# Introduction to the trace formula for people outside number theory

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.

• 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

There is the paper Theoretical aspects of the trace formula for $GL_2(2)$ by Knapp (available here, 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.