2024/08/15 by Henrik Russell, Russell, Henrik
Mathematics · #14G15 #14H30 #14L17 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2408.08175
openalex publication_date 2024/08/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The affine line and the punctured affine line over a finite field F are taken as benchmarks for the problem of describing geometric étale fundamental groups. To this end, using a reformulation of Tannaka duality we construct for a projective variety X a (non-commutative) universal affine pro-algebraic group Lu(X), such that for any given affine subvariety U of X any finite and étale Galois covering of U over F is a pull-back of a Galois covering of a quotient Lu(X,U) of Lu(X). Then the geometric fundamental group of U is a completion of the k-points of Lu(X,U), where k is an algebraic closure of F. We obtain explicit descriptions of the universal affine groups Lu(X,U) for U the affine line and the punctured affine line over F.