On $p$-adic $L$-functions of elliptic curves and the ideal class groups of the division fields

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and $F$ be $\mathbb{Q}$ or an imaginary quadratic field with certain conditions. In this article, we study the ideal class group $\mathrm{Cl}(F_E)$ of the $p$-division field $F_E:=F(E[p])$ of $E$ over $F$ for an odd prime number $p$. More precisely, we investigate the non-vanishing of the $E[p]$-component in the semi-simplification of $\mathrm{Cl}(F_E)/p\mathrm{Cl}(F_E)$ as an ${\mathbb{F}}_p[\mathrm{Gal}(F_E/F)]$-module when $E[p]$ is an irreducible $\mathrm{Gal}(F_E/F)$-module. When the analytic rank of $E$ over $F$ is $1$, we establish a new relationship between the non-vanishing of the $E[p]$-component and the $p$-divisibility of a certain $p$-adic analytic quantity associated with $E$. The quantity is defined by the leading coefficient of the cyclotomic $p$-adic $L$-function of $E$ when $F=\mathbb{Q}$ and by that of Bertolini--Darmon--Prasanna's anticyclotomic $p$-adic $L$-function of $E$ when $F$ is the imaginary quadratic field.

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