modeled on the theory of Grothendieck motives. Its purpose was to provide a general framework for formulating a conjectural description of the ${\bar{\mathbb{F}}_{p}}$-point set of Shimura variety which is *precise* enough to allow one to derive the Kottwitz formula from it.

Our proof of the Kottwitz formula (\[eq\_intro:Kottwitz\_formula\]) imitates these arguments of Langlands and Rapoport of deriving the Kottwitz formula from their conjecture. As such, their theory of Galois gerbs is a major ingredient in this work. Another essential ingredient is a recent work of Kisin [@Kisin17] on the aforementioned conjecture of Langlands and Rapoport. Previously, Kottwitz [@Kottwitz92] proved the formula (\[eq\_intro:Kottwitz\_formula\]) in PEL-type cases (of simple Lie type $A$ or $C$) by a method which is based on the Honda-Tate theory. This method however cannot be applied in general Hodge-type situations, and indeed our proof is different from his proof in his PEL-type cases.

In the next, we give more detailed discussion of the idea of our proof. We begin by introducing the work of Langlands and Rapoport [@LR87].

### Langlands-Rapoport conjecture

The conjecture of Langlands and Rapoport, which was stated in [@LR87] and a significant progress towards which was recently made by Kisin [@Kisin17], aims to give a 