The twenty-fifth AMS Summer Research Institute was devoted to automorphic
forms, representations and L-functions. It was held at Oregon State University,
Corvallis, from July 11 to August 5, 1977, and was financed by a grant from the
National Science Foundation. The Organizing Committee consisted of A. Borel,
W. Casselman (cochairmen:), P. Deligne, H. Jacquet, R. P. Langlands, and J. Tate.
The papers in this volume consist of the Notes of the Institute, mostly in revised
form, and of a few papers written later.
A main goal of the Institute was the discussion of the L-functions attached to
automorphic forms on, or automorphic representations of, reductive groups, the
local and global problems pertaining to them, and of their relations with the L-
functions of algebraic number theory and algebraic geometry, such as Artin L-
functions and Hasse-Weil zeta functions. This broad topic, which goes back to E.
Hecke, C. L. Siegel and others, has undergone in the last few years and is undergo-
ing even now a considerable development, in part through the systematic use of
infinite dimensional representations, in the framework of adelic groups. This devel-
opment draws on techniques from several areas, some of rather difficult access.
Therefore, besides seminars and lectures on recent and current work and open
problems, the Institute also featured lectures (and even series of lectures) of a more
introductory character, including background material on reductive groups, their
representations, number theory, as well as an extensive treatment of some relatively
The papers in this volume are divided into four main sections, reflecting to some
extent the nature of the prerequisites. I is devoted to the structure of reductive
groups and infinite dimensional representations of reductive groups over local
fields. Five of the papers supply some basic background material, while the others
are concerned with recent developments. II is concerned with automorphic forms
and automorphic representations, with emphasis on the analytic theory. The first
four papers discuss some basic facts and definitions pertaining to those, and the
passage from one to the other. Two papers are devoted to Eisenstein series and the
trace formula, first for GL2 and there in more general cases. In fact, the trace
formula and orbital integrals turned out to be recurrent themes for the whole
Institute and are featured in several papers in the other sections as well. The main
theme of the last four papers is the restriction of the oscillator representation of the
metaplectic group to dual reductive pairs of subgroups, first in general and then in
more special cases.
Ill begins with the background material on number theory, chiefly on Weil
groups and their L-functions. It then turns to the L-functions attached to automor-
phic representations, various ways to construct them, their (conjectured or proven)
properties and local and global problems pertaining to them. The remaining papers
are mostly devoted to the base change problem for GL2 and its applications to the
proof of holomorphy of certain nonabelian Artin series.
Finally, IV relates automorphic representations and arithmetical algebraic
geometry. Over function fields, it gives an introduction to the work of Drinfeld for