The geometric fundamental group of the affine line over a finite field

TL;DR

This paper constructs a universal affine pro-algebraic group framework to describe the geometric étale fundamental groups of the affine line and punctured affine line over finite fields, providing explicit descriptions for these cases.

Contribution

It introduces a non-commutative universal affine pro-algebraic group approach to characterize geometric fundamental groups over finite fields, extending Tannaka duality.

Findings

Explicit descriptions of Lu(X,U) for affine line and punctured affine line.

Universal affine groups serve as a framework for fundamental groups over finite fields.

Connection between Galois coverings and quotients of Lu(X,U).

Abstract

The affine line and the punctured affine line over a finite field F are taken as benchmarks for the problem of describing geometric \'etale 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 \'etale 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.