Ideal Class Groups of Number Fields and Bloch-Kato’s Tate-Shafarevich Groups for Symmetric Powers of Elliptic Curves

Abstract

For an elliptic curve $E$ over $\mathrm{\mathbb{Q}}$, putting $K=\mathrm{\mathbb{Q}}\left(E\right[p\left]\right)$ which is the $p$-th division field of $E$ for an odd prime $p$, we study the ideal class group ${\text{Cl}}_K$ of $K$ as a $\text{Gal}(K/\mathrm{\mathbb{Q}})$-module. More precisely, for any $j$ with $1⩽j⩽p-2$, we give a condition that ${\text{Cl}}_K\otimes {\mathbb{F}}_p$ has the symmetric power ${\text{Sym}}^{j}E\left[p\right]$ of $E\left[p\right]$ as its quotient $\text{Gal}(K/\mathrm{\mathbb{Q}})$-module, in terms of Bloch-Kato’s Tate-Shafarevich group of ${\text{Sym}}^j{V}_{p}E$. Here $V_{p}E$ denotes the rational $p$-adic Tate module of $E$. This is a partial generalization of a result of Prasad and Shekhar for the case $j=1$.