Tetragonal intermediate modular curves

TL;DR

This paper classifies intermediate modular curves $X_\Delta(N)$ with specific low gonality values over both $\,\mathbb{Q}$ and $\,\mathbb{C}$, providing a comprehensive gonality analysis for $N\leq 40$.

Contribution

It determines all $X_\Delta(N)$ with gonality 4 or 5 over $\,\mathbb{Q}$ and $\,\mathbb{C}$, and computes gonality for all such curves with $N\leq 40$.

Findings

Classified all $X_\Delta(N)$ with $\,\mathbb{Q}$-gonality 4 or 5.

Computed gonality for all $X_\Delta(N)$ with $N\leq 40$.

Provided explicit gonality values for 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 $\{\pm1\}\subsetneq \Delta \subsetneq (\mathbb{Z}/N\mathbb{Z})^\times$.