Morphisms on the modular curve $X_0(p)$ and degree $6$ points
Maarten Derickx, Petar Orlić
TL;DR
The paper addresses the problem of classifying non-constant morphisms from the modular curve $X_0(p)$ to curves of genus at least $2$, showing that for primes $p<3000$ any such morphism of degree $d>1$ factors through the Atkin–Lehner quotient $X_0^+(p)$. The authors combine Jacobian decomposition techniques, endomorphism algebras, polarization kernels, and the De Franchis–Severi framework with extensive computations to establish this result and formulate a conjecture for all primes. They then classify points of degree up to $25$ arising from maps to genus-$2$ or higher curves and determine which $X_0(p)$ have infinitely many degree-$6$ points, identifying a finite list of exceptions (notably $p=193$). The work further employs Kadets–Vogt density results and Frey bounds to prove finiteness of degree-$6$ points for most levels, while providing explicit constructions that yield infinite degree-$6$ points for several primes, supported by computational verification and modular-parametrization arguments. Overall, the paper advances understanding of gonality, degree-bounded points, and morphisms on $X_0(p)$ through a blend of theoretical and computational techniques.
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 $X_0(N)$.