TY - JOUR

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

AU - Iovita, Adrian

AU - Morrow, Jackson S.

AU - Zaharescu, Alexandru

N1 - Funding Information:
This article owes much to many people. We thank Robert Benedetto, Olivier Brinon, Henri Darmon, Eyal Goren, Ralph Greenberg, and Sean Howe for helpful conversations and email exchanges on topics related to this research. We are grateful to Pierre Colmez for sending us a sketch of the proof of Theorem and to Yeuk Hay Joshua Lam and Alexander Petrov for providing us with the proof presented in the Appendix. We also thank Pierre Colmez and Jan Nekovar for pointing out some errors in earlier drafts of this paper. We would like to thank the referees for their very helpful and detailed comments. This research began at the thematic semester on ‘Number Theory — Cohomology in Arithmetic’ at the Centre de Recherches Mathématiques. Finally, the first author was partially supported by an NSERC Discovery grant.
Funding Information:
This article owes much to many people. We thank Robert Benedetto, Olivier Brinon, Henri Darmon, Eyal Goren, Ralph Greenberg, and Sean Howe for helpful conversations and email exchanges on topics related to this research. We are grateful to Pierre Colmez for sending us a sketch of the proof of Theorem 5.5 and to Yeuk Hay Joshua Lam and Alexander Petrov for providing us with the proof presented in the Appendix. We also thank Pierre Colmez and Jan Nekovar for pointing out some errors in earlier drafts of this paper. We would like to thank the referees for their very helpful and detailed comments. This research began at the thematic semester on ‘Number Theory — Cohomology in Arithmetic’ at the Centre de Recherches Mathématiques. Finally, the first author was partially supported by an NSERC Discovery grant.
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 - abelian varieties

KW - Fontaine integration

KW - p-adic uniformization

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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 -