Towards a $p$-adic uniformization of abelian varieties with good reduction

Let $p$ be a rational prime, let $K$ denote the completion of the maximal unramified extension of $\mathbb{Q}_p$, let $\overline{K}$ be a fixed algebraic closure of $K$, and let $\mathbb{C}_p$ denote the completion of $\overline{K}$. Let $A$ be an abelian variety defined over $\mathbb{Q}_p$ with good reduction and base change it to $K$. Classically, the Fontaine integral was seen as a Hodge--Tate comparison morphism, i.e. as a map $\varphi_{A} \otimes 1_{\mathbb{C}_p}\colon T_p(A)\otimes_{\mathbb{Z}_p}\mathbb{C}_p\to \text{Lie}(A)(K)\otimes_K\mathbb{C}_p(1)$, 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, we conjecture that if $T_p(A)^{G_K} = 0$, then $\varphi_A$ is injective. We prove this conjecture for elliptic curves, for abelian varieties with good, ordinary reduction, and for abelian varieties where the formal group associated to the N\'eron model of $A$ satisfies a certain mod $p$ condition. Furthermore, after extending the Fontaine integral to a perfectoid like universal cover of $A$, we show that if $T_p(A)^{G_K} = 0$ and $\varphi_A$ is injective, then $A(\overline{K})$ has a type of $p$-adic uniformization, which resembles the classical complex uniformization.