Tetragonal intermediate modular curves

Abstract

For every group $\{\pm1\}\subseteq \Delta\subseteq (\mathbb{Z}/N\mathbb{Z})^\times$, 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 $N\leq 40$ and $\{\pm1\}\subsetneq \Delta \subsetneq (\mathbb{Z}/N\mathbb{Z})^\times$.