Interplay between weak Maass forms and modular forms, and, Statistical properties of number theoretic objects.

摘要

This thesis is divided into two parts. The first part deals with the study of harmonic weak Maass forms, which have played a crucial role in solving many problems in combinatorics, especially in the theory of partitions. We study the theory of harmonic weak Maass forms with an eye towards applications to the study of classical modular forms.;This work begins by systematically providing identities connecting Ramanujan's mock theta functions with classical modular forms. Such results are possible because of cancelation among the non-holomorphic parts of different harmonic weak Maass forms.;In later chapters we study arithmetic and analytic aspects of classical modular forms. In particular, we transfer Lehmer's conjecture about the non-vanishing of tau(n) to the study of the coefficients of a weakly holomorphic modular form living in a 'dual space'. Our study leads naturally to the resolution of a problem of Iwaniec about the relations between the classical cuspidal Poincare series. The resolution of these problems uses the regularized inner product of Borcherds and a pairing of Bruinier and Funke, extending the usual theory of the Petersson inner product and a pair of differential operators.;In the second part of this thesis we discuss questions concerning statistics of number theoretic objects. In particular, we show that the distribution of the size of the 2-Selmer groups of quadratic twists of the elliptic curve y2 = x3 -- x are governed by statistics arising from the rank of random symmetric matrices over F2 . Secondly, we show that the distribution of prime divisors of elements of Fq [t] are Poisson distributed, in some suitable sense. The techniques employed are drawn from the theory of L-functions and multiplicative number theory.