Relative $p$-class groups and $p$-Selmer groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929635393011712 |
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| author | De, Debajyoti Majumdar, Dipramit Mondal, Sudipa |
| author_facet | De, Debajyoti Majumdar, Dipramit Mondal, Sudipa |
| contents | Let $E$ be an elliptic curve with $j$-invariant $0$ or $1728$ and let $\widetilde{E}$ be a $k^{th}$ twist of $E$. We show that for any prime $p$ of good reduction of $\widetilde{E}$, a degree $k$ relative $p$-class group and the root number of $\widetilde{E}$ determines the dimension of the $p$-Selmer group of $\widetilde{E}$. As a consequence, we construct families of large rank $p$-class group. We also relate congruent number and cube sum problem with relative $p$-class group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_13022 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Relative $p$-class groups and $p$-Selmer groups De, Debajyoti Majumdar, Dipramit Mondal, Sudipa Number Theory 11R29, 11G05, 11G15, 11R23 Let $E$ be an elliptic curve with $j$-invariant $0$ or $1728$ and let $\widetilde{E}$ be a $k^{th}$ twist of $E$. We show that for any prime $p$ of good reduction of $\widetilde{E}$, a degree $k$ relative $p$-class group and the root number of $\widetilde{E}$ determines the dimension of the $p$-Selmer group of $\widetilde{E}$. As a consequence, we construct families of large rank $p$-class group. We also relate congruent number and cube sum problem with relative $p$-class group. |
| title | Relative $p$-class groups and $p$-Selmer groups |
| topic | Number Theory 11R29, 11G05, 11G15, 11R23 |
| url | https://arxiv.org/abs/2412.13022 |