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

Project Details

Description

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.

StatusFinished
Effective start/end date7/15/016/30/04

Funding

  • National Science Foundation: $69,190.00

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