LECTURES ON THE ARTHUR–SELBERG TRACE FORMULA

Steve Gelbart

Abstract. These are Notes prepared for nine lectures given at the Mathematical Sciences Research Institute, MSRI, Berkeley during the period January–March 1995. It is a pleasant duty to record here my gratitude to MSRI, and its staff, for making possible this 1994–95 Special Year in Automorphic Forms, and for providing such a setting for work.

The purpose of these Notes is to describe the contents of Arthur’s earlier, foundational papers on the trace formula. In keeping with the introductory nature of the lectures, we have sometimes illustrated the ideas of Arthur’s general theory by applying them in detail to the case of GL(2); we have also included a few lectures on the “simple trace formula” (and its applications), and on Jacquet’s relative trace formula.

The TEX preparation of these Notes I owe to Wendy McKay, who patiently and professionally transformed my messy scrawl into something readable; her expertise, and good cheer under the pressure of weekly deadlines, is something I shall not soon forget.

I wish to thank the auditors of these lectures for their interest, and , J. Bernstein, D. Goldberg, E. Lapid, C. Rader, S. Rallis, A. Reznikov, and D. Soudry, for their helpful suggestions and explanations. Finally, I wish to thank J. Arthur, H. Jacquet, and J. Rogawski for many tutorials on these and related topics over the past year; I hope they do not mind seeing some of their comments reappear in these Notes.

Table of Contents

Lecture I. Introduction to the Trace Formula

Lecture II. Arthur’s Modified Kernels I: The Geometric Terms

Lecture III. Arthur’s Modified Kernels II: The Spectral Terms

Lecture IV. More Explicit Forms of the Trace Formula

Lecture V. Simple Forms of the Trace Formula

Lecture VI. Applications of the Trace Formula

Lecture VII. (G, M)-Families and the Spectral J_{χ}(f)

Lecture VIII. Jacquet’s Relative Trace Formula

Lecture IX. Some Applications of Paley–Wiener, and Concluding Remarks

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