Lectures on the Arthur–Selberg Trace Formula
Steve Gelbart
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 ${\rm GL}(2)$; 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.
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 $J_\chi(f)$
Lecture VIII. Jacquet's Relative Trace Formula
Lecture IX. Some Applications of Paley–Wiener, and Concluding Remarks