Points of integral canonical models of Shimura varieties of preabelian type, p-divisible groups, and applications. First Part

Abstract

This work is the first part in a series of three dedicated to the foundations of integral aspects of Shimura varieties and of Fontaine's categories. It deals mostly with the unramified context of (arbitrary) mixed characteristic (0,p). Among the topics covered we mention: the generalization of the classical Serre-Tate theory of ordinary p-divisible groups and of their canonical lifts, the generalization of the classical Serre-Tate-Dwork-Katz theory of (crystalline) coordinates for ordinary abelian varieties, the strong form of the generalized Manin problem, global deformations in the generalized Shimura context, Dieudonné's theories (reobtained, simplified and extended), the main list of stratifications of special fibres of integral canonical models of Shimura varieties of preabelian type, the uniqueness of such models in mixed characteristic (0,2), the existence (in many situations) of such models in mixed characteristic (0,2), steps towards the classification of Shimura p-divisible groups over algebraically closed field of characteristic p (like the boundedness principle, the purity principle, Bruhat decompositions in the F-context, etc.), generalized Serre lemma, connections in Fontaine's categories, etc.