Galois gerbs and Lefschetz number formula for Shimura varieties of Hodge type

TL;DR

This paper proves a conjectured Lefschetz number formula for Shimura varieties of Hodge type using Galois gerb theory, extending previous results without assuming simply connected derived groups.

Contribution

It establishes the Lefschetz number formula for a broad class of Shimura varieties of Hodge type, verifying Kottwitz's conjecture and extending Langlands-Rapoport results to non-simply connected cases.

Findings

Proved the Lefschetz number formula for Shimura varieties of Hodge type.

Verified Kottwitz's conjecture on stabilization of the formula.

Extended Langlands-Rapoport results to more general cases at parahoric levels.

Abstract

For any Shimura variety of Hodge type with hyperspecial level at a prime $p$ and a lisse sheaf on it, we prove a formula, conjectured by Kottwitz \cite{Kottwitz90}, for the Lefschetz number of an arbitrary Frobenius-twisted Hecke correspondence acting on the compactly supported \'etale cohomology and verify another conjecture of Kottwitz \cite{Kottwitz90} on the stabilization of that formula. The main ingredients of our proof of the formula are a recent work of Kisin \cite{Kisin17} on Langlands-Rapoport conjecture and the theory of Galois gerbs developed by Langlands and Rapoport \cite{LR87}. Especially, we use the Galois gerb theory to establish an effectivity criterion of Kottwitz triple, and mimic the arguments of Langlands and Rapoport of deriving the Kottwitz formula from their conjectural description of the $\Fpb$-point set of Shimura variety (Langlands-Rapoport conjecture). We do not assume that the derived group is simply connected, and also obtain partial results at (special) parahoric levels under some condition at $p$. For that, in the first part of our work we extend the results of Langlands and Rapoport to such general cases.