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

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 $Gal({\mathbb Q}(E[p]/{\mathbb Q}) = GL_2({\mathbb Z}/p)$operatesbyitsstandard2dimensionalrepresentationover${\mathbb Z}/p$. We establish such a relationship in most cases.