Local and Global Aspects of Automorphic L-functions

Ohio State University, The, Columbus OH

Investigators

Abstract

The problem of Langlands' Functoriality is central in the theory of automorphic forms and representations. It is a ramification of Langlands' formulation of non-abelian class field theory, probably the most important problem in modern number theory, which can be tested in a self-contained manner within the context of the theory of automorphic representations. The successful approach to this problem taken by the proposer and his coworkers is via thetheory of L-functions of automorphic representations and the development of a Converse Theorem for these L-functions for the general linear group. These L-functions are analytic invariants that can be attached both arithmetic objects and analytic objects and are used to mediate between them; Langlands non-abelian class field theory is one such connection. Converse Theorems allow one to characterize the analytic side of this equation via the properties of these invariants. The problem of Functoriality comes from interpreting arithmetic phenomena on the analytic side in term of these L-function invariants. The main thrust of this proposal is to develop techniques that will allow for the extension of these efforts. The local projects are to extend the proposers previous work on Bessel functions and stability of local L-functions and related invariants and to develop techniques for computing local L-functions at ramified and infinite places. The global aspects of the project are to improve the Converse Theorem, which is the engine that drives the results on Functoriality. Besides applications to extending the proposers results on Functoriality, which is the main motivation, these results, local and global, should have applications to unveiling the arithmetic hidden in special values of L-functions. The projects in this proposal all fall under the broad rubric of analytic number theory. At its most basic level, number theory is interested in understanding the integers. Additively, the integers are quite simple, generated by 1, but from the point of view of multiplication and factoring they are quite complicated and mysterious. The multiplicative structure is generated by the prime numbers and a large swath of number theory is devoted to the study of prime numbers. This study is full of problems that are simple to state but with no apparent machinery with which to attack them. Over the ages a vast and subtle algebraic structure has been built around these problems -- this is algebraic number theory. But as with many problems, to bring in seemingly incongruous techniques from other areas can lead to new insights. One such ``incongruous'' area is analysis and the theory of group representations; this leads to the theory of automorphic forms, a type of analytic number theory. The connection between the two in its most basic guise is ``class field theory'' and is mediated by certain analytic invariants, called L-functions. Class field theory is a deep and hard problem and any light we can shed on this connection lets us bring the tools of analysis to bear on basic arithmetic problems. This proposal continues our investigations of these invariants, the L-functions, from both the algebraic and analytic points of view, in hopes of narrowing the gap between these two areas in the short term and impacting our understanding of class field theory in the long term.