Analysis of Whittaker periods and applications to automorphic forms

Princeton University, Princeton NJ

Investigators

Abstract

This proposal is motivated by the theory of automorphic forms and L-functions. The emphasis is on the interplay between harmonic analysis and ergodic theory. The primary focus is the analysis of Whittaker periods. Whittaker periods of automorphic forms occur very frequently in many problems. Our goal is to have a complete theory and to prove sharp results. The proposal contains three closely related projects. A project motivated by the geometry of number concerns the uniform distribution of the skewness of integral flags in euclidean space. A related analytic problem is to estimate global Whittaker periods. We shall improve significantly previous results in the literature using global methods from ergodic theory. The third problem concerns local Whittaker functions. We want to establish the conjectural asymptotic behavior of Whittaker functions at infinity. There are several outcomes of proving this conjecture as well as applications to automorphic forms. The PI will apply methods from geometry and representation theory. Number theory is among the oldest branches in mathematics and basic arithmetic is taught in elementary school. Applications to technology are prevalent: communication systems, data processing, cryptographic algorithms. L-functions capture fundamental information about prime numbers which appear everywhere. The Langlands program is a vast network of conjectures and results motivated by the interplay between representation theory and number theory. The proposed research will provide a new bridge between the Langlands program and several topics in analysis and representation theory. This will deepen our understanding and knowledge through identifying profound analogies and stimulate collaboration between experts in different fields. Because of its history the interface between analysis and number theory has plenty of longstanding problems; the analysis of periods and L-functions is a central theme and driving force. The proposed research provides theoretical results which can be used as prospective tools in many different problems: numerical investigations by different teams, vanishing of special values and arithmetic cycles, moments, period bounds and subconvexity problems. The PI will continue to teach and mentor students research projects: junior papers, senior and PhD thesis, disseminating knowledge and discoveries while promoting learning through the investigation of open problems.