On certain unitary group Shimura varieties. Variétés de Shimura, espaces de Rapoport-Zink et correspondances de Langlands locales

Abstract

In this paper, we study the local geometry at a prime p of a certain class of (PEL) type Shimura varieties. We begin by studying the Newton polygon stratification of the special fiber of a Shimura variety with good reduction at p. Each stratum can be described in terms of the products of the reduced fiber of the corresponding Rapoport-Zink space with some smooth varieties (we call the Igusa varieties), and of the action on them of a certain p-adic group T_α, which depends on the stratum. (The definition of the Igusa varieties in this context is based upon a result of Zink on the slope filtration of a Barsotti-Tate group and on the notion of Oort’s foliation.) In particular, we show that it is possible to compute the étale cohomology with compact supports of the Newton polygon strata, in terms of the étale cohomology with compact supports of the Igusa varieties and the Rapoport-Zink spaces, and of the group homology of T_α. Further more, we are able to extend Zariski locally the above constructions to characteristic zero and obtain an analoguous description for the étale cohomology of the Shimura varieties in both the cases of good and bad reduction at p. As a result of this analysis, we obtain a description of the l-adic cohomology of the Shimura varieties, in terms of the l-adic cohomology with compact supports of the Igusa varieties and of the Rapoport-Zink spaces.