The basic stratum of some simple Shimura varieties

TL;DR

This paper establishes a relation between the l-adic cohomology of the basic stratum of certain Shimura varieties and their complex counterparts, providing explicit point count formulas over finite fields.

Contribution

It introduces explicit formulas connecting the cohomology of the basic stratum of PEL-type Shimura varieties with their complex versions under simplifying hypotheses.

Findings

Derived explicit formulas for point counts over finite fields.

Linked l-adic cohomology of basic strata to complex Shimura variety cohomology.

Utilized trace formula and Kottwitz's formula for computations.

Abstract

Under simplifying hypotheses we prove a relation between the l-adic cohomology of the basic stratum of a Shimura variety of PEL-type modulo a prime of good reduction of the reflex field and the cohomology of the complex Shimura variety. In particular we obtain explicit formulas for the number of points in the basic stratum over finite fields. We obtain our results using the trace formula and truncation of the formula of Kottwitz for the number of points on a Shimura variety over a finite fields.