Relative $p$-class groups and $p$-Selmer groups

Debajyoti De; Dipramit Majumdar; Sudipa Mondal

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arXiv · 2412.13022

Relative $p$-class groups and $p$-Selmer groups

Debajyoti De·Dipramit Majumdar·Sudipa Mondal

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Abstract

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.

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BibTeX RIS

Debajyoti De, Dipramit Majumdar, Sudipa Mondal. 2024-12-17. Relative $p$-class groups and $p$-Selmer groups. https://arxiv.org/abs/2412.13022

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