Relating Tate-Shafarevich group of an elliptic curve with class group

TL;DR

This paper establishes a precise relationship between the Tate-Shafarevich group of an elliptic curve over Q and a quotient of the class group of a related number field, linking elliptic curve arithmetic with class field theory.

Contribution

It formulates and proves a relationship connecting the Tate-Shafarevich group of an elliptic curve with class groups of certain number fields, expanding understanding of their arithmetic connection.

Findings

Relationship established in most cases

Connection between Tate-Shafarevich group and class group quotient

Enhances understanding of elliptic curve arithmetic

Abstract

The paper formulates a precise relationship between the Tate-Shafarevich group of an elliptic curve $E$ over ${\mathbb Q}$ with a quotient of the classgroup of ${\mathbb Q}(E[p])$ on which $Gal({\mathbb Q}(E[p]/{\mathbb Q}) = GL_2({\mathbb Z}/p)$ operates by its standard 2 dimensional representation over ${\mathbb Z}/p$. We establish such a relationship in most cases.