Reconstruction anabélienne du squelette des courbes analytiques
[Anabelian reconstruction of the skeleton of analytic curves]
Annales de l'Institut Fourier, Online first, 86 p.

This work brings to light some anabelian behaviours of analytic curves in the context of Berkovich geometry. We show that the knowledge of the tempered fundamental group of some curves called analytically anabelian determines their analytic skeletons as graphs. The famous Drinfeld half-plane is an example of such a curve. The tempered fundamental group of a Berkovich space, introduced by André, enabled Mochizuki to prove the first anabelian result in Berkovich geometry, dealing with analytifications of hyperbolic curves over ¯ p . To that end, Mochizuki develops the language of semi-graphs of anabelioids and temperoids. This article consists in associating a semi-graph of anabelioids to a Berkovich curve equipped with a minimal triangulation and in adapting the results of Mochizuki in order to recover the analytic skeleton. The novelty here is that the curves we are interested in are not supposed anymore to be of algebraic nature.

Ce travail met en évidence certains comportements anabéliens des courbes analytiques au sens de la géométrie de Berkovich. Nous montrons que la connaissance du groupe fondamental tempéré de certaines courbes appelées analytiquement anabéliennes détermine leurs squelettes analytiques en tant que graphes. Le fameux demi-plan de Drinfeld en est un exemple. Le groupe fondamental tempéré d’un espace de Berkovich, introduit par André, a permis à Mochizuki de démontrer le premier résultat anabélien en géométrie de Berkovich, concernant les analytifiées de courbes hyperboliques sur ¯ p . À cette fin, Mochizuki développe le langage des semi-graphes d’anabélioïdes et des tempéroïdes. Cet article consiste à associer un semi-graphe d’anabélioïdes à une courbe analytique munie d’une triangulation minimale, puis s’inspirer des résultats de Mochizuki afin de retrouver le squelette analytique de la courbe. La nouveauté de ces résultats est que les courbes que nous étudions ne sont plus supposées de nature algébrique.

Online First:
DOI: 10.5802/aif.3548
Classification: 14H30,  12J25
Keywords: Anabelian geometry, Berkovich spaces, tempered fundamental group, analytic curves, semi-graph of anabelioids.
Gaulhiac, Sylvain 1

1 Department of Mathematical and Statistical Science University of Alberta Edmonton (Canada)
     author = {Gaulhiac, Sylvain},
     title = {Reconstruction anab\'elienne du squelette des courbes analytiques},
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Gaulhiac, Sylvain. Reconstruction anabélienne du squelette des courbes analytiques. Annales de l'Institut Fourier, Online first, 86 p.

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