Jean-François Dat

Théorie de Lubin-Tate non-abélienne et représentations elliptiques

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 GLd 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.