Generalized Serre--Tate Ordinary Theory

We study a generalization of Serre–Tate theory of ordinary abelian varieties and their deformation spaces. This generalization deals with abelian varieties equipped with additional structures. The additional structures can be not only an action of a semisimple algebra and a polarization, but more generally the data given by some crystalline Hodge cycles'' (a $p$-adic version of a Hodge cycle in the sense of motives). Compared to Serre--Tate ordinary theory, new phenomena appear in this generalized context. We give an application of this theory to the existence of good’’ integral models of those Shimura varieties whose adjoints are products of simple, adjoint Shimura varieties of $D_l^{\mathbf{H}}$ type with $l\ge 4$.