TITLE: Carayol via the trace formula
QUESTION [10 upvotes]: Hi,
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...
Thanks

REPLY [6 votes]: 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.).