The Langlands-Kottwitz method and deformation spaces of $p$-divisible groups

TL;DR

This paper extends Kottwitz's results on Shimura varieties over finite fields to bad reduction cases, using geometric test functions derived from deformation spaces of p-divisible groups.

Contribution

It introduces a geometric approach to defining test functions for twisted orbital integrals in bad reduction scenarios, expanding the Langlands-Kottwitz method.

Findings

Generalization of Kottwitz's point counting to bad reduction cases

Definition of test functions via deformation spaces of p-divisible groups

Enhanced understanding of Shimura varieties over finite fields

Abstract

We extend the results of Kottwitz on points of Shimura varieties over finite fields to cases of bad reduction. The "test function" whose twisted orbital integrals appear in the final expression is defined geometrically using deformation spaces of p-divisible groups.