On p-adic uniformization of abelian varieties with good reduction

Let Formula Presented be a rational prime, let Formula Presented denote a finite, unramified extension of Formula Presented, let Formula Presented be the maximal unramified extension of Formula Presented, Formula Presented some fixed algebraic closure of Formula Presented, and Formula Presented be the completion of Formula Presented. Let Formula Presented be the absolute Galois group of Formula Presented. Let Formula Presented be an abelian variety defined over Formula Presented, with good reduction. Classically, the Fontaine integral was seen as a Hodge-Tate comparison morphism, i.e. as a map Formula Presented, and as such it is surjective and has a large kernel. This paper starts with the observation that if we do not tensor Formula Presented with Formula Presented, then the Fontaine integral is often injective. In particular, it is proved that if Formula Presented, then Formula Presented is injective. As an application, we extend the Fontaine integral to a perfectoid like universal cover of Formula Presented and show that if Formula Presented, then Formula Presented has a type of Formula Presented-adic uniformization, which resembles the classical complex uniformization.