On the Distribution in Arithmetic Progressions of Primes of Various Properties Related to Elliptic Curves

The primary focus of this thesis is the examination of the distribution in arithmetic progressions of primes of various properties related to elliptic curves. Following the introduction, the thesis comprises three distinct papers contributing to this central theme. In Section 2, we investigate the necessary and sufficient conditions for the absence of primes of cyclic reduction for an elliptic curve within an arithmetic progression, under GRH. This chapter represents collaborative research conducted with my advisor, Nathan Jones. Section 3 explores the statistical distribution of primes of cyclic reduction and m-divisibility for elliptic curves over congruence classes modulo n. This work stands as an independent contribution. Finally, Section 4 delves into the constants related to the cyclicity conjecture and Koblitz’s conjecture within arithmetic progressions. We provide explicit computations of these constants for Serre curves and propose upper bounds for non-Serre curves. This chapter represents collaborative research with Jacob Mayle and Tian Wang.