Class field theory for function fields and finite abelian torsors

2025/07/03 by Cais, Bryden, Otabe, Shusuke
#14H30 #14H40 #14L15 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2507.02483

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.

Citations

• On the fundamental groups of commutative algebraic groups
• Geometric class field theory and Cartier duality
• The fundamental group of affine curves in positive characteristic

Related

• Class field theory for curves over p-adic fields
• Iwasawa Dieudonné theory of function fields
• Abelian geometric fundamental groups for curves over a p-adic field
• Geometric Class Field Theory
• Tame class field theory over local fields