Non-hyperelliptic modular curves of genus 3
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911456962805760 |
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| author | González-Jiménez, Enrique Oyono, Roger |
| author_facet | González-Jiménez, Enrique Oyono, Roger |
| contents | A curve $C$ defined over $\mathbb Q$ is modular of level $N$ if there exists a non-constant morphism from $X_1(N)$ onto $C$ defined over $\mathbb Q$ for some positive integer $N$. We provide a sufficient and necessary condition for the existence of a modular non-hyperelliptic curve $C$ of genus $3$ and level $N$ such that $\mathrm{Jac}{(C)}$ is $\mathbb Q$-isogenous to a given three dimensional $\mathbb Q$-quotient of $J_1 (N)$. Using this criterion, we present an algorithm to compute explicitly equations for modular non-hyperelliptic curves of genus $3$. Let $C$ be a modular curve of level $N$, we say that $C$ is new if the corresponding morphism between $J_1(N)$ and $\mathrm{Jac}{(C)}$ factors through the new part of $J_1(N)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_17167 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-hyperelliptic modular curves of genus 3 González-Jiménez, Enrique Oyono, Roger Number Theory Algebraic Geometry A curve $C$ defined over $\mathbb Q$ is modular of level $N$ if there exists a non-constant morphism from $X_1(N)$ onto $C$ defined over $\mathbb Q$ for some positive integer $N$. We provide a sufficient and necessary condition for the existence of a modular non-hyperelliptic curve $C$ of genus $3$ and level $N$ such that $\mathrm{Jac}{(C)}$ is $\mathbb Q$-isogenous to a given three dimensional $\mathbb Q$-quotient of $J_1 (N)$. Using this criterion, we present an algorithm to compute explicitly equations for modular non-hyperelliptic curves of genus $3$. Let $C$ be a modular curve of level $N$, we say that $C$ is new if the corresponding morphism between $J_1(N)$ and $\mathrm{Jac}{(C)}$ factors through the new part of $J_1(N)$. |
| title | Non-hyperelliptic modular curves of genus 3 |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2602.17167 |