Ideal class groups of division fields of elliptic curves and everywhere unramified rational points
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916287824789504 |
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| author | Dainobu, Naoto |
| author_facet | Dainobu, Naoto |
| contents | Let $E$ be an elliptic curve over $\mathbb{Q}$, $p$ an odd prime number and $n$ a positive integer. In this article, we investigate the ideal class group $\mathrm{Cl}(\mathbb{Q}(E[p^n]))$ of the $p^n$-division field $\mathbb{Q}(E[p^n])$ of $E$. We introduce a certain subgroup $E(\mathbb{Q})_{\mathrm{ur},p^n}$ of $E(\mathbb{Q})$ and study the $p$-adic valuation of the class number $\#\mathrm{Cl}(\mathbb{Q}(E[p^n]))$. In addition, when $n = 1$, we further study $\mathrm{Cl}(\mathbb{Q}(E[p]))$ as a $\mathrm{Gal}(\mathbb{Q}(E[p])/\mathbb{Q})$- module. More precisely, we study the semi-simplification $(\mathrm{Cl}(\mathbb{Q}(E[p]))\otimes \mathbb{Z}_p)^{\mathrm{ss}}$ of $\mathrm{Cl}(\mathbb{Q}(E[p]))\otimes \mathbb{Z}_p$ as a $\mathbb{Z}_p[\mathrm{Gal}(\mathbb{Q}(E[p])/\mathbb{Q})]$-module. We obtain a lower bound of the multiplicity of the $E[p]$-component in the semi-simplification when $E[p]$ is an irreducible $\mathrm{Gal}(\mathbb{Q}(E[p])/\mathbb{Q})$-module. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_05035 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Ideal class groups of division fields of elliptic curves and everywhere unramified rational points Dainobu, Naoto Number Theory Primary 11R29, 11G05, Secondary 11R34 Let $E$ be an elliptic curve over $\mathbb{Q}$, $p$ an odd prime number and $n$ a positive integer. In this article, we investigate the ideal class group $\mathrm{Cl}(\mathbb{Q}(E[p^n]))$ of the $p^n$-division field $\mathbb{Q}(E[p^n])$ of $E$. We introduce a certain subgroup $E(\mathbb{Q})_{\mathrm{ur},p^n}$ of $E(\mathbb{Q})$ and study the $p$-adic valuation of the class number $\#\mathrm{Cl}(\mathbb{Q}(E[p^n]))$. In addition, when $n = 1$, we further study $\mathrm{Cl}(\mathbb{Q}(E[p]))$ as a $\mathrm{Gal}(\mathbb{Q}(E[p])/\mathbb{Q})$- module. More precisely, we study the semi-simplification $(\mathrm{Cl}(\mathbb{Q}(E[p]))\otimes \mathbb{Z}_p)^{\mathrm{ss}}$ of $\mathrm{Cl}(\mathbb{Q}(E[p]))\otimes \mathbb{Z}_p$ as a $\mathbb{Z}_p[\mathrm{Gal}(\mathbb{Q}(E[p])/\mathbb{Q})]$-module. We obtain a lower bound of the multiplicity of the $E[p]$-component in the semi-simplification when $E[p]$ is an irreducible $\mathrm{Gal}(\mathbb{Q}(E[p])/\mathbb{Q})$-module. |
| title | Ideal class groups of division fields of elliptic curves and everywhere unramified rational points |
| topic | Number Theory Primary 11R29, 11G05, Secondary 11R34 |
| url | https://arxiv.org/abs/2304.05035 |