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I am looking for references on the trace formula, by which I mean the Selberg trace formula and its successor the Arthur-Selberg trace formula.

I am aware that there are "standard references" on the subject, and I have looked at some of those standard references. However, I am not necessarily looking for a "standard reference". I am looking for a reference that doesn't actually assume much number-theoretic background.

My own background is in complex algebraic geometry. I would like to have a reference that would be readable to someone who is comfortable with portions of either Griffiths-Harris or Hartshorne and with basic representation theory, but who may not have had much prior exposure to algebraic number theory.

Any advice would be appreciated it.

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I think Terras' book Harmonic Analysis on Symmetric Spaces devotes space to Selberg's trace formula. I can't give you a page reference because my order hasn't arrived yet. A review published in the Bull. AMS is here. This book approaches matters from an analytic background, which you may or may not be comfortable. –  Neal May 13 '13 at 5:21

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There is the paper Theoretical aspects of the trace formula for $GL_2(2)$ by Knapp (available here, for example, for example). It doesn't use much number theory, but requires familiarity with some analysis and rep'n theory. (It gets a bit adelic towards the end.)

The complications in the trace formula have to with the fact that congruence subgroups of $SL(2,\mathbb Z)$ don't give compact quotients of the upper half-plane, and so one has to deal with the issue that the traces you are trying to compute receive divergent contributions from the boundary.

The case of compact quotients is much simpler, and is treated at the beginning of section 4 of the article. You might want to begin there, since this gives the basic idea of the trace formula as non-abelian Fourier analysis.

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