# Lectures on the Arthur-Selberg Trace Formula

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*Stephen Gelbart*

The Arthur-Selberg trace formula is an equality between
two kinds of traces: the geometric terms given by the conjugacy classes
of a group and the spectral terms given by the induced
representations. In general, these terms require a truncation in order
to converge, which leads to an equality of truncated kernels. The
formulas are difficult in general and even the case of \(GL\)(2)
is nontrivial. The book gives proof of Arthur's trace formula of
the 1970s and 1980s, with special attention given to \(GL\)(2).
The problem is that when the truncated terms converge, they are also
shown to be polynomial in the truncation variable and expressed
as “weighted” orbital and “weighted”
characters. In some important cases the trace formula takes on a simple
form over \(G\). The author gives some examples of this, and
also some examples of Jacquet's relative trace formula.

This work offers for the first time a simultaneous treatment of
a general group with the case of \(GL\)(2). It also treats
the trace formula with the example of Jacquet's relative formula.

Features:

- Discusses why the terms of the geometric and spectral type must be truncated, and why the resulting truncations are polynomials in the truncation of value \(T\).
- Brings into play the significant tool of (\(G, M\)) families and how the theory of Paley-Weiner is applied.
- Explains why the truncation formula reduces to a simple formula involving only the elliptic terms on the geometric sides with the representations appearing cuspidally on the spectral side (applies to Tamagawa numbers).
- Outlines Jacquet's trace formula and shows how it works for \(GL\)(2).

#### Reviews & Endorsements

For graduate students or for seasoned researchers wishing to expand their repertoire, this book will be invaluable. Many basic facts … are collected in one convenient reference.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Lectures on the Arthur-Selberg Trace Formula

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface ix10 free
- Lecture I. Introduction to the Trace Formula 112 free
- Lecture II. Arthur's Modified Kernels I: The Geometric Terms 718 free
- Lecture III. Arthur's Modified Kernels II: The Spectral Terms 1728
- Lecture IV. More Explicit Forms of the Trace Formula 3142
- Lecture V. Simple Forms of the Trace Formula 4556
- Lecture VI. Applications of the Trace Formula 5364
- Lecture VII. (G,M)-Families and the Spectral J[sub(x)](f) 6374
- Lecture VIII. Jacquet's Relative Trace Formula 7586
- Lecture IX. Applications of Paley–Wiener, and Concluding Remarks 8798
- References 97108
- Back Cover Back Cover1111