Lifting the field of norms

Laurent Berger

doi:10.5802/jep.2

Laurent Berger ¹

¹ UMPA de l’ENS de Lyon, UMR 5669 du CNRS, IUF 46 allée d’Italie, 69007 Lyon, France

Journal de l’École polytechnique — Mathématiques, Volume 1 (2014), pp. 29-38.

Abstract
Résumé

Let $K$ be a finite extension of $Q_{p}$ . The field of norms of a $p$ -adic Lie extension $K_{\infty} / K$ is a local field of characteristic $p$ which comes equipped with an action of $Gal (K_{\infty} / K)$ . When can we lift this action to characteristic $0$ , along with a compatible Frobenius map? In this note, we formulate precisely this question, explain its relevance to the theory of $(ϕ, Γ)$ -modules, and give a condition for the existence of certain types of lifts.

Soit $K$ une extension finie de $Q_{p}$ . Le corps des normes d’une extension de Lie $p$ -adique $K_{\infty} / K$ est un corps local de caractéristique $p$ muni d’une action de $Gal (K_{\infty} / K)$ . Quand peut-on relever cette action en caractéristique nulle, en même temps qu’une application de Frobenius compatible ? Dans cette note, nous formulons de manière précise cette question, expliquons son intérêt pour la théorie des $(ϕ, Γ)$ -modules et donnons une condition pour l’existence de certains types de relèvements.

Received: 2013-10-09
Accepted: 2014-04-16
Published online: 2014-08-26

MR Zbl

DOI: 10.5802/jep.2

Classification: 11S15, 11S20, 11S25, 11S31, 11S82, 13F25
Keywords: Field of norms, $(\phi ,\Gamma )$-module, $p$-adic representation, anticyclotomic extension, Cohen ring, non-Archimedean dynamical system
Mot clés : Corps des normes, $(\phi ,\Gamma )$-module, représentation $p$-adique, extension anticyclotomique, anneau de Cohen, système dynamique non archimédien

Author's affiliations:

Laurent Berger ¹

¹ UMPA de l’ENS de Lyon, UMR 5669 du CNRS, IUF 46 allée d’Italie, 69007 Lyon, France

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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Laurent Berger. Lifting the field of norms. Journal de l’École polytechnique — Mathématiques, Volume 1 (2014), pp. 29-38. doi : 10.5802/jep.2. https://jep.centre-mersenne.org/articles/10.5802/jep.2/

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