Rational Points on a Family of Genus 3 Hyperelliptic Curves
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914104495570944 |
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| author | Hernandez, Roberto |
| author_facet | Hernandez, Roberto |
| contents | We compute the rational points on certain members of the following family of hyperelliptic curves \[C_a \colon y^2 = x^8 + (4-4a^4) x^6 + (8a^4 + 6)x^4 + (4-4a^4)x^2 + 1\] via the method first developed by Dem'yanenko \cite{dem1966rational} and then further generalized by Manin \cite{manin1969p}. In particular, we show that the method of Chabauty--Coleman, while applicable to certain members of this family, is not the most efficient way of computing $C_a(\mathbb{Q})$. We adapt the approach of \cite{kulesz1999application}, incorporating root numbers to further restrict the possible ranks of the elliptic curves arising in the Jacobian decomposition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17791 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rational Points on a Family of Genus 3 Hyperelliptic Curves Hernandez, Roberto Number Theory Algebraic Geometry 14G05 (Primary), 11Y50 (Secondary) We compute the rational points on certain members of the following family of hyperelliptic curves \[C_a \colon y^2 = x^8 + (4-4a^4) x^6 + (8a^4 + 6)x^4 + (4-4a^4)x^2 + 1\] via the method first developed by Dem'yanenko \cite{dem1966rational} and then further generalized by Manin \cite{manin1969p}. In particular, we show that the method of Chabauty--Coleman, while applicable to certain members of this family, is not the most efficient way of computing $C_a(\mathbb{Q})$. We adapt the approach of \cite{kulesz1999application}, incorporating root numbers to further restrict the possible ranks of the elliptic curves arising in the Jacobian decomposition. |
| title | Rational Points on a Family of Genus 3 Hyperelliptic Curves |
| topic | Number Theory Algebraic Geometry 14G05 (Primary), 11Y50 (Secondary) |
| url | https://arxiv.org/abs/2510.17791 |