Let be a complete discretely valued field with algebraically closed residue field and let be a smooth projective and geometrically connected algebraic -curve of genus . Assume that , so that there exists a minimal finite Galois extension of such that admits a semi-stable model. In this paper, we study the extension in terms of the minimal triangulation of , a distinguished finite subset of the Berkovich analytification of . We prove that the least common multiple of the multiplicities of the points of the minimal triangulation always divides the degree . Moreover, in the special case when is prime to the residue characteristic of , then we show that , 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.
Soit un corps discrètement valué complet avec corps résiduel algébriquement clos et soit une -courbe projective lisse, géométriquement connexe, de genre . Si l’on suppose il y a une extension Galoisienne finie minimale pour la propriété que le changement de base admet un modèle semistable. Nous étudions l’extension en utilisant la triangulation minimale de : un ensemble fini de points de l’analytification à la Berkovich de . Nous prouvons que le plus petit multiple commun des multiplicités des points de cette triangulation minimale divise le degré de l’extension . Lorsque est premier avec la caractéristique résiduelle de , nous démontrons que , 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.
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Keywords: Berkovich curves, Semi-stable reduction, monodromy, Galois theory
Mot clés : Courbes de Berkovich, réduction semistable, monodromie, théorie de Galois
Fantini, Lorenzo 1; Turchetti, Daniele 2
@article{AIF_2023__73_2_695_0, author = {Fantini, Lorenzo and Turchetti, Daniele}, title = {Triangulations of non-archimedean curves, semi-stable reduction, and ramification}, journal = {Annales de l'Institut Fourier}, pages = {695--746}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {73}, number = {2}, year = {2023}, doi = {10.5802/aif.3536}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3536/} }
TY - JOUR AU - Fantini, Lorenzo AU - Turchetti, Daniele TI - Triangulations of non-archimedean curves, semi-stable reduction, and ramification JO - Annales de l'Institut Fourier PY - 2023 SP - 695 EP - 746 VL - 73 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3536/ DO - 10.5802/aif.3536 LA - en ID - AIF_2023__73_2_695_0 ER -
%0 Journal Article %A Fantini, Lorenzo %A Turchetti, Daniele %T Triangulations of non-archimedean curves, semi-stable reduction, and ramification %J Annales de l'Institut Fourier %D 2023 %P 695-746 %V 73 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3536/ %R 10.5802/aif.3536 %G en %F AIF_2023__73_2_695_0
Fantini, Lorenzo; Turchetti, Daniele. Triangulations of non-archimedean curves, semi-stable reduction, and ramification. Annales de l'Institut Fourier, Volume 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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