The square sieve and the lang-trotter conjecture

Abstract

Let E be an elliptic curve defined over ℚ and without complex multiplication. Let K be a fixed imaginary quadratic field. We find nontrivial upper bounds for the number of ordinary primes p ≤ x for which ℚ(πp) = K, where πp denotes the Frobenius endomorphism of E at p. More precisely, under a generalized Riemann hypothesis we show that this number is OE(x17/18 log x), and unconditionally we show that this number is OE,K (x(log log x) 13/12/(log x)25/24). We also prove that the number of imaginary quadratic fields K, with - disc K ≤ x and of the form K = ℚ(πp), is ≫E log log log x for x ≥ x 0(E). These results represent progress towards a 1976 Lang-Trotter conjecture. ©Canadian Mathematical Society 2005.