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arXiv · 2111.14532

Lefschetz number formula for Shimura varieties of Hodge type

Dong Uk Lee
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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 \cite{Kottwitz90}, 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 \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.

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Dong Uk Lee. 2021-11-29. Lefschetz number formula for Shimura varieties of Hodge type. https://arxiv.org/abs/2111.14532

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