IAS /Park City Mathematics Series
Volume 12 , 2002£-functions and
Converse Theorems for GLn
James W. Cogdell
Introdu ction
The purpose of these notes is to develop the analytic theory of L-functions for
cuspidal automorphic representations of GLn over a global field. There are two
approaches to L-functions of GLn: via integral representations or through analysis
of Fourier coefficients of Eisenstein series. In these notes we develop the theory via
integral representations.
The theory of L-functions of automorphic forms (or modular forms) via integral
representations has its origin in the paper of Riemann on the (-function [72]. How-
ever the theory was really developed in the classical context of L-functions of mod-
ular forms for congruence subgroups of SL 2 (Z) by Hecke and his school [34]. Much
of our current theory is a direct outgrowth of Hecke's. L-functions of automorphic
representations were first developed by Jacquet and Langlands for GL 2 [30,37,39].
Their approach followed Hecke combined with the local-global techniques of Tate's
thesis [91]. The theory for GLn was then developed a long the same lines in a long
series of papers by various combinations of Jacquet, Piatetski-Shapiro, and Sha-
lika [40-4 7 , 64 , 66, 85]. In addition to associating an L-function to an automorphic
form, Hecke also gave a criterion for a Dirichlet series to come from a modular
form, the so called Converse Theorem of Hecke [35]. In the context of automor-
phic representations, the Converse Theorem for GL 2 was developed by Jacquet
and Langlands [39], extended and significantly strengthened to GL3 by Jacquet,
Piatetski-Shapiro, and Shalika [40], and then extended to GLn [9, 12].
What we have attempted to present here is a synopsis of this work and in doing
so present the paradigm for the analysis of automorphic L-functions via integral
representations. Lecture 1 deals with the Fourier expansion of automorphic forms
on GLn and the related Multiplicity One and Strong Multiplicity One Theorems.
Lecture 2 then develops the theory of Eulerian integrals for GLn. In Lecture 3
we turn to the loca l theory of L-functions for GLn, in both the archimedean and
non-archimedean local contexts, which comes out of the Euler factors of the global
(^1) Department of Mathematics Oklahoma State University, Stillwater, OK 74078 , USA.
E-mail address: cogdell©math. okstate. edu.
The author was supported in part by the NSA and the C lay Mathematics Institute.
@2007 American Math ematical Society
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