Frobenius fields for elliptic curves

Let $E/{\Bbb Q}$ be an elliptic curve over the field of rational numbers, with ${\mathop{\rm End}}_{\bar{{\Bbb Q}}}(E) = {\Bbb Z}$. Let $K$ be a fixed imaginary quadratic field over ${\Bbb Q}$, and $x$ a positive real number. For each prime $p$ of good reduction for $E$, let $\pi_p(E)$ be a root of the characteristic polynomial of the Frobenius endomorphism of $E$ over the finite field ${\Bbb F}_p$. Let $\Pi_{E}(K; x)$ be the number of primes $p \leq x$ such that the field extension ${\Bbb Q}(\pi_p(E))$ is the fixed imaginary quadratic field $K$. We present upper bounds for $\Pi_{E}(K; x)$ 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, $\Pi_E(K; x) \ll x^{4/5}/(\log x)^{1/5}$, 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.