First-order definability of Darmon points in number fields
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912846957248512 |
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| author | De Rasis, Juan Pablo Handley, Hunter |
| author_facet | De Rasis, Juan Pablo Handley, Hunter |
| contents | For a given number field $K$, we give a $\forall\exists\forall$-first order description of affine Darmon points over $\mathbb{P}^1_K$, and show that this can be improved to a $\forall\exists$-definition in a remarkable particular case. Darmon points, which are a geometric generalization of perfect powers, constitute a non-linear set-theoretical filtration between $K$ and its ring of $S$-integers, the latter of which can be defined with universal formulas, as has been progressively proven by Koenigsmann, Park, and Eisenträger & Morrison. We also show that our formulas are uniform with respect to all possible $S$, with a parameter-free uniformity, and we compute the number of quantifiers and a bound for the degree of the defining polynomial. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_03033 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | First-order definability of Darmon points in number fields De Rasis, Juan Pablo Handley, Hunter Number Theory Algebraic Geometry Logic 11U05 (Primary), 11R04, 11R52 (Secondary) For a given number field $K$, we give a $\forall\exists\forall$-first order description of affine Darmon points over $\mathbb{P}^1_K$, and show that this can be improved to a $\forall\exists$-definition in a remarkable particular case. Darmon points, which are a geometric generalization of perfect powers, constitute a non-linear set-theoretical filtration between $K$ and its ring of $S$-integers, the latter of which can be defined with universal formulas, as has been progressively proven by Koenigsmann, Park, and Eisenträger & Morrison. We also show that our formulas are uniform with respect to all possible $S$, with a parameter-free uniformity, and we compute the number of quantifiers and a bound for the degree of the defining polynomial. |
| title | First-order definability of Darmon points in number fields |
| topic | Number Theory Algebraic Geometry Logic 11U05 (Primary), 11R04, 11R52 (Secondary) |
| url | https://arxiv.org/abs/2410.03033 |