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 ≤ 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.