98 J.W. COGDELL, £-FUNCTIONS FOR GLnintegrals. In Lecture 4 we finally combine the global Eulerian integrals with the
definition and analysis of the lo cal L-functions to define the global L-function of an
automorphic representation and derive their major analytic properties. In Lecture
5 we turn to the various Converse Theorems for GLn. Lecture 6 is devoted to
the application of the Converse Theorem to questions of Functoriality, that is, the
lifting or transfer of automorphic representations from a group H to GLn.
We have tried to keep the tone of the notes informal for the most part. We
have tried to provide complete proofs where feasible, at least sketches of most major
results, and references for technical facts.
There is another body of work on integral representations of L-functions for
GLn which developed out of the classical work on zeta functions of algebras. This
is the theory of principal L-functions for GLn as developed by Godement and
Jacquet [31, 37]. This approach is related to the one pursued here, but we have
not attempted to present it here.
The other approach to these L-functions is via the Fourier coefficients of Eisen-
stein series. In the context of automorphic representations, and in a broader context
than GLn, this approach was originally laid out by Langlands [60] but then most
fruitfully pursued by Shahidi. Some of the major papers of Shahidi on this subject
are [74-8 4 ]. In particular, in [77] he shows that the two approaches give the same
L-functions for GLn. We will not pursue this approach in these notes, but the
interested reader should consult Shahidi's lectures in this volume [84].
For a balanced presentation of all three methods we recommend the book of
Gelbart and Shahidi [24]. They treat not only L-functions for GLn but L-functions
of automorphic representations of other groups as well.
We have not discussed the arithmetic theory of automorphic representations
and L-functions. For the connections with motives, we recommend the surveys of
Clozel [5] and Ramakrishnan [68].
The original version of these notes was prepared for and distributed at the
School on Automorphic Forms on GL(n) held at The Abdus Salam International
Centre for Theoretical Physics (ICTP) in Trieste, Italy, 31 July - 18 August 2000.
That version, entitled "Notes on L-functions for GLn", is available on the ICTP
web site. Since then I have used the ICTP notes in conjunction with lectures given
in the Programme on Lie Groups 2001 at the Institute of Mathematical Research of
Hong Kong University, 20 May - 26 June 2001, and most recently the IAS/PCMI
Graduate Summer School on Automorphic Forms held in Park City, Utah, 30 June- 20 July 2002. In the version of these notes presented here Lectures 1- 4 are
essentially the same as in the ICTP notes, with some corrections and updates.
Lecture 5 has been rewritten to conform with the presentation at the PCMI school.
Lecture 6 is new and was added to give an exposition of the application of the
Converse Theorem to the question of Functoriality, which was one of the points of
emphasis for the PCMI school. The Lectures as presented in these notes, of which
there are 6, do not coincide with the actual lectures I gave at Park City, where I
gave only 4 lectures. Lectures 1 and 2 here were covered in one lecture at Park City,
Lectures 3 and 4 were covered in one lecture, and Lectures 5 and 6 were alloted one
lecture each.
Most of what I know about L-functions for GLn I have learned through my
years of work with Piatetski-Shapiro. I owe him a great debt of gratitude for all
that he has taught me. For several years Piatetski-Shapiro and I have envisioned