On $p$-adic uniformization of abelian varieties with good reduction

Abstract: Let $p$ be a rational prime, let $F$ denote a finite, unramified extension of $\mathbb{Q}p$, $K$ the maximal unramified extension of $\mathbb{Q}_p$, $\overline{K}$ some fixed algebraic closure of $K$, and $\mathbb{C}_p$ the completion of $\overline{K}$. Let $G_F$ 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 $\varphi{A} \otimes 1_{\mathbb{C}p}\colon T_p(A)\otimes{\mathbb{Z}_p}\mathbb{C}_p\to \text{Lie}(A)(F)\otimes_F\mathbb{C}_p(1)$, and as such it is surjective and has a large kernel. The present article starts with the observation that if we do not tensor $T_p(A)$ with $\mathbb{C}_p$, then the Fontaine integral is often injective. In particular, it is proved that if $T_p(A)^{G_K} = 0$, then $\varphi_A$ is injective. As an application, we extend the Fontaine integral to a perfectoid like universal cover of $A$ and show that if $T_p(A)^{G_K} = 0$, then $A(\overline{K})$ has a type of $p$-adic uniformization, which resembles the classical complex uniformization.