Groupe fondamental des champs algébriques, inertie et action galoisienne

Lochak, Pierre; Vaquié, Michel

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 1, pp. 199-264.

Nous étudions l’action du groupe de Galois arithmétique sur l’inertie géométrique attachée au groupe fondamental d’un (1-)champ algébrique, dans un contexte modéré. La situation est analogue mais aussi foncièrement distincte et plus complexe que celle, essentiellement bien comprise, qui concerne l’étude des groupes d’inertie procycliques associés aux composantes d’un diviseur à croisements normaux. Une grande partie du texte est consacrée à mettre en place les outils nécessaires à cette étude, elle-même en partie motivée par et appliquée à l’exemple important des champs de modules de courbes, dans lequel les groupes d’inertie en question correspondent aux automorphismes des courbes algébriques du type classifié par le champ.

We study the action of the arithmetic Galois group on the geometric inertia subgroups of the fundamental group, in a tame but typically stacky context. The problem is analogous but more involved than the by now fairly well-understood situation of the procyclic inertia subgroups associated with the components of a divisor with normal crossings. A significant part of the text is devoted to introducing the necessary tools for a study which is in part motivated by and applied to the important example of the moduli stacks of curves, where the geometric inertia groups correspond to the automorphisms of algebraic curves of the type classified by the stack.

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