Abstract

Let E be an elliptic curve without complex multiplication defined over Q . Let [Special characters omitted.] be a fixed imaginary field where D > 0 is a nonsquare integer. Let [straight phi] p denote the Frobenius endomorphism of E at p . The Frobenius field of [straight phi] p is denoted by Q ( s p ). In this thesis, we find upper bounds for the number of primes p whose Frobenius fields Q ( s p ) equal a fixed imaginary quadratic field [Special characters omitted.] . In The Square Sieve and the Lang-Trotter Conjecture , A. Cojocaru, E. Fouvry and M. Murty found upper bounds for this problem by applying the Chebotarev Density Theorem on the torsion fields Q ( E [ m ]) associated with E . Based on Serre's theorem, those fields have Galois groups GL 2 ( Z/mZ ). In this thesis, we improve their result by considering smaller extensions F E [ m ] ✹ Q ( E [ m ]) over Q with Galois groups PGL 2 ( Z/mZ ) and by applying an explicit version of the Chebotarev Density Theorem to those fields. More precisely, we show that the bound obtained by A. Cojocaru, E. Fouvry and M. Murty under the Generalized Riemann Hypothesis, can be improved from O ( x 17/18 log x ) to O ( x 13/14 log x ). Under the additional condition of Artin's Holomorphy Conjecture, the bound obtained by A. Cojocaru, E. Fouvry and M. Murty can be improved from O ( x 13/14 log x ) to O ( x 7/8 log x ).