Class field theory for function fields and finite abelian torsors

TL;DR

This paper extends classical geometric class field theory to classify finite abelian torsors over algebraic curves in positive characteristic, linking them to generalized Jacobians and fundamental group schemes.

Contribution

It generalizes classical theory to classify fppf G-torsors via isogenies of generalized Jacobians for any finite abelian group scheme G.

Findings

Classification of fppf G-torsors in terms of Jacobian isogenies

Description of abelianized Nori fundamental group scheme

Recovery of known fundamental group descriptions for projective curves

Abstract

Let $U$ be a smooth and connected curve over an algebraically closed field of positive characteristic, with smooth compactification $X$. We generalize classical Geometric Class Field theory to provide a classification of fppf $G$-torsors over $U$ in terms of isogenies of generalized Jacobians, for any finite abelian group scheme $G$. We then apply this classification to give a novel description of the abelianized Nori fundamental group scheme of $U$ in terms of the Serre--Oort fundamental groups of generalized Jacobians of $X$; when $U=X$ is projective, we recover a well known description of the abelianized fundamental group scheme of $X$ as the projective limit of all torsion subgroup schemes of its Jacobian.