Non-hyperelliptic modular curves of genus 3

TL;DR

This work develops a criterion to realize non-hyperelliptic genus $3$ modular curves C over $\mathbb{Q}$ with ${\rm Jac}(C)\stackrel{\mathbb{Q}}{\sim} A$, where $A$ is a 3-dimensional quotient of $J_1(N)$. The criterion uses a basis $f_1,f_2,f_3$ of $S_2(A)$ and a degree-4 form $F$ with $F(f_1,f_2,f_3)=0$, together with the constant-ness of the associated modular function $\psi_F$, to produce a plane quartic model for $C$. An explicit computational algorithm then constructs $F$ from $q$-expansions, enabling explicit equations for the curves; applied to a range of levels, this yields 44 new non-hyperelliptic genus-$3$ modular curves with verified Jacobians. The paper also analyzes automorphisms of new curves, distinguishes new vs non-new cases, and discusses limitations and open questions on finiteness and completeness of the modular-genus-$3 landscape.

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 $J_1(N)$.

Theorems & Definitions (20)

• Definition 1
• Definition 2
• Theorem 1
• Proposition 1
• Lemma 1
• Proposition 2
• Lemma 2
• Remark 1
• Remark 2
• Lemma 3