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.