Lefschetz number formula for Shimura varieties of Hodge type

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 [Kot90], for the Lefschetz numbers of Frobenius-twisted Hecke correspondences acting on the compactly supported étale cohomology. Our proof is an adaptation of the arguments of Langlands and Rapoport [LR87] of deriving the Kottwitz’s formula from their conjectural description of the set of mod-$p$ points of Shimura variety (Langlands–Rapoport conjecture), but replaces their Galois gerb theoretic arguments by more standard group-theoretic ones, using Kisin’s geometric work [Kis17]. We also prove a generalization of Honda–Tate theorem in the context of Shimura varieties and fix an error in the Kisin’s work. We do not assume that the derived group is simply connected.

Keywords

Shimura varieties, Lefschetz number formula, Langlands-Kottwitz method

2010 Mathematics Subject Classification

11G18, 11G25, 11S40

Full Text (PDF format)

Received 1 December 2021

Accepted 5 September 2023

Published 7 August 2024