Non-hyperelliptic modular curves of genus 3

TL;DR

This paper establishes a criterion for the existence of non-hyperelliptic modular curves of genus 3 and provides an algorithm to explicitly compute their equations, advancing understanding of their structure and classification.

Contribution

It introduces a necessary and sufficient condition for such curves and develops an explicit computational method for their equations.

Findings

Criteria for existence of modular non-hyperelliptic genus 3 curves.

Algorithm for computing explicit equations of these curves.

Insight into the structure of Jacobians related to modular curves.

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 $J_1(N)$.