Theorie de Lubin-Tate non-abelienne et representations elliptiques

Details Theorie de Lubin-Tate non-abelienne et representations elliptiques Theorie de Lubin-Tate non-abelienne et representations elliptiques

2006-04-03

arxiv.org/abs/math/0604020v1

Mathematics

Algebraic Geometry

54 pages

Scientific paper

10.1007/s00222-007-0044-3

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.

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