Isolated $j$-invariants arising from the modular curve $X_0(n)$

TL;DR

This paper introduces an algorithm to identify isolated rational non-CM $j$-invariants on modular curves $X_0(n)$, extending previous work on $X_1(n)$, and analyzes their relationship.

Contribution

It provides a new algorithm for testing isolated points on $X_0(n)$ and compares these points with those on $X_1(n)$, revealing their non-inclusion relationship.

Findings

The set of isolated $j$-invariants on $X_1(n)$ is neither a subset nor a superset of those on $X_0(n)."

The algorithm effectively distinguishes isolated points on $X_0(n)$.

Implementation results demonstrate the differences in isolated points between $X_0(n)$ and $X_1(n)$.

Abstract

An isolated point of degree $d$ is a closed point on an algebraic curve which does not belong to an infinite family of degree $d$ points that can be parameterized by some geometric object. We provide an algorithm to test whether a rational, non-CM $j$-invariant gives rise to an isolated point on the modular curve $X_0(n)$, for any $n \in \mathbb{Z}^+$, using key results from Menendez and Zywina. This work is inspired by the prior algorithm of Bourdon et al. which tests whether a rational, non-CM $j$-invariant gives rise to an isolated point on any modular curve $X_1(n)$. From the implementation of our algorithm, we determine that the set of $j$-invariants corresponding to isolated points on $X_1(n)$ is neither a subset nor a superset of those corresponding to isolated points on $X_0(n)$.