Special Semester Program on Automorphic Forms, Shimura Varieties and L-functions; January 1-May 31, 2003, Fields Institute, Toronto, Canada

Purdue University, West Lafayette IN

Investigators

Abstract

The theory of automorphic forms is a wide and deep subject touching many areas of mathematics, such as number theory, harmonic analysis and geometry. Langlands' program is an ambitious plan to develop the theory of automorphic forms in a systematic way. It opened a new frontier in Mathematics, and has given new insights and techniques in solving old problems. In fact, the solution of Fermat's last theorem by Andrew Wiles is one of the achievements of the Langlands' program. There have been many exciting new developments in the theory recently: the global Langlands correspondence for GL(n) over a functiona field by L. Lafforgue, using the ideas of Drinfeld; the local Langlands correspondence for GL(n) over a p-adic field by M. Harris and R. Taylor, using Shimura varieties, and by G. Henniart, using L-functions; Langlands' functoriality for symmetric cube of cusp form of GL(2) by H. Kim and F. Shahidi, and symmetric fourth of cusp form of GL(2) by H. Kim, using The Langlands-Shahidi method and the converse theorem of Cogdell-Piatetski-Shapiro. It is an ideal time to have a special program in automorphic forms to review these new developments. These have far-reaching applications in classical number theory. Especially, Langlands' functoriality of symmetric cube and symmetric fourth have direct applications to analytic number theory. In fact, one of the goals of the program is to bring experts in automorphic forms and analytic number theory to find applications of automorphic forms in analytic number theory, vice versa. One of the most important mathematical ideas of the second half of the 19th century is that an analytic formula often encodes discrete information. For example, one might want to count the number of solutions of a particular equation, but discover that the solutions are very hard to find. On the other hand, one might want to count the number of solutions to a sequence of equations and have the answers as a sequence. Mathematicians call these kinds of problems 'discrete'. The functions that appear in calculus, and for which calculus works so well, are not 'discrete', but 'analytic.' Amazingly, the right kinds of analytic functions often provide the answers to the discrete problems. An L-function is a type of generating function formed out of data associated with either a geometric object that is related to number theory or with a kind of 'analytic' function, called an automorphic form, which are determined by groups of linear transformations. These L-functions provide the right kinds of analytic functions. In the past thirty years, the study of these examples of L-functions has been systematized into a branch of number theory. In recent years many people have found new ways that L-functions encode discrete information, though there is still much to be discovered. The chief purpose of this program is to bring together experts in analytic number theory, automorphic forms, and geometry so that we may find more of the ways that arithmetic information is encoded by L-functions.