Non-hyperelliptic modular curves of genus 3

Abstract

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 $J_1(N)$.

Typos in the labels of the curves in lines 31, 35, 36, 38, 39, 41, 42, and 43 of Table 1 have been fixed. The equation in line 41 has also been corrected, with respect to the published version

References in corpus (1)

• Modular Curves Of Genus 2

Cited by in corpus (1)

• On smooth plane models for modular curves of Shimura type