Tate-Shafarevich Groups and Frobenius Fields of Reductions of Elliptic Curves

Abstract

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