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 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.

Theorems & Definitions (75)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Theorem 2.1
• Lemma 2.2
• proof
• Remark 2.3
• proof : Proof of Theorem \ref{['thm:CFT']}