New Directions in the Theory of Automorphic Forms

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

This research project concerns number theory, the oldest branch of mathematics. More specifically, it focuses on the study of automorphic forms. Automorphic forms are a very special class of functions that form an important bridge connecting the discrete objects of algebraic number theory and the continuous objects of analytic number theory. Automorphic forms are used as tools to study number theoretic functions, e.g. by measuring their rates of growth, discovering formulas for them, or proving relations that they satisfy. Classically, special values of automorphic forms provided the solution to Kronecker's problem for extensions of the rational numbers that do not lie in the real line. More recently, automorphic forms played a pivotal role in the proof of Fermat's Last Theorem. This project aims to further explore the connection of automorphic forms to other mathematical structures. The main object of the proposed research is to further understand and exploit the relationship between automorphic forms and quadratic number fields. This relationship is exceedingly rich and unites the study of diverse objects such as Heegner points, closed geodesics, the hyperbolic Laplacian, Kloosterman sums, L-functions and class fields. The theory for imaginary quadratic fields is in general better developed and simpler, especially in relation to class field theory. The investigators will concentrate mostly on the real quadratic case. In one direction, they will study some new geometric invariants associated to real quadratic fields that were introduced recently. These invariants are certain surfaces that are bounded by modular closed geodesics. The investigators will study the distribution of the areas of the surfaces, especially as this relates to ideal classes and also investigate some new geometric problems about the closed geodesics. They plan to express various invariants for real quadratic fields (such as surface integrals of modular functions) in terms of the Fourier coefficients of automorphic forms. This naturally leads to problems involving sums of Kloosterman sums and to extensions of recent work on uniform estimates for such sums.