version of a conjecture of Coates--Sujatha. We further prove that the fine Selmer group is finite (respectively zero) if the separable $p$-primary torsion of the abelian variety is finite (respectively zero). These results are then generalized to certain ramified $p$-adic Lie extensions. |
| title | Finiteness and cofiniteness of fine Selmer groups over function fields |
| topic | Number Theory Algebraic Geometry 11R23 (Primary) 11G10, 11R58, 14G17, 14K15 (Secondary) |
| url | https://arxiv.org/abs/2408.06938 |