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Is there a proof of the result that Carayol proves in "Sur les representations l-adiques..." using the Langlands-Kottwitz method of comparing the Lefschetz trace formula and the Selberg trace formula? As far as I can see, the proof in Carayol's paper is done in a more case-by-case fashion...


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The Langlands-Kottwitz method, until Scholze, would only tell you about the behaviour at the primes of good reduction. Carayol's proof in the paper you reference deals with local-global at all places including the ones of bad reduction. Scholze has generalized the method, but my gut feeling (perhaps wrong) is that he might not be able to say much about monodromy just via these methods, so again one will have to get one's hands dirty via monodromy filtration etc to prove Carayol's result in full. Go and look at Scholze's papers (Peter Scholze Bonn will find him on google). – Kevin Buzzard Jun 4 '11 at 15:09
Thanks, Kevin. That clears up my confusion. – Nicolás Jun 4 '11 at 15:14
Just to add to Kevin's comment, Carayol's method (which actually goes back to a letter of Deligne to PS) is (in some sense) more systematic than you might think, e.g. it was generalized to GL_n by Harris and Taylor. Regards, Matthew – Emerton Jun 4 '11 at 17:14
Just to add to Emerton's comment: The letter from Deligne to Piatetski-Shapiro may be found on Jared Weinstein's website. – David Hansen Jun 4 '11 at 18:52
Just to add to David's comment: actually, Jared's website only contains a translation of the letter ;-) – Kevin Buzzard Jun 4 '11 at 20:16

In Langlands's Antwerp article (LNM 349) he proves local-global compatibility (what you are refererring to as "Carayol's result") in the case of principal series or special representations (so not just good reduction), using a comparison of trace formulas. He wasn't able to get the supercuspidal case in this way (and that is what led Deligne to write his letter to P.S.).

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