Tetragonal modular quotients of $X_0(N)$

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 $X_0(N)$. Let $B(N)$ be the group of all such involutions. In this paper we determine all $\mathbb C$ and $\mathbb Q$-tetragonal quotient curves $X_0(N)/W_N$, where $W_N\subseteq B(N)$ such that $4\leq|W_N|\leq 2^{\omega(N)-1}$, thus completing the classification of all $\mathbb C$-tetragonal quotients of $X_0(N)$ by Atkin-Lehner involutions.