Tetragonal modular quotients $X_0^{+d}(N)$

TL;DR

This paper completes the classification of tetragonal quotients of the modular curve $X_0(N)$ in the Atkin–Lehner setting by determining all pairs $(N,d)$ with $d|N$, $(d,N/d)=1$, for which $X_0^{+d}(N)=X_0(N)/w_d$ has $ ext{gon}_{\mathbb{Q}}=4$ or $\text{gon}_{\mathbb{C}}=4$. The authors combine lower bounds from finite-field gonality via Ogg-type inequalities, explicit degree-4 maps constructed from quotient curves and divisors, Castelnuovo–Severi-type bounds for $\mathbb{C}$-gonality, and Green’s conjecture–type Betti-number obstructions to exclude many cases, supported by extensive Magma computations. They provide comprehensive tables and a public codebase, delivering a near-complete landscape for levels up to $N\le 806$ and offering a framework for tackling remaining instances. The results advance understanding of how Atkin–Lehner quotients constrain possible maps to $\mathbb{P}^1$, with implications for arithmetic and geometric properties of modular curves and their quotients.

Abstract

Let $N$ be a positive integer. For every $d | N$ such that $(d, N/d) = 1$ there exists an Atkin-Lehner involution $w_d$ of the modular curve $X_0(N)$. In this paper we determine all quotient curves $X_0(N)/w_d$ whose $\mathbb{Q}$-gonality is equal to $4$ and all quotient curves $X_0(N)/w_d$ whose $\mathbb{C}$-gonality is equal to $4$.

Theorems & Definitions (39)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5
• proof