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On p-adic uniformization of abelian varieties with good reduction

Part of: Algebraic groups Abelian varieties and schemes Arithmetic algebraic geometry

Published online by Cambridge University Press:  05 September 2022

Adrian Iovita ,
Jackson S. Morrow  and
Alexandru Zaharescu
Adrian Iovita
Affiliation:
Department of Mathematics and Statistics, Concordia University, Montréal, Québec, Canada adrian.iovita@concordia.ca Dipartimento di Matematica, Universita degli Studi di Padova, Padova, Italy
Jackson S. Morrow
Affiliation:
Department of Mathematics, University of California, 749 Evans Hall, Berkeley, CA 94720, USA jacksonmorrow@berkeley.edu
Alexandru Zaharescu
Affiliation:
Department of Mathematics, University of Illinois Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801, USA zaharesc@illinois.edu “Simion Stoilow” Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania
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Abstract

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.

Keywords

p-adic uniformizationabelian varietiesFontaine integration

MSC classification

Primary: 11G10: Abelian varieties of dimension $> 1$
Secondary: 14K20: Analytic theory; abelian integrals and differentials 11G25: Varieties over finite and local fields 14L05: Formal groups, $p$-divisible groups
Type
Research Article
Information
Compositio Mathematica , Volume 158 , Issue 7 , July 2022 , pp. 1449 - 1476
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Copyright
© 2022 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

With an appendix by Yeuk Hay Joshua Lam and Alexander Petrov

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