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.
