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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.04365 |
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| _version_ | 1866913640573042688 |
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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 consider projective, irreducible, non-singular curves over an algebraically closed field $\k$. A cover $Y \to X$ of such curves corresponds to an extension $Ω/Σ$ of their function fields and yields an isomorphism $\A_{Y} \simeq \A_{X} \otimes_Σ Ω$ of their geometric adele rings. The primitive element theorem shows that $\A_{Y}$ is a quotient of $\A_{X}[T]$ by a polynomial.
In general, we may look at quotient algebras $\AXp{\p} = \A_{X}[T]/(\p(T))$ where $\p(T) \in \A_{X}[T]$ is monic and separable over $\A_{X}$, and try to characterize the field extensions $Ω/Σ$ lying in $\AXp{\p}$ which arise from covers as above. We achieve this topologically, namely, as those $Ω$ which embed discretely in $\AXp{\p}$, and in terms of an additive analog of the product formula for global fields, a result which is reminiscent of classical work of Artin-Whaples and Iwasawa.
The technical machinery requires studying which topology on $\AXp{\p}$ is natural for this problem. Local compactness no longer holds, but instead we have linear topologies defined by commensurability of $\k$-subspaces which coincide with the restricted direct product topology with respect to integral closures. The content function is given as an index measuring the discrepancy in commensurable subspaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04365 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Characterization of subfields of adelic algebras by a product formula Vicente, Luis Manuel Navas Martin, Francisco J. Plaza Rings and Algebras Number Theory 14H05 (Primary), 12J20, 13B02, 13A18, 13J99 (Secondary) We consider projective, irreducible, non-singular curves over an algebraically closed field $\k$. A cover $Y \to X$ of such curves corresponds to an extension $Ω/Σ$ of their function fields and yields an isomorphism $\A_{Y} \simeq \A_{X} \otimes_Σ Ω$ of their geometric adele rings. The primitive element theorem shows that $\A_{Y}$ is a quotient of $\A_{X}[T]$ by a polynomial. In general, we may look at quotient algebras $\AXp{\p} = \A_{X}[T]/(\p(T))$ where $\p(T) \in \A_{X}[T]$ is monic and separable over $\A_{X}$, and try to characterize the field extensions $Ω/Σ$ lying in $\AXp{\p}$ which arise from covers as above. We achieve this topologically, namely, as those $Ω$ which embed discretely in $\AXp{\p}$, and in terms of an additive analog of the product formula for global fields, a result which is reminiscent of classical work of Artin-Whaples and Iwasawa. The technical machinery requires studying which topology on $\AXp{\p}$ is natural for this problem. Local compactness no longer holds, but instead we have linear topologies defined by commensurability of $\k$-subspaces which coincide with the restricted direct product topology with respect to integral closures. The content function is given as an index measuring the discrepancy in commensurable subspaces. |
| title | Characterization of subfields of adelic algebras by a product formula |
| topic | Rings and Algebras Number Theory 14H05 (Primary), 12J20, 13B02, 13A18, 13J99 (Secondary) |
| url | https://arxiv.org/abs/2501.04365 |