Automorphic L-functions, Boundary Distributions, and the Trace Formula

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

The research of the PI (Miller) focuses on the analytic theory of automorphic forms. The proposed research concentrates on two analytic tools in the subject. The first is the trace formula. The PI hopes to sharpen Arthur's trace formula for GL(n) so that it can be applied to analytic questions in the same fashion that Selberg's original trace formula for GL(2) has been. One of the expected applications is to counting various types of automorphic forms, for example a general Weyl law for arithmetic quotients of reductive Lie groups. The second tool is that of automorphic L-functions. The PI and his coworker Wilfried Schmid (Harvard University) are engaged in a research program to study automorphic forms using the boundary distributions of eigenfunctions on symmetric spaces. These boundary techniques allow new constructions of L-functions, and offer a new way to investigate many problems in automorphic forms. The proposed research involves developing this technique and its applications. The study of automorphic forms slices across many important areas of modern mathematical research, including number theory, representation theory, geometry, analysis, and mathematical physics. Through L-functions, Langlands has conjectured many deep and interesting structural relationships between automorphic forms which have implications in the above areas. As an example, the work of Wiles et al demonstrates the link between certain automorphic forms and the ancient problem of solving equations between squares and cubes. The proposed research aims to apply and develop new tools for automorphic forms and L-functions from analysis, which is the branch of mathematics expanding calculus, and representation theory, the concrete study of symmetry. Current applications of automorphic forms and L-functions are manifest in constructing the sophisticated codes which enable high-speed and secure transactions over the internet.