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.
Categories: math.NT math.AG