$p$-converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910678528294912 |
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| author | Keller, Timo Yin, Mulun |
| author_facet | Keller, Timo Yin, Mulun |
| contents | Let $E/\mathbf{Q}$ be an elliptic curve and $p\geq 3$ be a prime. We prove the $p$-converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes (i.e., such that the residual representation $E[p]$ is reducible) when the $p$-Selmer rank is $0$ or $1$. The key step is to obtain the anticyclotomic Iwasawa Main Conjectures for an auxiliary imaginary quadratic field $K$ where $E$ does not have CM similar to those in [CGLS22] and descent to $\mathbf{Q}$.
As an application we get improved proportions for the number of elliptic curves in quadratic twist families having rank $0$ or $1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23241 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $p$-converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes Keller, Timo Yin, Mulun Number Theory 11G40 (Primary) 11G05, 11G10, 14G10 (Secondary) Let $E/\mathbf{Q}$ be an elliptic curve and $p\geq 3$ be a prime. We prove the $p$-converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes (i.e., such that the residual representation $E[p]$ is reducible) when the $p$-Selmer rank is $0$ or $1$. The key step is to obtain the anticyclotomic Iwasawa Main Conjectures for an auxiliary imaginary quadratic field $K$ where $E$ does not have CM similar to those in [CGLS22] and descent to $\mathbf{Q}$. As an application we get improved proportions for the number of elliptic curves in quadratic twist families having rank $0$ or $1$. |
| title | $p$-converse theorems for elliptic curves of potentially good ordinary reduction at Eisenstein primes |
| topic | Number Theory 11G40 (Primary) 11G05, 11G10, 14G10 (Secondary) |
| url | https://arxiv.org/abs/2410.23241 |