2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/217415We show that the number of copies of ${\Bbb Q}_p/{\Bbb Z}_p$ in the Tate-Shafarevich group of an elliptic curve $E$ over ${\Bbb Q}$ with complex multipication, is at most $2p - g$, where $g$ is the rank of $E({\Bbb Q})$, and for all sufficiently large good ordinary primes $p$.Number Theory11G05;14G40text