MODULAR FORMS AND ANALYTIC NUMBER THEORY

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

This project proposes to investigate several sets of problems that lie in the intersection of the theory of modular forms and analytic number theory. One set of problems concerns real quadratic analogues of singular moduli. Classical singular moduli are special values of the modular j-function at imaginary quadratic irrationalities and have well-known importance for class field theory and the theory of half-integral weight weakly holomorphic modular forms. The real quadratic analogues are defined through certain cycle integrals of the j-function. It is proposed to understand their possible relations to real quadratic fields as well as the asymptotic behavior of their ``traces'', which occur as Fourier coefficients of a new kind of mock modular form. Also to be investigated are new connections between the period functions of modular integrals and certain orthogonal polynomials. One problem here is to apply Riemann-Hilbert analysis to obtain strong asymptotics for the orthogonal polynomials associated to rational period functions. These polynomials are perturbations of Jacobi polynomials and generalize Atkin's polynomials. Another research topic is to study the distribution of roots of polynomial congruences using higher rank automorphic forms. Number theory, which is one of the oldest parts of mathematics, continues to enjoy remarkable advances today. The theory of modular forms occupies a central position within number theory and has proven to be a wellspring of new ideas both within number theory and in other parts of mathematics. The research proposed here is intended to contribute in a meaningful way to the theory of modular forms by developing new connections to analytic number theory as well as to other parts of analysis. An important component of this proposal is the integration of research and education at the postdoctoral, graduate and undergraduate levels.Several research problems that are suitable for undergraduates are proposed, and these will also involve the participation of the PI, graduate students, and postdocs. The aim is to help introduce the undergraduates to research and to develop the mentoring skills of the graduate students and postdocs. Other planned activities are designed to elevate the research experience of mathematics teachers who will be educating students at pre-research stages. These are intended to be cost-effective ways to benefit future research in mathematics in the United States.