Frobenius fields for Drinfeld modules of rank 2

Let φ be a Drinfeld module of rank 2 over the field of rational functions F = Fq(T ), with EndF̄ (φ) = Fq[T ]. Let K be a fixed imaginary quadratic field over F and d a positive integer. For each prime p of good reduction for φ, let πp(φ) be a root of the characteristic polynomial of the Frobenius endomorphism of φ over the finite field Fq[T ]/p. Let Πφ(K; d) be the number of primes p of degree d such that the field extension F (πp(φ)) is the fixed imaginary quadratic field K. We present upper bounds for Πφ(K; d) obtained by two different approaches, inspired by similar ones for elliptic curves. The first approach, inspired by the work of Serre, is to consider the image of Frobenius in a mixed Galois representation associated to K and to the Drinfeld module φ. The second approach, inspired by the work of Cojocaru, Fouvry and Murty, is based on an application of the square sieve. The bounds obtained with the first method are better, but depend on the fixed quadratic imaginary field K. In our application of the second approach, we improve the results of Cojocaru, Murty and Fouvry by considering projective Galois representations.