Tetragonal modular quotients $X_0^*(N)$

TL;DR

The paper classifies the tetragonal quotients $X_0^*(N)=X_0(N)/B(N)$ of modular curves by combining genus-based analyses for low-genus cases, $ ext{F}_p$-gonality bounds, Betti-number obstructions, and explicit degree-$4$ morphisms obtained from Atkin–Lehner involutions. It reports a complete determination of levels with $ ext{C}$-gonality $4$ and all levels with $ ext{Q}$-gonality $4, except $N=378$ for the latter, using extensive Magma computations to model curves and construct maps. The approach integrates classical bounds (Abramovich–Kim–Sarnak, Ogg) with modern computational techniques (newform decompositions, Betti numbers, and quotient involutions) to exhaustively test candidate levels and produce explicit degree-$4$ maps when possible. The work extends the tetragonal classification of $X_0^*(N)$ and clarifies the status of several previously unresolved cases, while identifying $N=378$ as the sole outstanding genus-5 instance and outlining avenues to resolve it.

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 $X_0(N)$. The curve $X_0^*(N)$ is a quotient curve of $X_0(N)$ by $B(N)$, the group of all involutions $w_d$. In this paper we determine all quotient curves $X_0^*(N)$ whose $\mathbb C$-gonality is equal to $4$. We also determine all curves $X_0^*(N)$ whose $\mathbb Q$-gonality is equal to $4$ with the exception of level $N=378$.

Theorems & Definitions (52)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1: Poonen2007
• Theorem 2.2
• Corollary 2.3
• proof
• Lemma 2.4: Ogg
• Lemma 2.5
• Proposition 3.1
• proof