The Tate-Shafarevich group for elliptic curves with complex multiplication II

TL;DR

This paper investigates the Tate-Shafarevich group of elliptic curves with complex multiplication over Q, establishing bounds on its p-primary subgroup's corank and providing numerical evidence for its vanishing at many primes.

Contribution

It proves an upper bound on the p-primary subgroup's corank for large primes and offers extensive numerical data supporting the subgroup's triviality at primes below 30,000.

Findings

Bound t_{E/Q, p} by (1/2+ε)p for large primes p

Numerical evidence shows t_{E/Q, p} = 0 for primes p < 30,000 in tested cases

Almost all tested primes p have trivial p-primary Tate-Shafarevich subgroup

Abstract

Let E be an elliptic curve over Q with complex multiplication. The aim of the present paper is to strengthen the theoretical and numerical results of \cite{CZS}. For each prime p, let t_{E/Q, p} denote the Z_p-corank of the p-primary subgroup of the Tate-Shafarevich group of E/Q. For each \epsilon 0, we prove that t_{E/Q, p} is bounded above by (1/2+\epsilon)p for all sufficiently large good ordinary primes p. We also do numerical calculations on one such E of rank 3, and 5 such E of rank 2, showing in all cases that t_{E/Q, p} = 0 for all good ordinary primes p < 30,000. In fact, we show that, with the possible exception of one good ordinary prime in this range for just one of the curves of rank 2, the p-primary subgroup of the Tate-Shafarevich group of the curve is zero (always supposing p is a good ordinary prime).