Note sur la d\'etermination alg\'ebrique du groupe fondamental pro-r\'esoluble d'une courbe affine

TL;DR

This paper provides an algebraic proof for the structure of the pro-solvable quotient of the étale fundamental group of an affine algebraic curve, extending known transcendental results to an algebraic context.

Contribution

It introduces an algebraic method to determine the pro-solvable quotient of the fundamental group of affine curves, complementing existing transcendental approaches.

Findings

Proves the algebraic structure of the pro-solvable quotient

Extends known results from transcendental to algebraic methods

Applicable to affine algebraic curves over algebraically closed fields

Abstract

Let X be a smooth projective algebraic curve of genus g minus $r\geq 1$ points defined over an algebraically closed field k of characteristic $p\geq 0$. The structure of the largest prime to p quotient of the \'etale 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.