Summer School entitled “Harmonic Analysis, the Trace Formula, and Shimura

The main goal of the School was to introduce graduate students and young mathematicians to three broad and interrelated areas in the theory of automorphic forms. Much of the volume is comprised of the articles of Arthur, Kottwitz, and Milne. Although these articles are based on lectures given at the school, the authors have chosen to go well beyond what was discussed there, in order to provide both a sense of the underlying structure of the subject and a working knowledge of some of its techniques. They were written to be self-contained in some places, and to be used in conjunction with given references in others. We hope the volume will convey the depth and beauty of this challenging field, in which there yet remains so much to be discovered—perhaps some of it by you, the reader! The theory of automorphic forms is formulated in terms of reductive algebraic groups. This is sometimes a serious obstacle for mathematicians whose background does not include Lie groups and Lie algebras. The monograph is by no means intended to exclude such mathematicians, even though the theory of reductive groups was an informal prerequisite for the Summer School. Some modest familiarity with the language of algebraic groups is often sufficient, at least to get started. For this reason, we have generally resisted the temptation to work with specific matrix groups. The short article of Murnaghan contains a summary of some of the basic properties of reductive algebraic groups that are used elsewhere in the monograph. Much of the modern theory of automorphic forms is governed by two fundamental problems that are at the heart of the Langlands program. One is Lang-lands' principle of functoriality. The other is the general analogue of the Shimura-Taniyama-Weil conjecture on modular elliptic curves. (See [A] and [L, §2].) These problems are among the deepest questions in mathematics. It is premature to try to guess what various techniques will play a role in their ultimate resolution. However , the trace formula and the theory of Shimura varieties are both likely to be an essential part of the story. They have already been used to establish significant special cases. The trace formula has perhaps been more closely identified with the first problem. Special cases of functoriality arise naturally from the conjectural theory of endoscopy, in which a comparison of trace formulas would be used to characterize the internal structure of the …