Lemme fondamental et endoscopie, une approche géométrique  [ Fundamental lemma and endoscopy, a geometric approach ]
Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque no. 307  (2006), Talk no. 940, p. 71-112

The “principle of functoriality”, as conjectured by Langlands by the late sixties, is a very suggestive way to express the deep links between automorphic forms, arithmetic and algebraic geometry. Despite its rather simple formulation, the few techniques which are used to try and solve it are quite involved. One of them, based on the trace formula has been faced for 25 years with a conjecture of harmonical analysis on p-adic groups: the “fundamental lemma”. We will survey a recent geometric approach by Laumon and Ngô, which has already lead to a proof of this “lemma”for unitary groups.

Le “principe de fonctorialité”, conjecturé par Langlands à la fin des années 60, est un moyen remarquablement synthétique d’unifier et exprimer certains liens profonds entre formes automorphes, arithmétique et géométrie algébrique. Son apparente simplicité contraste fortement avec la difficulté des techniques utilisées pour l’aborder. Parmi celles-ci, la stabilisation de la formule des traces d’Arthur-Selberg bute depuis 25 ans sur une conjecture d’analyse harmonique sur des groupes p-adiques : le “lemme fondamental”. Nous présenterons une approche géométrique de ce “lemme”, par Laumon et Ngô, qui les a déjà conduits à une preuve dans le cas des groupes unitaires.

Classification:  14D20,  20G30
Keywords: automorphic, trace formula, endoscopy, fundamental lemma G-torsors, Hitchin pairs
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     author = {Dat, Jean-Fran\c cois},
     title = {Lemme fondamental et endoscopie, une approche g\'eom\'etrique},
     booktitle = {S\'eminaire Bourbaki : volume 2004/2005, expos\'es 938-951},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {307},
     year = {2006},
     note = {talk:940},
     pages = {71-112},
     zbl = {1175.22013},
     mrnumber = {2296416},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2004-2005__47__71_0}
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Dat, Jean-François. Lemme fondamental et endoscopie, une approche géométrique, in Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Talk no. 940, pp. 71-112. http://www.numdam.org/item/SB_2004-2005__47__71_0/

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