Relating the Tate–Shafarevich group of an elliptic curve with the class group

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