of the main results
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The main results of this work are the following two descriptions of the Lefschetz numbers of Frobenius-twisted Hecke correspondences on the compactly supported cohomology of Shimura varieties of Hodge type with hyperspecial level, which were conjectured by Kottwitz [@Kottwitz90], (3.1) and Thm. 7.2. The first one is a consequence of the second one.

Fix two distinct primes $p$, $l$ of ${\mathbb Q}$. Let $(G,X)$ be a Shimura datum of Hodge type, $K_p$ a hyperspecial subgroup of $G({{\mathbb Q}_p})$, $K^p$ a (sufficiently small) compact open subgroup of $G({\mathbb A}_f^p)$; put $K=K_pK^p$. Let $Sh_K(G,X)$ be (the canonical model over the reflex field $E(G,X)$ of) the associated Shimura variety, and ${\mathscr{F}}_K$ a $\lambda$-adic lisse sheaf on it defined by a finite-dimensional algebraic representation $\xi$ of $G$ over a number field $L$ and choice of a place $\lambda$ of $L$ over $l$. For $i\in{\mathbb N}_{\geq0}$, let $H_c^i(Sh_K(G,X)_{{\overline{\mathbb Q}}},{\mathscr{F}}_K)$ be the compactly supported cohomology of ${\mathscr{F}}_K$. Let $\wp$ be a prime of $E=E(G,X)$ above $p$ and $\Phi\in {\mathrm{Gal}}({\overline{\mathbb{Q}}_p}/E_{\wp})$ be a geometric Frobenius for an embedding $E_{\wp}\hookrightarrow {\overline{\mathbb{Q}}_p}$. It is known [@Kisin10], [@Vasiu99] that $Sh_K(G,X)$ has a canonical smooth integral model over the ring of integers $\mathcal{O}_{E_{\wp}}$ and ${\mathscr{F}}_K$ extends over 