  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).
