Class field theory for function fields and finite abelian torsors
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912973247741952 |
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| author | Cais, Bryden Otabe, Shusuke |
| author_facet | Cais, Bryden Otabe, Shusuke |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_02483 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Class field theory for function fields and finite abelian torsors Cais, Bryden Otabe, Shusuke Algebraic Geometry 14H30, 14H40, 14L15 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. |
| title | Class field theory for function fields and finite abelian torsors |
| topic | Algebraic Geometry 14H30, 14H40, 14L15 |
| url | https://arxiv.org/abs/2507.02483 |