Lefschetz number formula for Shimura varieties of Hodge type

TL;DR

This paper proves a formula for Lefschetz numbers of Frobenius-twisted Hecke correspondences on Shimura varieties of Hodge type, extending previous conjectures and generalizing key theorems without assuming simply connected derived groups.

Contribution

It provides a proof of Kottwitz's conjectured Lefschetz number formula for Shimura varieties of Hodge type, using geometric methods and generalizing the Honda-Tate theorem.

Findings

Proved the Lefschetz number formula for Frobenius-twisted Hecke correspondences.

Generalized Honda-Tate theorem in the context of Shimura varieties.

Fixed an error in Kisin's work on related topics.

Abstract

For any Shimura variety of Hodge type with hyperspecial level at a prime $p$ and automorphic lisse sheaf on it, we prove a formula, conjectured by Kottwitz \cite{Kottwitz90}, for the Lefschetz numbers of Frobenius-twisted Hecke correspondences acting on the compactly supported \'etale cohomology. Our proof is an adaptation of the arguments of Langlands and Rapoport \cite{LR87} of deriving the Kottwitz's formula from their conjectural description of the set of mod-$p$ points of Shimura variety (Langlands-Rapoport conjecture), which replaces their Galois gerb theoretic arguments by more geometric ones. We also prove a generalization of Honda-Tate theorem in the context of Shimura varieties and fix an error in Kisin's work \cite{Kisin17}. We do not assume that the derived group is simply connected.