TY - JOUR

T1 - On p-adic uniformization of abelian varieties with good reduction

AU - Iovita, Adrian

AU - Morrow, Jackson S.

AU - Zaharescu, Alexandru

N1 - Publisher Copyright:
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence.

PY - 2022/7/5

Y1 - 2022/7/5

N2 - Let Formula Presented be a rational prime, let Formula Presented denote a finite, unramified extension of Formula Presented, let Formula Presented be the maximal unramified extension of Formula Presented, Formula Presented some fixed algebraic closure of Formula Presented, and Formula Presented be the completion of Formula Presented. Let Formula Presented be the absolute Galois group of Formula Presented. Let Formula Presented be an abelian variety defined over Formula Presented, with good reduction. Classically, the Fontaine integral was seen as a Hodge-Tate comparison morphism, i.e. as a map Formula Presented, and as such it is surjective and has a large kernel. This paper starts with the observation that if we do not tensor Formula Presented with Formula Presented, then the Fontaine integral is often injective. In particular, it is proved that if Formula Presented, then Formula Presented is injective. As an application, we extend the Fontaine integral to a perfectoid like universal cover of Formula Presented and show that if Formula Presented, then Formula Presented has a type of Formula Presented-adic uniformization, which resembles the classical complex uniformization.

AB - Let Formula Presented be a rational prime, let Formula Presented denote a finite, unramified extension of Formula Presented, let Formula Presented be the maximal unramified extension of Formula Presented, Formula Presented some fixed algebraic closure of Formula Presented, and Formula Presented be the completion of Formula Presented. Let Formula Presented be the absolute Galois group of Formula Presented. Let Formula Presented be an abelian variety defined over Formula Presented, with good reduction. Classically, the Fontaine integral was seen as a Hodge-Tate comparison morphism, i.e. as a map Formula Presented, and as such it is surjective and has a large kernel. This paper starts with the observation that if we do not tensor Formula Presented with Formula Presented, then the Fontaine integral is often injective. In particular, it is proved that if Formula Presented, then Formula Presented is injective. As an application, we extend the Fontaine integral to a perfectoid like universal cover of Formula Presented and show that if Formula Presented, then Formula Presented has a type of Formula Presented-adic uniformization, which resembles the classical complex uniformization.

KW - Fontaine integration

KW - abelian varieties

KW - p-adic uniformization

UR - http://www.scopus.com/inward/record.url?scp=85138267308&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85138267308&partnerID=8YFLogxK

U2 - 10.1112/S0010437X22007643

DO - 10.1112/S0010437X22007643

M3 - Article

AN - SCOPUS:85138267308

SN - 0010-437X

VL - 158

SP - 1449

EP - 1476

JO - Compositio Mathematica

JF - Compositio Mathematica

IS - 7

ER -