Ramification groups  in Artin-Schreier-Witt extensions

Thomas, Lara

doi:10.5802/jtnb.514

Ramification groups in Artin-Schreier-Witt extensions

Thomas, Lara ^1,²

¹ Equipe GRIMM Université Toulouse II 5, allées A. Machado 31058 Toulouse, France
² Chaire de Structures Algébriques et Géométriques Ecole Polytechnique Fédérale de Lausanne SB - IMB (Bâtiment MA) Station 8 CH-1015 Lausanne

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 2, pp. 689-720.

Résumé
Abstract

Soit $K$ un corps local de caractéristique $p > 0$ . L’objectif de cet article est de décrire les groupes de ramification des pro- $p$ extensions abéliennes de $K$ à travers la théorie d’Artin-Schreier-Witt. Dans le cadre usuel de la théorie du corps de classes local, cette étude est menée entièrement et conduit à un accouplement non-dégénéré que nous définissons en détail, généralisant ainsi la formule de Schmid pour les vecteurs de Witt de longueur $n$ . Au passage, on retrouve un résultat de Brylinski avec des arguments plus explicites nécessitant moins d’outils techniques. La dernière partie aborde le cas plus général où le corps résiduel de $K$ est parfait.

Let $K$ be a local field of characteristic $p > 0$ . The aim of this paper is to describe the ramification groups for the pro- $p$ abelian extensions over $K$ with regards to the Artin-Schreier-Witt theory. We shall carry out this investigation entirely in the usual framework of local class field theory. This leads to a certain non-degenerate pairing that we shall define in detail, generalizing in this way the Schmid formula to Witt vectors of length $n$ . Along the way, we recover a result of Brylinski but with a different proof which is more explicit and requires less technical machinery. A first attempt is finally made to extend these computations to the case where the perfect field of $K$ is merely perfect.

MR Zbl | 2 citations dans Numdam

DOI : 10.5802/jtnb.514

Affiliations des auteurs :

Thomas, Lara ^{1, 2}

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Thomas, Lara. Ramification groups  in Artin-Schreier-Witt extensions. Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 2, pp. 689-720. doi : 10.5802/jtnb.514. https://www.numdam.org/articles/10.5802/jtnb.514/

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