Class field theory for function fields and finite abelian torsors

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 fundmental 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.