Mazur's main conjecture at Eisenstein primes
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| Acceso en línea: | |
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| _version_ | 1866912649121366016 |
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| author | Castella, Francesc Grossi, Giada Skinner, Christopher |
| author_facet | Castella, Francesc Grossi, Giada Skinner, Christopher |
| contents | Let $E/\mathbb{Q}$ be an elliptic curve, let $p>2$ be a prime of good reduction for $E$, and assume that $E$ admits a rational $p$-isogeny with kernel $\mathbb{F}_p(ϕ)$. In this paper we prove the cyclotomic Iwasawa main conjecture for $E$, as formulated by Mazur in 1972, when $ϕ\vert_{G_p}\neq 1,ω$, where $G_p$ is a decomposition group at $p$ and $ω$ is the Teichmüller character. Our proof is based on a study of the anticyclotomic Iwasawa theory of $E$ over an imaginary quadratic field $K$ in which $p$ splits, and a congruence argument exploiting the cyclotomic Euler system of Beilinson--Flach classes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_04373 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Mazur's main conjecture at Eisenstein primes Castella, Francesc Grossi, Giada Skinner, Christopher Number Theory Let $E/\mathbb{Q}$ be an elliptic curve, let $p>2$ be a prime of good reduction for $E$, and assume that $E$ admits a rational $p$-isogeny with kernel $\mathbb{F}_p(ϕ)$. In this paper we prove the cyclotomic Iwasawa main conjecture for $E$, as formulated by Mazur in 1972, when $ϕ\vert_{G_p}\neq 1,ω$, where $G_p$ is a decomposition group at $p$ and $ω$ is the Teichmüller character. Our proof is based on a study of the anticyclotomic Iwasawa theory of $E$ over an imaginary quadratic field $K$ in which $p$ splits, and a congruence argument exploiting the cyclotomic Euler system of Beilinson--Flach classes. |
| title | Mazur's main conjecture at Eisenstein primes |
| topic | Number Theory |
| url | https://arxiv.org/abs/2303.04373 |