Modular curves $X_0(N)$ with infinitely many quartic points

TL;DR

This paper classifies all modular curves $X_0(N)$ that have infinitely many quartic points by analyzing rational morphisms to elliptic curves using a new quadratic form pairing.

Contribution

It introduces a novel pairing that helps determine the degrees of rational morphisms from $X_0(N)$ to elliptic curves, leading to a complete classification.

Findings

Identifies all $X_0(N)$ with infinitely many quartic points.

Develops a pairing inducing a quadratic form for morphism degrees.

Provides a method to analyze rational points on modular curves.

Abstract

We determine all modular curves $X_0(N)$ with infinitely many quartic points. To do this, we define a pairing that induces a quadratic form representing all possible degrees of a rational morphism from $X_0(N)$ to a positive rank elliptic curve.