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 approach to this problem considered here is via the theory 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. Much progress on the problem of Functoriality has been made by the proposer in collaboration with Kim, Piatetski-Shapiro, and Shahidi and has set the paradigm for proving such results with these techniques. The main thrust of this proposal is to continue these efforts. It includes projects toimprove the Converse Theorem, projects to extend certain technical results on L-functions and develop new techniques that are more widely applicable, and finally projects aimed towards applications to arithmetic.
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 investigates 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.
