Geometric Methods in the Theory of Automorphic Forms

Harvard University, Cambridge MA

Investigators

Abstract

Geometric methods in the theory of automorphic forms. Abstract. This proposal consists of 2 parts. In the first (and main) part we introduce certain functions on the set of isomorphism classes of irreducible representations of a reductive group over a local field. We call them gamma-functions. The existence of such functions is a corollary of the local Langlands conjecture. We formulate some conjectures about explicit construction of these gamma-functions and explain how they can be applied to study the problem of Langlands lifting itself (both locally and globally). In the second part we suggest how to generalize the results of author's joint paper with D.Gaitsgory entitled "Geometric Eisenstein series". In particular we plan to study intersection cohomology sheaves on parabolic Drinfelds' spaces and also generalize our previous results to the case of tamely ramified Eisenstein series. This proposal is in the part of mathematics known as the Langlands program. The Langlands program is part of number theory. Number theory is the study of the properties of the whole numbers and is the oldest branch of mathematics. From the beginning problems in number theory have furnished a driving force in creating new mathematics in other diverse parts of the discipline. The Langlands program is a general philosophy that connects number theory with calculus; it embodies the modern approach to the study of whole numbers. One aspect of this proposal is to explore the applications of geometric techniques within the Langlands program.