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Cyclicity and generation of points mod p on elliptic curves

In this paper we study the group of points modulo p of elliptic curves defined over Q. In particular, we are interested in the frequency with which this group is cyclic and with which it is generated by a fixed set of global rational points. Let E be an elliptic curve over Q and for each prime p where E has good reduction, let/~(0:p) be the group of rational points on the reduction of E modulo p. In [14], Serre raised the question of how often this group is cyclic, and following Hooley's work [6-1 on Artin's primitive root conjecture, showed that the number of such p < x is ~ cx/log x for some constant c, assuming the Generalized Riemann Hypothesis (GRH). The second author removed this hypothesis for curves with complex multiplication (CM) in [9-1, and also demonstrated unconditionally the existence of infinitely many primes for which/~([]:p) is cyclic in [10], for certain non-CM elliptic curves. Though the method of [93 establishes an asymptotic formula, the method of [10] does not give a good lower bound for the number of such primes. We will prove unconditionally: