Triangulations of non-archimedean curves, semi-stable reduction, and ramification

Fantini, Lorenzo; Turchetti, Daniele

doi:10.5802/aif.3536

Triangulations of non-archimedean curves, semi-stable reduction, and ramification
[Triangulations des courbes non archimédiennes, réduction semistable et ramification]

Fantini, Lorenzo ¹ ; Turchetti, Daniele ²

¹ Centre de Mathématiques Laurent Schwartz, École Polytechnique, CNRS, Palaiseau, France
² University of Warwick, Mathematics Institute, Zeeman Building, Coventry CV4 7AL, United Kingdom

Annales de l'Institut Fourier, Tome 73 (2023) no. 2, pp. 695-746.

Résumé
Abstract

Soit $K$ un corps discrètement valué complet avec corps résiduel algébriquement clos et soit $ℭ$ une $K$ -courbe projective lisse, géométriquement connexe, de genre $g$ . Si l’on suppose $g \geq 2$ il y a une extension Galoisienne finie $L | K$ minimale pour la propriété que le changement de base $ℭ_{L}$ admet un modèle semistable. Nous étudions l’extension $L | K$ en utilisant la triangulation minimale de $C$ : un ensemble fini de points de l’analytification à la Berkovich $C$ de $ℭ$ . Nous prouvons que le plus petit multiple commun $d$ des multiplicités des points de cette triangulation minimale divise le degré de l’extension $[L : K]$ . Lorsque $d$ est premier avec la caractéristique résiduelle de $K$ , nous démontrons que $d = [L : K]$ , en déduisant une nouvelle preuve d’un théorème classique de T. Saito. Nous discutons ensuite le cas des courbes avec points marqués, ce qui nous permet d’obtenir des résultats analogues pour les courbes elliptiques et de décrire complètement les triangulations de ces dernières dans le cas modérément ramifié. Nous terminons en expliquant grâce à nos résultats pourquoi les extensions plus naturelles du théorème de Saito ne peuvent pas être vraies dans le cas sauvagement ramifié.

Let $K$ be a complete discretely valued field with algebraically closed residue field and let $ℭ$ be a smooth projective and geometrically connected algebraic $K$ -curve of genus $g$ . Assume that $g ⩾ 2$ , so that there exists a minimal finite Galois extension $L$ of $K$ such that $ℭ_{L}$ admits a semi-stable model. In this paper, we study the extension $L ∣ K$ in terms of the minimal triangulation of $C$ , a distinguished finite subset of the Berkovich analytification $C$ of $ℭ$ . We prove that the least common multiple $d$ of the multiplicities of the points of the minimal triangulation always divides the degree $[L : K]$ . Moreover, in the special case when $d$ is prime to the residue characteristic of $K$ , then we show that $d = [L : K]$ , obtaining a new proof of a classical theorem of T. Saito. We then discuss curves with marked points, which allows us to prove analogous results in the case of elliptic curves, whose minimal triangulations we describe in full in the tame case. In the last section, we illustrate through several examples how our results explain the failure of the most natural extensions of Saito’s theorem to the wildly ramified case.

Reçu le : 2020-01-07
Révisé le : 2020-12-22
Accepté le : 2021-01-25
Publié le : 2023-05-12

DOI : 10.5802/aif.3536

Classification : 14D10, 14G22, 14E22
Keywords: Berkovich curves, Semi-stable reduction, monodromy, Galois theory
Mot clés : Courbes de Berkovich, réduction semistable, monodromie, théorie de Galois

Affiliations des auteurs :

Fantini, Lorenzo ¹ ; Turchetti, Daniele ²

¹ Centre de Mathématiques Laurent Schwartz, École Polytechnique, CNRS, Palaiseau, France
² University of Warwick, Mathematics Institute, Zeeman Building, Coventry CV4 7AL, United Kingdom

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Triangulations of non-archimedean curves, semi-stable reduction, and ramification},
     journal = {Annales de l'Institut Fourier},
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Fantini, Lorenzo; Turchetti, Daniele. Triangulations of non-archimedean curves, semi-stable reduction, and ramification. Annales de l'Institut Fourier, Tome 73 (2023) no. 2, pp. 695-746. doi : 10.5802/aif.3536. https://aif.centre-mersenne.org/articles/10.5802/aif.3536/

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