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I am particularly interested in the comparison of the trace formula part of Jacquet-Langlands. But I found the original text hard to read.

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Take a look at Jacquet and Gelbart article in Corvallis. They give a very nice account of the trace formula part of Jacquet-Langlands. –  Aakumadula Feb 16 '13 at 2:02
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A nice summary of the Jacquet-Langlands Lecture Notes, is Alain Robert 's Bourbaki seminar archive.numdam.org/ARCHIVE/SB/SB_1971-1972__14_/… –  Alain Valette Feb 16 '13 at 22:58
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How about Gelbart's book "Automorphic Forms on Adele Groups"? Especially section 10 where he explains the comparison of the trace formulas. –  Judith Ludwig Feb 22 '13 at 10:15
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2 Answers

I think that you may like the notes by Ioan Badulescu: see http://www-math.univ-poitiers.fr/~badulesc/pub/JLtata1.pdf

They are a 20 pages modern essentially complete exposition of the proof of Jacquet-Langlands not only for $Gl_2$ but for higher higher rank linear groups and their inner form as well.

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@Joel: thanks for the notes. I am at the Tata Institute but I did not know about them! I looked through the notes. They treat only the local correspondence; not the global one. I believe that most of the last chapter of Jacquet-Langlands is about the $global$ correspondence. Of course, the "local" statement uses some global ideas, but I think these notes do not treat completely the global case. –  Aakumadula Feb 17 '13 at 5:19
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I suggest Gelbart Lecture notes Arthur trace formula in addition to Gelbart Jacquet (Aakumadula`s reference).

Here is a site with references: http://www.charlesli.org/math/trace.html

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The LN only treat the number field case though, where as JL consider arbitrary global fields. –  plusepsilon.de Jul 3 '13 at 12:10
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