Reconstruction anabélienne du squelette des courbes analytiques
[Anabelian reconstruction of the skeleton of analytic curves]
Annales de l'Institut Fourier, Volume 73 (2023) no. 3, pp. 999-1084.

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.

Published online:
DOI: 10.5802/aif.3548
Classification: 14H30, 12J25
Mot clés : Géométrie anabélienne, espaces de Berkovich, groupe fondamental tempéré, courbes analytiques, semi-graphe d’anabélioïdes.
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)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Gaulhiac, Sylvain. Reconstruction anabélienne du squelette des courbes analytiques. Annales de l'Institut Fourier, Volume 73 (2023) no. 3, pp. 999-1084. doi : 10.5802/aif.3548.

[1] Alon, Gil Automorphisms of products of Drinfeld half planes (2017) (

[2] André, Yves On a geometric description of Gal(Q ¯ p /Q p ) and a p-adic avatar of GT ^, Duke Math. J., Volume 119 (2003) no. 1, pp. 1-39 | DOI | MR | Zbl

[3] André, Yves; Katō, Fumiharu; Tsuzuki, Nobuo Period mappings and differential equations. From to p , MSJ Memoirs, 12, Mathematical Society of Japan, Tokyo, 2003, viii+246 pages | MR

[4] Berkovich, Vladimir G. Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, Providence, RI, 1990, x+169 pages | DOI | MR | Zbl

[5] Berkovich, Vladimir G. Étale cohomology for non-Archimedean analytic spaces, Inst. Hautes Études Sci. Publ. Math. (1993) no. 78, p. 5-161 (1994) | DOI | Numdam | MR | Zbl

[6] Berkovich, Vladimir G. Smooth p-adic analytic spaces are locally contractible, Invent. Math., Volume 137 (1999) no. 1, pp. 1-84 | DOI | MR | Zbl

[7] Cadoret, Anna Galois categories, Arithmetic and geometry around Galois theory (Progr. Math.), Volume 304, Birkhäuser/Springer, Basel, 2013, pp. 171-246 | DOI | MR | Zbl

[8] Ducros, Antoine Triangulation et cohomologie étale sur une courbe analytique (2005) (

[9] Ducros, Antoine La structure des courbes analytiques (2014) (Manuscript available at

[10] Ducros, Antoine Families of Berkovich spaces, Astérisque, Société mathématique de France, 2018 no. 400, vii+262 pages | MR | Zbl

[11] Grothendieck, Alexander; Raynaud, Michele Revêtements étales et groupe fondamental (SGA 1) (2002) (

[12] Grothendieck, Alexandre Geometric Galois actions, 1. Around Grothendieck’s Esquisse D’un Programme, London Math. Soc. Lecture Note Ser., 242, Cambridge University Press, Cambridge, 1997

[13] de Jong, A. J. Étale fundamental groups of non-Archimedean analytic spaces, Compositio Math., Volume 97 (1995) no. 1-2, pp. 89-118 (Special issue in honour of Frans Oort) | Numdam | MR | Zbl

[14] Lepage, Emmanuel Tempered fundamental group and metric graph of a Mumford curve, Publ. Res. Inst. Math. Sci., Volume 46 (2010) no. 4, pp. 849-897 | MR | Zbl

[15] Lepage, Emmanuel Tempered fundamental group and graph of the stable reduction, The arithmetic of fundamental groups—PIA 2010 (Contrib. Math. Comput. Sci.), Volume 2, Springer, Heidelberg, 2012, pp. 209-224 | DOI | MR | Zbl

[16] Lepage, Emmanuel Resolution of nonsingularities for Mumford curves, Publ. Res. Inst. Math. Sci., Volume 49 (2013) no. 4, pp. 861-891 | DOI | MR | Zbl

[17] Mochizuki, Shinichi The absolute anabelian geometry of hyperbolic curves, Galois theory and modular forms (Dev. Math.), Volume 11, Kluwer Acad. Publ., Boston, MA, 2004, pp. 77-122 | DOI | MR | Zbl

[18] Mochizuki, Shinichi The geometry of anabelioids, Publ. Res. Inst. Math. Sci., Volume 40 (2004) no. 3, pp. 819-881 | DOI | MR | Zbl

[19] Mochizuki, Shinichi Semi-graphs of anabelioids, Publ. Res. Inst. Math. Sci., Volume 42 (2006) no. 1, pp. 221-322 | DOI | MR | Zbl

[20] Mochizuki, Shinichi Comments on “Semi-graphs of Anabelioids”, 2020

[21] Nakamura, Hiroaki Galois rigidity of pure sphere braid groups and profinite calculus, J. Math. Sci. Univ. Tokyo, Volume 1 (1994) no. 1, pp. 71-136 | MR | Zbl

[22] Temkin, M. On local properties of non-Archimedean analytic spaces. II, Israel J. Math., Volume 140 (2004), pp. 1-27 | DOI | MR | Zbl

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