Infinitely many cubic points on $X_0(N)/\langle w_d\rangle$ with $N$ square-free

TL;DR

This paper classifies modular curves of the form $X_0(N)/\langle w_d\rangle$ with square-free $N$ that have infinitely many cubic points over $\mathbb{Q}$, providing a complete characterization.

Contribution

It provides a complete classification of such modular curves with infinitely many cubic points over $\mathbb{Q}$ for square-free levels.

Findings

Identifies all $X_0(N)/\langle w_d\rangle$ with infinite cubic points

Characterizes the structure of these modular curves

Advances understanding of rational points on modular curves

Abstract

We determine all modular curves $X_0(N)/\langle w_d\rangle$ that admit infinitely many cubic points over the rational field $\mathbb{Q}$, when $N$ is square-free.