Frobenius fields for elliptic curves

Let E/ Q be an elliptic curve over the field of rational numbers, with End_ Q(E) = Z. Let K be a fixed imaginary quadratic field over Q, and x a positive real number. For each prime p of good reduction for E, let πₚ(E) be a root of the characteristic polynomial of the Frobenius endomorphism of E over the finite field Fₚ. Let ΠE(K; x) be the number of primes p ≤ x such that the field extension Q(πₚ(E)) is the fixed imaginary quadratic field K. We present upper bounds for ΠE(K; x) obtained using two different approaches. The first one, inspired from work of Serre, is to consider the image of Frobenius in a mixed Galois representation associated to K and to the elliptic curve E. The second one, inspired from work of Cojocaru, Fouvry and Murty, is based on an application of the square sieve. The bounds obtained using the first approach are better, ΠE(K; x) x4/5/(log x)1/5, and are the best known so far. The bounds obtained using the second approach are weaker, but are independent of the number field K, a property which is essential for other applications. All results are conditional upon GRH.