I gave a short series of lectures (at graduate student level) about the trace formula. The lectures were 1200-1315 or so, on Wednesdays, starting 12/05/10, missing 02/06/10 because of the London-Paris Number Theory Seminar, and finishing 23/06/10 when the seminar finishes. Here's what I wrote about it at the time:
Syllabus---my main aim, which might be too optimistic, is to at least say something about how one can use the trace formula to prove theorems of Jacquet-Langlands type for automorphic forms on the group GL_2, although I will restrict to the case of compact quotient when the going gets tough. Along the way I'll give an introduction to (what little I know about) the trace formula and do some toy examples; I will also set up some of the theory of automorphic forms on GL_2 and their relations to modular forms.
References:
1) The book "Automorphic forms on GL(2)" by Jacquet and Langlands, although I will very much be picking and choosing from this, rather than working through it.
2) The first 5 or 6 sections of David Whitehouse's notes "An introduction to the trace formula", which taught me a great deal. There are typos in these notes; I told David about them a while ago but he never fixed them :-/ Try here (update: link changed to a direct link to the paper on Whitehouse's site). (BREAKING NEWS: Wed 12th May: Whitehouse tells me that he's updated the pdf fixing some typos.)
After the course, some natural next places to go are:
3) Gelbart's lectures on the non-compact case at ArXiv here.
4) Michael Harris' introductory article on endoscopy here.
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The syllabus is here.
The current state of the slides for the course is here (rescaled to fit several slides on a page). The date which I last modified the slides is shown at the top of the 1st page of that pdf file.
I can't imagine that in general people would be interested in the actual pdfs of the slides, but they are here. NOTE THAT THIS IS OVER 100 PAGES AND COUNTING! DON'T PRINT THIS OUT! PRINT OUT THE THING FROM THE LINK IN THE ABOVE PARA!
Here's an example sheet! (24/10/08). And here's another one! (14/11/08). And another! (14/11/08). And yet another! (09/12/08).