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 ${\phi }_A\otimes 1_{{\mathbb{C}}_p}:T_p(A){\otimes }_{{\mathbb{Z}}_p}{\mathbb{C}}_p\to \text{Lie}(A)(F){\otimes }_F{\mathbb{C}}_p(1)$, 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 ${\phi }_A$ is injective. As an application, we extend the Fontaine integral to a perfectoid like universal cover of $A$ and show that if $T_p(A{)}^{G_K}=0$, then $A(\overline{K})$ has a type of $p$-adic uniformization, which resembles the classical complex uniformization.

v3: 25 pages, final version