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(π p ) = K , where π p denotes the Frobenius endomorphism of E at p . More precisely, under a generalized Riemann hypothesis we show that this number is O E ( x 17/18 log x ), and unconditionally we show that this number is We also prove that the number of imaginary quadratic fields K , with −disc K ≤ x and of the form K = ℚ(π p ), is ≫ E log log log x for x ≥ x 0 ( E ). These results represent progress towards a 1976 Lang–Trotter conjecture.