The theory of L-functions of automorphic forms (or modular forms) via integral representations has its origin in the paper of Riemann on the zeta-function. However the theory was really developed in the classical context of L-functions of modular forms for congruence subgroups of SL(2,Z) by Hecke and his school. 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). Their approach followed Hecke combined with the local-global techniques of Tate's thesis. The theory for GL(n) was then developed along the same lines in a long series of papers by various combinations of Jacquet, Piatetski-Shapiro, and Shalika. 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. In the context of automorphic representations, the Converse Theorem for GL(2) was developed by Jacquet and Langlands, extended and significantly strengthened to GL(3) by Jacquet, Piatetski-Shapiro, and Shalika, and then extended to GL(n).
In these lectures we hope to present a synopsis of this work and in doing so present the paradigm for the analysis of general automorphic L-functions via integral representations. We will begin with the classical theory of Hecke and then a description of its translation into automorphic representations of GL(2) by Jacquet and Langlands. We will then turn to the theory of automorphic representations of GL(n), particularly cuspidal representations. We will first develop the Fourier expansion of a cusp form and present results on Whittaker models since these are essential for defining Eulerian integrals.We will then develop integral representations for L-functions for GL(n) × GL(m) which have nice analytic properties (meromorphic continuation, boundedness in vertical strips, functional equations) and have Eulerian factorization into products of local integrals.
We next turn to the local theory of L-functions for GL(n), in both the archimedean and non-archimedean local contexts, which comes out of the Euler factors of the global integrals. We finally combine the global Eulerian integrals with the definition and analysis of the local L-functions to define the global L-function of an automorphic representation and derive their major analytic properties.
We will then turn to the various Converse Theorems for GL(n). We will begin with the simple inversion of the integral representation. Then we will show how to proceed from this to the proof of the basic Converse Theorems, those requiring twists by cuspidal representations of GL(m) with m at most n-1. We will then discuss how one can reduce the twisting to m at most n-2. Finally we will consider what is conjecturally true about the amount of twisting necesssary for a Converse Theorem.
We will end with a description of the applications of these Converse Theorems to new cases of Langlands Functoriality. We will discuss both the basic paradigm for using the Converse Theorem to establish liftings to GL(n) and the specifics of the lifts from the split classical groups SO(2n+1), SO(2n), and Sp(2n) to the appropriate GL(N).
