Sporadic points on $X_0(N)$

TL;DR

This paper classifies all integers N for which the modular curve X_0(N) has sporadic points, including CM and non-CM points, extending previous classifications of elliptic curve isogenies.

Contribution

It generalizes the classification of sporadic points on X_0(N) to include points of any degree, not just degree 1, broadening the understanding of modular curves.

Findings

Identifies all N with sporadic CM points on X_0(N).

Identifies all N with sporadic points (CM or non-CM) on X_0(N).

Extends Mazur and Kenku's classification to arbitrary degree points.

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.