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Models of Shimura varieties in mixed characteristic. (English) Zbl 0864.14015
Let $$X$$ be a principally polarized abelian scheme over the ring $$W$$ of Witt vectors of the algebraic closure $$\overline\mathbb{F}_p$$ of the finite field $$\mathbb{F}_p$$. Let $$T_{\text{DR}} (X/W)$$ be a tensor product of de Rham cohomology groups of $$X$$ and their duals twisted to make its weight zero, and choose a class $$\xi\in \text{Fil}^0T_{DR} (X/W)$$. Let $$X_0$$ be the special fiber of $$X$$, and let $${\mathcal N}$$ be the subgroup of the formal deformation space of $$X_0$$ parametrizing principally polarized liftings $$Y/W$$ of $$X_0$$ such that the image of $$\xi$$ in $$T_{\text{DR}} (X/W)$$ is a Tate class lying in $$\text{Fil}^0T_{\text{DR}} (X/W)$$. In this paper the author shows that one can find such an $${\mathcal N}$$ with properties analogous to those of a Shimura variety, and applies this result to the study of models of Shimura varieties in mixed characteristic.

##### MSC:
 14G35 Modular and Shimura varieties 14K15 Arithmetic ground fields for abelian varieties 11G18 Arithmetic aspects of modular and Shimura varieties