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.


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