Comparaison entre cohomologie cristalline et cohomologie \'etale $p$-adique sur certaines vari\'et\'es de Shimura

Abstract

Let $X$ be an integral model at a prime $p$ of a Shimura variety of PEL type having good reduction, associated to a reductive group $G$. To $\mathbb{Z}_p$ reprsententations of the group $G$ can be associated two kinds of sheaves : crystals on the special fiber of $X$, and locally constant \'etale sheaves on the generic fiber. We establish a comparison between the cohomology of these two kinds of sheaves.