Note sur la détermination algébrique du groupe fondamental pro-résoluble d'une courbe affine

Abstract

Reçu le 14 avril 2008Disponible sur Internet le 30 juin 2008Communiqué par Laurent Moret-Bailly RésuméSoit X une courbe projective et lisse de genre g privée de r 1 points sur un corps algébriquement clos k de caractéristique p 0. La structure du plus grand quotient d'ordre premier à p du groupe fondamental étale de X est bien connu par des méthodes transcendantes : il est isomorphe au plus grand quotient d'ordre premier à p d'un groupe pro-fini libre à 2g + r − 1 générateurs. On montre que l'on peut retrouver ce résultat par des méthodes purement algébriques pour le plus grand quotient pro-résoluble de ces groupes. AbstractLet X be a smooth projective algebraic curve of genus g minus r 1 points defined over an algebraically closed field k of characteristic p 0. The structure of the largest prime to p quotient of the étale fundamental group is well known by transcendental methods: it is isomorphic to the largest prime to p quotient of a free pro-finite group on 2g + r − 1 generators. We show that, with purely algebraic means, we can prove the corresponding result for the largest pro-solvable quotient of these groups.

Generators and Relations for the Etale Fundamental Group

“…Proof of Theorem 1.4(ii) 6.1. The main result of Borne and Emsalem [1] (building on work of Serre [16]) gives an isomorphism between π ′ 1 (P 1 \ {0, 1, ∞}) sol and F ′sol 2 . Given any punctured curve C ⊂ C, choosing a generic map C → P 1 and removing sufficiently many points from the base yields an open subcurve U ⊂ C mapping by a finite étale morphism to an open subset V of P 1 .…”

“…Lemma 4.4. Let F 2 be the free profinite group on generators γ 0 and γ 1 , and let τ : F 2 → π 1 (A 1 , η) be the map sending γ 0 to ρ 0 (1) and γ 1 to ρ 1 (1). Then the kernel of the composite map…”

“…Remark 1.8. With notation as in Problem 1.2, the main result of Borne and Emsalem in [1] gives an abstract isomorphism between the prime-to-p prosolvable quotient F ′ solv 2g+n−1 and π ′ 1 ( C, x) solv . Their method, however, does not show that the generators can be chosen to be ramification elements in the sense of §3.…”