Class Numbers of Quadratic Imaginary Fields and the Sato-Tate Conjecture

Abstract

In the first chapter we study the distribution of class numbers of quadratic imaginary fields of the form Q( √ −p), where p is a prime in a fixed arithmetic progression. Building upon known results for the arithmetic progression of primes given by p ≡ 3 (mod 4), we establish equidistribution for arbitrary arithmetic progressions of primes for distributions coming from random L-functions. In the second chapter we find effective bounds on some formulations of the Sato-Tate conjecture for elliptic curves over Q. Specifically, assuming a slight strengthening of known potential automorphy results we derive effective bounds on the convergence of character sums, moments, and distribution functions associated to elliptic curves over Q known to satisfy the Sato-Tate conjecture. The material presented in the two chapters is largely independent and can be read in any order.