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SUR LES COMPOSANTES CONNEXES D’UNE FAMILLE D’ESPACES ANALYTIQUES ${P}$-ADIQUES

Part of: Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  29 May 2014

JÉRÔME POINEAU
JÉRÔME POINEAU*
Affiliation:
Institut de recherche mathématique avancée, 7, rue René Descartes, 67084 Strasbourg, France; [email protected]

Résumé

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Soit $X=\mathcal{M}(\mathscr{A})$ un espace affinoïde et soient $f,g\in \mathscr{A}$. Nous étudions les ensembles de composantes connexes des espaces définis par une inégalité de la forme $|f|\le r\, |g|$, avec $r\ge 0$. Nous montrons qu’il existe une partition finie de $\mathbf{R}_{+}$ en intervalles sur lesquels ces ensembles sont canoniquement en bijection et que les bornes de ces intervalles appartiennent à $\sqrt{\rho (\mathscr{A})}$.

Abstract

On the connected components of a family of$\boldsymbol {p}$-adic analytic spaces. Let $X=\mathcal{M}(\mathscr{A})$ be an affinoid space and let $f,g\in \mathscr{A}$. We study the sets of connected components of the spaces defined by an inequality of the form $|f|\le r\, |g|$, with $r\ge 0$. We prove that there exists a finite partition of $\mathbf{R}_{+}$ into intervals where those sets are canonically in bijection and that the bounds of those intervals belong to $\sqrt{\rho (\mathscr{A})}$.

MSC classification

Secondary: 14G22: Rigid analytic geometry
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e14
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author 2014

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