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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
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