On the adjoint Selmer groups of semi-stable elliptic curves and Flach's zeta elements
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912590259552256 |
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| author | Kim, Chan-Ho |
| author_facet | Kim, Chan-Ho |
| contents | We explicitly construct the rank one primitive Stark (equivalently, Kolyvagin) system extending a constant multiple of Flach's zeta elements for semi-stable elliptic curves. As its arithmetic applications, we obtain the equivalence between a specific behavior of the Stark system and the minimal modularity lifting theorem, and we also discuss the cyclicity of the adjoint Selmer groups. This Stark system construction yields a more refined interpretation of the collection of Flach's zeta elements than the "geometric Euler system" approach due to Flach, Wiles, Mazur, and Weston. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01500 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the adjoint Selmer groups of semi-stable elliptic curves and Flach's zeta elements Kim, Chan-Ho Number Theory We explicitly construct the rank one primitive Stark (equivalently, Kolyvagin) system extending a constant multiple of Flach's zeta elements for semi-stable elliptic curves. As its arithmetic applications, we obtain the equivalence between a specific behavior of the Stark system and the minimal modularity lifting theorem, and we also discuss the cyclicity of the adjoint Selmer groups. This Stark system construction yields a more refined interpretation of the collection of Flach's zeta elements than the "geometric Euler system" approach due to Flach, Wiles, Mazur, and Weston. |
| title | On the adjoint Selmer groups of semi-stable elliptic curves and Flach's zeta elements |
| topic | Number Theory |
| url | https://arxiv.org/abs/2403.01500 |