Monodromy of the perverse sheaf of vanishing cycles of some simple Shimura varieties and applications

Abstract

In the geometric situation of the simple Shimura varieties of Kottwitz studied in Harris and Taylor's book, we describe the monodromy filtration of the vanishing cycles complex and the spectral sequence associated to it. We prove in particular that this filtration coincides with the weight one up to shift. Thanks to the Berkovich-Fargues' theorem, we deduce the description of the local monodromy filtration of the Deligne-Carayol model. In application, we obtain the description of the local components of cohomological automorphic representations of certains unitary groups, a global Jacquet-Langlands correspondence between two such groups and a proof of the weight-monodromy conjecture for the cohomology of these Shimura varieties.