Lefschetz number formula for Shimura varieties of Hodge type

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 [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.