Fluctuations in the distribution of Hecke eigenvalues

Abstract

A famous conjecture of Sato and Tate (now a celebrated theorem of Taylor et al) predicts that the normalised p-th Fourier coeffcients of a non-CM Hecke eigenform follow the Sato-Tate distribution as we vary the primes p. In 1997, Serre obtained a distribution law for the vertical analogue of the Sato-Tate family, where one fixes a prime p and considers the family of p-th coefficients of Hecke eigenforms. In this thesis, we address a situation in which we vary the primes as well as families of Hecke eigenforms. In the same year, Conrey, Duke and Farmer obtained distribution measures for Fourier coefficients of Hecke eigenforms in these families. Later, in 2006, Nagoshi obtained similar results under weaker conditions. We consider another quantity, namely the number of primes p for which the p-th Fourier coefficient of a Hecke eigenform lies in a fixed interval I. On averaging over families of Hecke eigenforms, we derive an expression for the uctuations in the distribution of these eigenvalues about the Sato-Tate measure. Further, the uctuations are shown to follow a Gaussian distribution. In this way, we obtain a conditional Central Limit Theorem for the quantity in question. Similar results are also proved in the setting of Maass forms. This extends a result of Wang (2014), who proved that the Sato-Tate theorem holds on average in the context of Maass forms. In a separate project, we consider a classical result in number theory: Dirichlet's theorem on the density of primes in an arithmetic progression. We prove i a similar result for numbers with exactly k prime factors for k > 1. Building upon a proof by E.M. Wright in 1954, we compute the asymptotic density of such numbers where each prime satisfies a congruence condition. As an application, we obtain the density of squarefree n ≤ x with k prime factors such that a fixed quadratic equation has exactly 2k solutions modulo n.