Cyclic covers of an algebraic curve from an adelic viewpoint
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866917336715362304 |
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| author | Vicente, Luis Manuel Navas Martin, Francisco J. Plaza |
| author_facet | Vicente, Luis Manuel Navas Martin, Francisco J. Plaza |
| contents | We propose an algebraic method for the classification of branched Galois covers of a curve $X$ focused on studying Galois ring extensions of its geometric adele ring $\A_{X}$. As an application, we deal with cyclic covers; namely, we determine when a given cyclic ring extension of $\A_{X}$ comes from a corresponding cover of curves $Y \to X$, which is reminiscent of a Grunwald-Wang problem, and also determine when two covers yield isomorphic ring extensions, which is known in the literature as an equivalence problem. This completely algebraic method permits us to recover ramification, certain analytic data such as rotation numbers, and enumeration formulas for covers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04355 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cyclic covers of an algebraic curve from an adelic viewpoint Vicente, Luis Manuel Navas Martin, Francisco J. Plaza Algebraic Geometry Rings and Algebras 14H30 (Primary) 13B05 14H05, 11R56 (Secondary) We propose an algebraic method for the classification of branched Galois covers of a curve $X$ focused on studying Galois ring extensions of its geometric adele ring $\A_{X}$. As an application, we deal with cyclic covers; namely, we determine when a given cyclic ring extension of $\A_{X}$ comes from a corresponding cover of curves $Y \to X$, which is reminiscent of a Grunwald-Wang problem, and also determine when two covers yield isomorphic ring extensions, which is known in the literature as an equivalence problem. This completely algebraic method permits us to recover ramification, certain analytic data such as rotation numbers, and enumeration formulas for covers. |
| title | Cyclic covers of an algebraic curve from an adelic viewpoint |
| topic | Algebraic Geometry Rings and Algebras 14H30 (Primary) 13B05 14H05, 11R56 (Secondary) |
| url | https://arxiv.org/abs/2501.04355 |