Iwasawa theory of fine Selmer groups over global fields
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arXiv
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| Format: | Preprint |
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2022
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| author | Ghosh, Sohan Jha, Somnath Shekhar, Sudhanshu |
| author_facet | Ghosh, Sohan Jha, Somnath Shekhar, Sudhanshu |
| contents | The $p^\infty$-fine Selmer group of an elliptic curve $E$ over a number field $F$ is a subgroup of the classical $p^\infty$-Selmer group of $E$ over $F$. Fine Selmer group is closely related to the 1st and 2nd Iwasawa cohomology groups. Coates-Sujatha observed that the structure of the fine Selmer group of $E$ over a $p$-adic Lie extension of a number field is intricately related to some deep questions in classical Iwasawa theory; for example, Iwasawa's classical $μ$-invariant vanishing conjecture. In this article, we study the properties of the $p^\infty$-fine Selmer group of an elliptic curve over certain $p$-adic Lie extensions of a number field. We also define and discuss $p^\infty$-fine Selmer group of an elliptic curve over function fields of characteristic $p$ and also of characteristic $\ell \neq p.$ We relate our study with a conjecture of Jannsen. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_01751 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Iwasawa theory of fine Selmer groups over global fields Ghosh, Sohan Jha, Somnath Shekhar, Sudhanshu Number Theory 11R23, 11G05, 11S25, 11R60 The $p^\infty$-fine Selmer group of an elliptic curve $E$ over a number field $F$ is a subgroup of the classical $p^\infty$-Selmer group of $E$ over $F$. Fine Selmer group is closely related to the 1st and 2nd Iwasawa cohomology groups. Coates-Sujatha observed that the structure of the fine Selmer group of $E$ over a $p$-adic Lie extension of a number field is intricately related to some deep questions in classical Iwasawa theory; for example, Iwasawa's classical $μ$-invariant vanishing conjecture. In this article, we study the properties of the $p^\infty$-fine Selmer group of an elliptic curve over certain $p$-adic Lie extensions of a number field. We also define and discuss $p^\infty$-fine Selmer group of an elliptic curve over function fields of characteristic $p$ and also of characteristic $\ell \neq p.$ We relate our study with a conjecture of Jannsen. |
| title | Iwasawa theory of fine Selmer groups over global fields |
| topic | Number Theory 11R23, 11G05, 11S25, 11R60 |
| url | https://arxiv.org/abs/2201.01751 |