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

Abstract

Let p be a rational prime, let K denote the completion of the maximal unramified extension of Q p , let K be a fixed algebraic closure of K, and let C p denote the completion of K. Let A be an abelian variety defined over 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 ϕ, 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 C p , then the Fontaine integral is often injective. In particular, we conjecture that if T p (A) G K = 0, then ϕ 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 Néron 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 ϕ A is injective, then A(K) has a type of p-adic uniformization, which resembles the classical complex uniformization.Let us first suppose K is archimedean. In this case we have K = R or K = C, K = C = C, G K = {1, τ}, with τ the complex conjugation, or G K = {1}. The exponential map is everywhere defined and defines the exact sequence:From this, we obtain the complex uniformization of A, i.e. an isomorphism of complex Lie-groups:where Λ is the image of the lattice H 1 (A(C), Z).

Ramification of $p$-power torsion points of formal groups

“…By using previous work of the authors [IMZ21], we are able to prove that the O (1) -points of the Tate module of A are trivial, which implies that following theorem. Theorem A.…”

“…The kernel of the Fontaine integral. In [IMZ21], we studied the kernel of ϕ A . As noted in Remark 2.7, we have that T p (A) G K lies in ker(ϕ A ), and in [IMZ21, Theorem 4.5, Theorem A.4], we showed that T p (A) G K = ker(ϕ A ).…”

“…Outline of paper. In Section 2, we recall the definition of the Fontaine integral, our previous work on the kernel of the Fontaine integral [IMZ21], and a different perspective on the Fontaine integral via work of Wintenberger. In Section 3, we prove Theorem A.…”