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

摘要

In the geometric situation of the simple Shimura varieties of Kottwitzstudied in Harris and Taylor's book, we describe the monodromy filtration ofthe vanishing cycles complex and the spectral sequence associated to it. Weprove in particular that this filtration coincides with the weight one up toshift. Thanks to the Berkovich-Fargues' theorem, we deduce the description ofthe local monodromy filtration of the Deligne-Carayol model. In application, weobtain the description of the local components of cohomological automorphicrepresentations of certains unitary groups, a global Jacquet-Langlandscorrespondence between two such groups and a proof of the weight-monodromyconjecture for the cohomology of these Shimura varieties.