Rank growth of elliptic curves over S3 extensions with fixed quadratic resolvents
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866910819952885760 |
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| author | Keliher, Daniel Park, Sun Woo |
| author_facet | Keliher, Daniel Park, Sun Woo |
| contents | We study the probability with which an elliptic curve $E/k$, subject to some technical conditions, gains rank upon base extension to an $S_3$-cubic extension $K/k$ with quadratic resolvent field $F/k$, all three fields of which are subject to some mild technical conditions. To do so, we determine the distribution (under a non-standard ordering) of Selmer ranks of an auxiliary abelian variety associated to $E$ and $S_3$-cubic extensions $K/k$ following ideas of Klagsbrun, Mazur, and Rubin. One corollary of this distribution is that $E$ gains rank by at most one upon base extension to $K$ with probability at least $31.95\%$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05705 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rank growth of elliptic curves over S3 extensions with fixed quadratic resolvents Keliher, Daniel Park, Sun Woo Number Theory 11G05 We study the probability with which an elliptic curve $E/k$, subject to some technical conditions, gains rank upon base extension to an $S_3$-cubic extension $K/k$ with quadratic resolvent field $F/k$, all three fields of which are subject to some mild technical conditions. To do so, we determine the distribution (under a non-standard ordering) of Selmer ranks of an auxiliary abelian variety associated to $E$ and $S_3$-cubic extensions $K/k$ following ideas of Klagsbrun, Mazur, and Rubin. One corollary of this distribution is that $E$ gains rank by at most one upon base extension to $K$ with probability at least $31.95\%$. |
| title | Rank growth of elliptic curves over S3 extensions with fixed quadratic resolvents |
| topic | Number Theory 11G05 |
| url | https://arxiv.org/abs/2502.05705 |