Modular curves $X_0(N)$ of density degree $5$

TL;DR

This work determines all modular curves $X_0(N)$ with density degree $5$, proving that min(δ$(X_0(N)/\,\mathbb{Q}$)) = 5 holds iff $N=109$ and identifying a large explicit set of levels with infinitely many quintic points. The authors combine multiple techniques: ruling out positive-rank pentaelliptic maps, applying Kadets–Vogt-type structure theorems, analyzing translates of abelian varieties under specialization, and constructing explicit modular parametrizations to elliptic curves. They show that for many $N$ the curve has only finitely many degree-$5$ points, while for others (including many small levels) quintic points proliferate via degree-$5$ functions or degeneracy maps to $b P^1$. A remaining 30 levels remain open, with the open cases organized around abelian-variety translates and Debarre–Fahlaoui-type obstructions; overall, the results advance the classification of quintic points on $X_0(N)$ and lay groundwork for resolving the remaining levels under the density-degree framework.

Abstract

We determine all modular curves $X_0(N)$ with density degree $5$, i.e. all curves $X_0(N)$ with infinitely many points of degree $5$ and only finitely many points of degree $d\leq4$. As a consequence, the problem of determining all curves $X_0(N)$ with infinitely many points of degree $5$ remains open for only $30$ levels $N$.

Theorems & Definitions (73)

• Theorem 1.1: Faltings' Theorem
• Theorem 1.2: Harris, Silverman: HarrisSilverman91
• Definition 1.3
• Theorem 1.4: KV2025
• Corollary 1.5
• proof
• Theorem 1.6: Bars Bars99
• Theorem 1.7: Jeon Jeon2021
• Theorem 1.8: Hwang, Jeon; Derickx, Orlić Hwang2023DerickxOrlic23
• Theorem 1.9