Morphisms on the modular curve $X_0(p)$ and degree $6$ points

TL;DR

This paper investigates morphisms from modular curves $X_0(p)$ to higher genus curves, proving that for primes less than 3000, such morphisms of degree greater than one are essentially quotient maps, and classifies low-degree points on these curves.

Contribution

It establishes that for primes under 3000, all non-trivial morphisms from $X_0(p)$ to higher genus curves are quotient maps, supporting a conjecture for all primes, and classifies degree ≤25 points.

Findings

All degree >1 morphisms for p<3000 are quotient maps.

Classified all points of degree ≤25 on $X_0(p)$.

Identified all $X_0(p)$ with infinitely many degree 6 points, except for p=193.

Abstract

Let $p$ be a prime. We study non-constant morphisms $f:X_0(p)_\mathbb \to Y$, where $Y/\mathbb Q$ is a curve of genus $\geq 2$. We prove that for $p<3000$ such an $f$ of degree $d>1$ must be isomorphic to the quotient map $X_0(p)\to X_0^+(p)$. Supported by computational and theoretical evidence, we also conjecture that this is true for all primes $p$. These results allow us to classify all points of degree $\leq 25$ on $X_0(p)$ that come from a map to some curve of genus $\geq 2$. As an application, we were able to determine all curves $X_0(p)$ with infinitely many points of degree $6$ over $\mathbb Q$ except for $p=193$, continuing the previous results on small degree points on $X_0(N)$.