Ideal class groups of number fields and Bloch-Kato's Tate-Shafarevich groups for symmetric powers of elliptic curves

TL;DR

This paper investigates the structure of ideal class groups of certain number fields generated by elliptic curves, linking their properties to Bloch-Kato's Tate-Shafarevich groups for symmetric powers of the curves.

Contribution

It provides new conditions relating ideal class groups of $p$-division fields of elliptic curves to Bloch-Kato's Tate-Shafarevich groups for symmetric powers, generalizing previous results.

Findings

Identifies conditions for the class group to have specific symmetric power modules as quotients.

Connects algebraic number theory with arithmetic geometry via Tate-Shafarevich groups.

Extends prior work from the case j=1 to higher symmetric powers.

Abstract

For an elliptic curve $E$ over $\mathbb{Q}$, putting $K=\mathbb{Q}(E[p])$ which is the $p$-th division field of $E$ for an odd prime $p$, we study the ideal class group $\mathrm{Cl}_K$ of $K$ as a $\mathrm{Gal}(K/\mathbb{Q})$-module. More precisely, for any $j$ with $1\leqslant j \leqslant p-2$, we give a condition that $\mathrm{Cl}_K\otimes \mathbb{F}_p$ has the symmetric power $\mathrm{Sym}^j E[p]$ of $E[p]$ as its quotient $\mathrm{Gal}(K/\mathbb{Q})$-module, in terms of Bloch-Kato's Tate-Shafarevich group of $\mathrm{Sym}^j V_p E$. 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 $j=1$.