Integral models of Shimura varieties with parahoric level structure

Abstract

For a prime $p > 2$ , we construct integral models over $p$ for Shimura varieties with parahoric level structure, attached to Shimura data $(G,X)$ of abelian type, such that $G$ splits over a tamely ramified extension of ${\mathbf {Q}}_{\,p}$ . The local structure of these integral models is related to certain “local models”, which are defined group theoretically. Under some additional assumptions, we show that these integral models satisfy a conjecture of Kottwitz which gives an explicit description for the trace of Frobenius action on their sheaf of nearby cycles.