On the cohomology of certain PEL-type Shimura varieties

Abstract

In this article we study the local geometry at a prime $p$ of PEL-type Shimura varieties for which there is a hyperspecial level subgroup. We consider the Newton polygon stratification of the special fiber at $p$ of Shimura varieties and show that each Newton polygon stratum can be described in terms of the products of the reduced fibers of the corresponding PEL-type Rapoport-Zink spaces with certain smooth varieties (which we call Igusa varieties) and of the action on them of a $p$-adic group that depends on the stratum. We then extend our construction to characteristic zero and, in the case of bad reduction at $p$, use it to compare the vanishing cycle sheaves of the Shimura varieties to those of the Rapoport-Zink spaces. As a result of this analysis, in the case of proper Shimura varieties 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 for any prime $l\ne p$.