Tetragonal modular quotients of $X_0(N)$

TL;DR

The paper classifies all tetragonal quotients of the modular curve $X_0(N)$ obtained by Atkin-Lehner involutions $W_N$ with $4 \leq |W_N| \leq 2^{\omega(N)-1}$. It integrates multiple techniques—bounds from $\mathbb{F}_p$-gonality, Betti-number analyses, Castelnuovo-Severi inequalities, explicit rational maps, and isomorphisms among quotient curves—to determine both $\mathbb{Q}$- and $\mathbb{C}$-gonality, producing a complete list of $\,\mathbb{Q}$-tetragonal and $\,\mathbb{C}$-tetragonal cases and confirming that no new $\,\mathbb{C}$-tetragonal examples exist beyond those for which the $\mathbb{Q}$-gonality is also $4$. The work highlights substantial computational use (Magma) for point counts and map constructions, resolves several prior ambiguities (e.g., certain trigonal cases and isomorphisms), and delivers a comprehensive table of tetragonal quotients, advancing the classification program for gonality of modular 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 $X_0(N)$. 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 $4\leq|W_N|\leq 2^{ω(N)-1}$, thus completing the classification of all $\mathbb C$-tetragonal quotients of $X_0(N)$ by Atkin-Lehner involutions.

Theorems & Definitions (50)

• Proposition 1.1: Poonen2007
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1: The Tower Theorem, Poonen2007
• Corollary 2.2
• Proposition 2.3
• proof
• Proposition 3.1
• proof
• Lemma 3.2