Foreword

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

simple cases.

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

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