The structure of Selmer groups and the Iwasawa main conjecture for elliptic curves
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909609671786496 |
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| author | Kim, Chan-Ho |
| author_facet | Kim, Chan-Ho |
| contents | We reveal a new and refined application of (a weaker statement than) the Iwasawa main conjecture for elliptic curves to the structure of Selmer groups of elliptic curves of arbitrary rank. For a large class of elliptic curves, we obtain the following arithmetic consequences.
1. Kato's Kolyvagin systems is non-trivial. It is the cyclotomic analogue of the Kolyvagin conjecture.
2. The structure of Selmer groups of elliptic curves over the rationals is completely determined in terms of certain modular symbols. It is a structural refinement of Birch and Swinnerton-Dyer conjecture.
3. The rank zero $p$-converse, the $p$-parity conjecture, and a new upper bound of the ranks of elliptic curves are obtained.
4. The conjecture of Kurihara on the semi-local description of mod $p$ Selmer groups is confirmed.
5. An application of the $p$-adic Birch and Swinnerton-Dyer conjecture to the structure of Iwasawa modules is discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_12159 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The structure of Selmer groups and the Iwasawa main conjecture for elliptic curves Kim, Chan-Ho Number Theory We reveal a new and refined application of (a weaker statement than) the Iwasawa main conjecture for elliptic curves to the structure of Selmer groups of elliptic curves of arbitrary rank. For a large class of elliptic curves, we obtain the following arithmetic consequences. 1. Kato's Kolyvagin systems is non-trivial. It is the cyclotomic analogue of the Kolyvagin conjecture. 2. The structure of Selmer groups of elliptic curves over the rationals is completely determined in terms of certain modular symbols. It is a structural refinement of Birch and Swinnerton-Dyer conjecture. 3. The rank zero $p$-converse, the $p$-parity conjecture, and a new upper bound of the ranks of elliptic curves are obtained. 4. The conjecture of Kurihara on the semi-local description of mod $p$ Selmer groups is confirmed. 5. An application of the $p$-adic Birch and Swinnerton-Dyer conjecture to the structure of Iwasawa modules is discussed. |
| title | The structure of Selmer groups and the Iwasawa main conjecture for elliptic curves |
| topic | Number Theory |
| url | https://arxiv.org/abs/2203.12159 |