# Harmonic Analysis, the Trace Formula, and Shimura Varieties

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*James Arthur; David Ellwood; Robert Kottwitz*

A co-publication of the AMS and Clay Mathematics Institute

The modern theory of automorphic forms, embodied in what has
come to be known as the Langlands program, is an extraordinary
unifying force in mathematics. It proposes fundamental relations that
tie arithmetic information from number theory and algebraic geometry
with analytic information from harmonic analysis and group
representations. These "reciprocity laws", conjectured by Langlands,
are still largely unproved. However, their capacity to unite large
areas of mathematics insures that they will be a central area of study
for years to come.

The goal of this volume is to provide an entry point into this
exciting and challenging field. It is directed, on the one hand, at
graduate students and professional mathematicians who would like to
work in the area. The longer articles in particular represent an
attempt to enable a reader to master some of the more difficult
techniques. On the other hand, the book will also be useful to
mathematicians who would like simply to understand something of the
subject. They will be able to consult the expository portions of the
various articles.

The volume is centered around the trace formula and Shimura
varieties. These areas are at the heart of the subject, but they have
been especially difficult to learn because of a lack of expository
material. The volume aims to rectify the problem. It is based on the
courses given at the 2003 Clay Mathematics Institute Summer
School. However, many of the articles have been expanded into
comprehensive introductions, either to the trace formula or the theory
of Shimura varieties, or to some aspect of the interplay and
application of the two areas.

This book is suitable for independent study.

Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).

#### Readership

Graduate students and research mathematicians interested in number theory, automorphic forms, and group representations.