Let
$p$ be a rational prime, let
$F$ denote a finite, unramified extension of
${{\mathbb {Q}}}_p$, let
$K$ be the maximal unramified extension of
${{\mathbb {Q}}}_p$,
${{\overline {K}}}$ some fixed algebraic closure of
$K$, and
${{\mathbb {C}}}_p$ be the completion of
${{\overline {K}}}$. 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
$\varphi _{A} \otimes 1_{{{\mathbb {C}}}_p}\colon T_p(A)\otimes _{{{\mathbb {Z}}}_p}{{\mathbb {C}}}_p\to \operatorname {Lie}(A)(F)\otimes _F{{\mathbb {C}}}_p(1)$, 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
${{\mathbb {C}}}_p$, 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.