Kottwitz-Rapoport and p-rank strata in the reduction of Shimura varieties of PEL type

TL;DR

This paper investigates the structure of Shimura varieties of PEL type with Iwahori level, establishing the constancy of p-rank on KR strata, providing explicit formulas, and analyzing the density and dimension of special loci.

Contribution

It generalizes previous results by proving p-rank constancy on KR strata and deriving explicit formulas in symplectic and unitary cases.

Findings

p-rank is constant on each KR stratum

Explicit formulas for p-rank in symplectic and unitary cases

Results on density of the ordinary locus and dimension of p-rank 0 locus

Abstract

We study the reduction of certain integral models of Shimura varieties of PEL type with Iwahori level structure. On these spaces we have the Kottwitz-Rapoport and the $p$-rank stratification. We show that the $p$-rank is constant on a KR stratum, generalizing a result of Ng\^o and Genestier. We prove an abstract, uniform formula for the $p$-rank on a KR stratum. In the symplectic and in the unitary case we derive explicit formulas for its value. We apply these formulas to the question of the density of the ordinary locus and to the question of the dimension of the $p$-rank 0 locus.