Curve equations from expansions of 1-forms at a nonrational point
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866915347082248192 |
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| author | van Bommel, Raymond Costa, Edgar Poonen, Bjorn Srinivasan, Padmavathi |
| author_facet | van Bommel, Raymond Costa, Edgar Poonen, Bjorn Srinivasan, Padmavathi |
| contents | We exhibit an algorithm to compute equations of an algebraic curve over a computable characteristic 0 field from the power series expansions of its regular 1-forms at a nonrational point of the curve, extending a 2005 algorithm of Baker, González-Jiménez, González, and Poonen for expansions at a rational point. If the curve is hyperelliptic, the equations present it as an explicit double cover of a smooth plane conic, or as a double cover of the projective line when possible. If the curve is nonhyperelliptic, the equations cut out the canonical model. The algorithm has been used to compute equations over $\mathbb{Q}$ for many hyperelliptic modular curves without a rational cusp in the L-functions and Modular Forms Database. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_14026 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Curve equations from expansions of 1-forms at a nonrational point van Bommel, Raymond Costa, Edgar Poonen, Bjorn Srinivasan, Padmavathi Number Theory 14Q05 (Primary) 11G18, 14G35 (Secondary) We exhibit an algorithm to compute equations of an algebraic curve over a computable characteristic 0 field from the power series expansions of its regular 1-forms at a nonrational point of the curve, extending a 2005 algorithm of Baker, González-Jiménez, González, and Poonen for expansions at a rational point. If the curve is hyperelliptic, the equations present it as an explicit double cover of a smooth plane conic, or as a double cover of the projective line when possible. If the curve is nonhyperelliptic, the equations cut out the canonical model. The algorithm has been used to compute equations over $\mathbb{Q}$ for many hyperelliptic modular curves without a rational cusp in the L-functions and Modular Forms Database. |
| title | Curve equations from expansions of 1-forms at a nonrational point |
| topic | Number Theory 14Q05 (Primary) 11G18, 14G35 (Secondary) |
| url | https://arxiv.org/abs/2506.14026 |