Cyclicity of elliptic curves modulo p and elliptic curve analogues of Linnik?s problem

Abstract

Let E be an elliptic curve defined over and of conductor N. For a prime we denote by the reduction of E modulo p. We obtain an asymptotic formula for the number of primes p ≤ x for which is cyclic, assuming a certain generalized Riemann hypothesis. The error terms that we get are substantial improvements of earlier work of J-P. Serre and M. Ram Murty. We also consider the problem of finding the size of the smallest prime p = p E for which the group is cyclic and we show that, under the generalized Riemann hypothesis, p E = ((log N)4 + ɛ) if E is without complex multiplication, and p E = ((log N)2 + ɛ) if E is with complex multiplication, for any 0 < ɛ < 1.