Average Frobenius Distributions of Elliptic Curves

Let E be an elliptic curve defined over Q . For p a prime of good reduction, let a p (E) be the trace of the Frobenius morphism of the reduced curve E p . The Lang-Trotter conjecture predicts the number of primes with a fixed value of a p (E). We prove in this paper average estimates which fit the estimate predicted by the conjecture, with an explicit constant confirming the conjectural constant of Lang and Trotter. 1 Introduction. Let E be an elliptic curve defined over the rationals. For any prime p of good reduction, let E p be the elliptic curve over F p obtained by reducing E mod p. Let a p (E) be the trace of the Frobenius morphism of E=F p . Then, #E(F p ) = p + 1 \Gamma a p (E), and ja p (E)j 2 p p. The case a p (E) = 0 corresponds to supersingular reduction mod p. For a fixed r 2 Z, what can be said about the number of primes p such that a p (E) = r? If E has complex multiplication, Deuring showed that half of the primes are primes of supersingular reduction [3]. More p..