Ideal class groups of division fields of elliptic curves and everywhere unramified rational points

Abstract

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.