  Let $\E/\Q$ be a fixed elliptic curve over $\Q$ which does not have complex
multiplication. Assuming the Generalized Riemann Hypothesis, A. C. Cojocaru and
W. Duke have obtained an asymptotic formula for the number of primes $p\le x$
such that the reduction of $\E$ modulo p has a trivial Tate-Shafarevich group.
Recent results of A. C. Cojocaru and C. David lead to a better error term. We
introduce a new argument in the scheme of the proof which gives further
improvement.
