Theorie de Lubin-Tate non-abelienne et representations elliptiques

Jean-Francois Dat (1)

1 LAGA - Laboratoire Analyse, Géométrie et Applications

Harris and Taylor proved that the supercuspidal part of the cohomology of the Lubin-Tate tower realizes both the local Langlands and Jacquet-Langlands correspondences, as conjectured by Carayol. Recently, Boyer computed the remaining part of the cohomology and exhibited two defects : first, the representations of GL_d which appear are of a very particular and restrictive form ; second, the Langlands correspondence is not realized anymore. In this paper, we study the cohomology complex in a suitable equivariant derived category, and show how it encodes Langlands correspondance for all elliptic representations. Then we transfer this result to the Drinfeld tower via an enhancement of a theorem of Faltings due to Fargues. We deduce that Deligne's weight-monodromy conjecture is true for varieties uniformized by Drinfeld's coverings of his symmetric spaces.

Mots clés: correspondance de Langlands; monodromie; espaces de Lubin-Tate

Domaines: Géométrie algébrique [math.AG]; Théorie des nombres [math.NT]; Théorie des représentations [math.RT]

Jean-Francois Dat. Theorie de Lubin-Tate non-abelienne et representations elliptiques. 2006. ⟨hal-00022116⟩