Sporadic points on $X_0(N)$

TL;DR

This work completely classifies when the modular curves $X_0(N)$ admit sporadic points, distinguishing sporadic CM points and general sporadic points. It combines Kadets–Vogt low-degree bounds, Abramovich gonality, and Mordell–Weil sieving to rule out low-degree points, and then treats CM points degree-by-degree, computing $\delta(X_0(N))$ and exploiting involutions to exclude non-CM sporadic points in critical cases. The results show $X_0(N)$ has a sporadic CM point iff $N\notin S_{CM}$, and a sporadic point (CM or non-CM) iff $N\notin (S_{CM}\setminus\{15,17,21,37\})$, extending Mazur–Kenku’s degree-1 classification to arbitrary degrees. Notable conclusions include explicit determinations for $X_0(144)$ and $X_0(720)$ and a finite, verification-heavy analysis that completes the characterization for all $N$.

Abstract

We determine all integers $N$ for which the modular curve $X_0(N)$ admits a sporadic CM point (of any degree), as well as all $N$ for which $X_0(N)$ admits a sporadic point, whether CM or non-CM. In a sense, our results generalize the classification of isogenies of elliptic curves over $\Q$ due to Mazur and Kenku: their work determines the $X_0(N)$ with degree 1 sporadic points, whereas we classify all $X_0(N)$ that have a sporadic point of arbitrary degree.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Theorem 3.1: KadetsVogt25
• Lemma 3.2
• proof
• Remark 3.3
• Remark 3.4
• Proposition 4.1
• proof
• Proposition 4.2