Lefschetz number formula for Shimura varieties of Hodge type
Abstract
For any Shimura variety of Hodge type with hyperspecial level at a prime 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- 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.
Keywords
Cite
@article{arxiv.2111.14532,
title = {Lefschetz number formula for Shimura varieties of Hodge type},
author = {Dong Uk Lee},
journal= {arXiv preprint arXiv:2111.14532},
year = {2021}
}
Comments
This is a minor revision of (the first part of) our previous work with the same title (submitted in 2018, July). The latter was itself a revision of another earlier work (arXiv:1801.03057v2), giving a Galois-gerb-free new proof