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For any Shimura variety of Hodge type with hyperspecial level at a prime $p$and a lisse sheaf on it, we prove a formula, conjectured by Kottwitz\cite{Kottwitz90}, for the Lefschetz number of an arbitrary Frobenius-twistedHecke correspondence acting on the compactly supported \'etale cohomology andverify another conjecture of Kottwitz \cite{Kottwitz90} on the stabilization ofthat formula. The main ingredients of our proof of the formula are a recentwork of Kisin \cite{Kisin17} on Langlands-Rapoport conjecture and the theory ofGalois gerbs developed by Langlands and Rapoport \cite{LR87}. Especially, weuse the Galois gerb theory to establish an effectivity criterion of Kottwitztriple, and mimic the arguments of Langlands and Rapoport of deriving theKottwitz formula from their conjectural description of the $\Fpb$-point set ofShimura variety (Langlands-Rapoport conjecture). We do not assume that thederived group is simply connected, and also obtain partial results at (special)parahoric levels under some condition at $p$. For that, in the first part ofour work we extend the results of Langlands and Rapoport to such general cases.

Galois gerbs and Lefschetz number formula for Shimura varieties of Hodge type