Tate Safarevich groups of elliptic curves with complex multiplication

TL;DR

This paper establishes an upper bound on the number of certain p-adic components in the Tate-Shafarevich group of elliptic curves with complex multiplication over Q, relating it to the curve's rank and prime size.

Contribution

It provides a new upper bound on the p-adic part of the Tate-Shafarevich group for CM elliptic curves over Q, connecting it to the rank and prime size.

Findings

Number of copies of Q_p/Z_p in Sha is at most 2p - g.

Bound holds for all sufficiently large good ordinary primes p.

Results apply specifically to elliptic curves with complex multiplication.

Abstract

We 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 $p$.