On p-adic uniformization of abelian varieties with good reduction

Let p be a rational prime, let F denote a finite, unramified extension of Qp, let K be the maximal unramified extension of Q(p), (K) over bar some fixed algebraic closure of K, and Cp be the completion of (K) over bar. Let G(F) be the absolute Galois group of F. Let A be an abelian variety defined over F, with good reduction. Classically, the Fontaine integral was seen as a Hodge-Tate comparison morphism, i.e. as a map phi A circle times 1(Cp) : T-p(A)circle times Z(p) C-p -> Lie(A)(F)circle times(F) C-p(1), and as such it is surjective and has a large kernel. This paper starts with the observation that if we do not tensor T-p(A) with C-p, then the Fontaine integral is often injective. In particular, it is proved that if T-p(A)(GK) = 0, then phi A is injective. As an application, we extend the Fontaine integral to a perfectoid like universal cover of A and show that if T-p(A)(GK) - 0, then A((K) over bar) has a type of p-adic uniformization, which resembles the classical complex uniformization.