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

Abstract

Let E be an elliptic curve over $Q$, p an odd prime number and n a positive integer. In this article, we investigate the ideal class group $\mathrm{Cl}\left(Q\right(E\left[p^n\right]\left)\right)$ of the $p^n$-division field $Q\left(E\right[p^n\left]\right)$ of E. We introduce a certain subgroup $E{\left(Q\right)}_{\mathrm{ur},p^n}$ of $E\left(Q\right)$ and study the p-adic valuation of the class number $\#\mathrm{Cl}\left(Q\right(E\left[p^n\right]\left)\right)$.

In addition, when $n=1$, we further study $\mathrm{Cl}\left(Q\right(E\left[p\right]\left)\right)$ as a $\mathrm{Gal}\left(Q\right(E\left[p\right])/Q)$-module. More precisely, we study the semi-simplification ${\left(\mathrm{Cl}\right(Q\left(E\right[p\left]\right))\otimes Z_p)}^{\mathrm{ss}}$ of $\mathrm{Cl}\left(Q\right(E\left[p\right]\left)\right)\otimes Z_p$ as a $Z_p\left[\mathrm{Gal}\right(Q\left(E\right[p\left]\right)/Q\left)\right]$-module. We obtain a lower bound of the multiplicity of the $E\left[p\right]$-component in the semi-simplification when $E\left[p\right]$ is an irreducible $\mathrm{Gal}\left(Q\right(E\left[p\right])/Q)$-module.