2021/11/29 by Dong Uk Lee, Lee, Dong Uk
Mathematics · #11G18 (Primary) 11S40 #11G25 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2111.14532
openalex publication_date 2021/11/29 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
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 \citeKottwitz90, 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 \citeLR87 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 \citeKisin17. We do not assume that the derived group is simply connected.