Tate-Shafarevich groups and Frobenius fields of reductions of elliptic curves

Let E/ℚ be a fixed elliptic curve over ℚ which does not have complex multiplication. Assuming the Generalized Riemann Hypothesis, Cojocaru and Duke have obtained an asymptotic formula for the number of primes p ≤ x such that the reduction of E modulo p has a trivial Tate–Shafarevich group. Recent results of Cojocaru and David lead to a better error term. We introduce a new argument in the scheme of the proof, which gives a further improvement.9 page(s