Tetragonal intermediate modular curves

Abstract

For every group $\{\pm 1\}\subseteq \Delta \subseteq (\mathbb{Z}/N\mathbb{Z}{)}^{\times }$, there exists an intermediate modular curve $X_{\Delta }(N)$. In this paper we determine all curves $X_{\Delta }(N)$ whose $\mathbb{Q}$-gonality is equal to $4$, all curves $X_{\Delta }(N)$ whose $\mathbb{C}$-gonality is equal to $4$, and all curves $X_{\Delta }(N)$ whose $\mathbb{Q}$-gonality is equal to $5$. We also determine the $\mathbb{Q}$-gonality of all curves $X_{\Delta }(N)$ for $N\le 40$ and $\{\pm 1\}⊊\Delta ⊊(\mathbb{Z}/N\mathbb{Z}{)}^{\times }$.