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My question is about the correct order to read the papers by Arthur on trace formula. Arthur's papers are perfectly well-written, but maybe a little too hard for me to go through easily.

I would like to start with unrefined trace formula, but there are bunches of papers already. There is a paper on trace formula for rank-1 groups (1974), and there are two papers on trace formula for general reductive groups (1978 & 1980). Also there is a paper in Corvallis Proceedings (1979), and a notes on trace formula (2005) which has become the standard introductory notes on this subject. I am seeking for any advice on how to read these references.

I wish to at least go through some details to feel how the formula is proved. Now I have read the notes by Whitehouse on trace formula for $\operatorname{GL}(2)$ (which covers a great many details about computation), and read a small part of Arthur's paper on rank-1 groups (1974). My question is that whether it is better to take 1974 paper first to gain more intuition, or instead start with the more general setting of reductive groups (like 1978 & 1980 papers), where maybe things are actually more clarified.

Any recommendation of other references is also sincerely welcomed. I am not aware of any other introductory references on trace formula on general reductive groups.

Thank you very much for reading this lengthy question! I am totally new to this community, so you are welcomed to point out any mistake.

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  • $\begingroup$ What is your goal? If your goal is to understand all of Arthur's work on the trace formula, that's a serious undertaking. $\endgroup$
    – Kimball
    May 6, 2021 at 14:44
  • $\begingroup$ @Kimball My aim is two-fold. One is to understand some "modern" side of trace formula (vaguely speaking), which might be useful for my future research. Another more practical aim is to write my bachelor thesis actually(๑•﹏•). So I hope it is based on more serious paper rather than introductory notes. $\endgroup$
    – Yang Yang
    May 6, 2021 at 15:39
  • $\begingroup$ The theory is significantly more complicated for general reductive groups, so if you just want to understand the GL(2) case I would suggest focusing on that, but in this case I wouldn't call it Arthur's trace formula. For GL(2), in addition to Knapp's notes, I would recommend Gelbart's book on GL(2) and his lecture notes on the trace formula. Knightly-Li may also be useful. $\endgroup$
    – Kimball
    May 6, 2021 at 17:17
  • $\begingroup$ @Kimball Thank you very much for your comment! I will go check these notes. I have to say my interest goes beyond GL(2), and I may also try the notes by Arthur. $\endgroup$
    – Yang Yang
    May 7, 2021 at 2:26

2 Answers 2

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One of the best places to learn about trace formula, other than David Whitehouse's wonderful notes, are the notes by Erez Lapid Introductory notes on the trace formula.

Arthur's trace formula relies on Langlands's work on Eisenstein series and Spectral decomposition from 1962–64. This stuff is extremely difficult for general reductive groups.

It would be near insanity to directly go to Arthur's papers or notes without mastering the spectral decomposition of automorphic forms on GL(2) and the Selberg trace formula. It is not wise to learn trace formula without learning about spectral decomposition.

For the general case, Borel's notes Automorphic forms on reductive groups are a reasonable beginning.

My advisor Paul Garrett's book Modern analysis of automorphic forms by example is, of course, my favorite place. They treat several examples, but do not do any higher rank case completely.

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I agree with @kimball's suggestions. The first thing to do is to get a handle on what it looks like for GL(2), then upgrade to general $G$ only as needed. For a basic introductory references, starting with Knapp's Theoretical Aspects of the Trace Formula for GL(2) and Gelbart's Lectures on the Arthur–Selberg Trace Formula. Then depending on your needs, I find Arthur's own An Introduction to the Trace Formula an excellent guide to his own work for the general case.

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    $\begingroup$ Thanks for adding the links! @LSpice $\endgroup$
    – Tian An
    Sep 9, 2022 at 16:05

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