Aspects géométriques du Lemme Fondamental de Langlands-Shelstad
Gérard LaumonUniversité Paris-Sud, Orsay, France
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In order to compute the Hasse–Weil zeta functions of the Shimura varieties or to establish some cases of the Langlands functoriality one first needs to stabilize the Arthur–Selberg trace formula. This stabilization can be done only if some combinatorial identities between orbital integrals over p-adic reductive groups are satisfied. The series of these conjectural identities form the so-called “Fundamental Lemma”. We present here some key points of the geometric approaches which have been used by Goresky, Kottwitz and MacPherson on the one hand and Ngô Bao Châu and myself one the other hand, to prove some cases of the Fundamental Lemma.