Théorie de Lubin-Tate non-abélienne et représentations elliptiques

• Autores: Jean-François Dat
• Localización: Inventiones mathematicae, ISSN 0020-9910, Vol. 169, Nº. 1, 2007, págs. 75-152
• Idioma: inglés
• DOI: 10.1007/s00222-007-0044-3
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• Resumen

  • Non abelian Lubin-Tate theory studies the cohomology of some moduli spaces for p-divisible groups, the broadest definition of which is due to Rapoport-Zink, aiming both at providing explicit realizations of local Langlands functoriality and at studying bad reduction of Shimura varieties. In this paper we consider the most famous examples ; the so-called Drinfeld and Lubin-Tate towers. In the Lubin-Tate case, Harris and Taylor proved that the supercuspidal part of the cohomology 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 correspondence for 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. This completes the computation of local L-factors of some unitary Shimura varieties.