Tetragonal modular quotients $X_0^*(N)$

TL;DR

This paper classifies certain modular quotient curves $X_0^*(N)$ with gonality equal to 4 over complex numbers and rationals, expanding understanding of their geometric properties.

Contribution

It determines all $X_0^*(N)$ curves with gonality 4 over $ ext{C}$ and $ ext{Q}$, except for one specific level, providing a comprehensive classification.

Findings

Identified all $X_0^*(N)$ with complex gonality 4.

Identified all $X_0^*(N)$ with rational gonality 4, excluding $N=378$.

Enhanced understanding of the structure of modular quotient curves.

Abstract

Let $N$ be a positive integer. For every $d\mid N$ such that $(d,N/d)=1$ there exists an Atkin-Lehner involution $w_d$ of the modular curve $X_0(N)$. The curve $X_0^*(N)$ is a quotient curve of $X_0(N)$ by $B(N)$, the group of all involutions $w_d$. In this paper we determine all quotient curves $X_0^*(N)$ whose $\mathbb C$-gonality is equal to $4$. We also determine all curves $X_0^*(N)$ whose $\mathbb Q$-gonality is equal to $4$ with the exception of level $N=378$.