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

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 $Gal(\mathbb{Q}(E[p]/\mathbb{Q})=G{L}_2(\mathbb{Z}/p)$ operates by its standard 2 dimensional representation over $\mathbb{Z}/p$. We establish such a relationship in most cases.

Cited by in corpus (2)

• Diophantine stability for elliptic curves on average
• Class group statistics for torsion fields generated by elliptic curves