Tate Safarevich groups of elliptic curves with complex multiplication

Abstract

We show that the number of copies of ${\mathbb{Q}}_p/{\mathbb{Z}}_p$ in the Tate-Shafarevich group of an elliptic curve $E$ over $\mathbb{Q}$ with complex multipication, is at most $2p-g$, where $g$ is the rank of $E(\mathbb{Q})$, and for all sufficiently large good ordinary primes $p$.