A unified finiteness theorem for curves
Fuente:
arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911559002882048 |
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| author | Janbazi, Fatemehzahra Sajadi, Fateme |
| author_facet | Janbazi, Fatemehzahra Sajadi, Fateme |
| contents | We study the arithmetic of Galois-invariant sets of points on algebraic curves with controlled reduction behavior. Let $C$ be a smooth projective curve with a smooth proper model $\mathcal{C}$ over $\mathcal{O}_{K,S}$. We define $Ω_n$ as the set of $n$-element subsets of $C(\overline{K})$ that are invariant under $\text{Gal}(\overline{K}/K)$ and such that no two points in the set become identified modulo any prime $\mathfrak{p} \notin S$. Our main result establishes that $Ω_n$ breaks into finitely many orbits under the action of $\text{Aut}_{\mathcal{O}_{K,S}}(\mathcal{C})$, generalizing finiteness theorems of Birch--Merriman, Siegel, and Faltings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09804 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A unified finiteness theorem for curves Janbazi, Fatemehzahra Sajadi, Fateme Number Theory Algebraic Geometry We study the arithmetic of Galois-invariant sets of points on algebraic curves with controlled reduction behavior. Let $C$ be a smooth projective curve with a smooth proper model $\mathcal{C}$ over $\mathcal{O}_{K,S}$. We define $Ω_n$ as the set of $n$-element subsets of $C(\overline{K})$ that are invariant under $\text{Gal}(\overline{K}/K)$ and such that no two points in the set become identified modulo any prime $\mathfrak{p} \notin S$. Our main result establishes that $Ω_n$ breaks into finitely many orbits under the action of $\text{Aut}_{\mathcal{O}_{K,S}}(\mathcal{C})$, generalizing finiteness theorems of Birch--Merriman, Siegel, and Faltings. |
| title | A unified finiteness theorem for curves |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2505.09804 |