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

This paper investigates non-constant morphisms from the modular curve $X_0(p)$ to curves of genus at least 2, proving that for certain primes $p$ , the only such morphism is the quotient map to $X_0^+(p)$ , and uses this result to classify low-degree points on $X_0(p)$ and identify curves with infinitely many degree 6 points.

Original authors: Maarten Derickx, Petar Orlić

Published 2026-02-12

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Maarten Derickx, Petar Orlić

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to map out all the secret, hidden paths in a massive, ancient labyrinth. This labyrinth is a mathematical structure called a Modular Curve ( $X_0(p)$ ).

These labyrinths are incredibly complex, but they have a special property: they are built out of "modular forms," which are like the DNA of number theory. Mathematicians want to know how many "shortcuts" exist through these labyrinths—specifically, how many ways you can travel through them using only a certain number of "steps" (which we call the degree of a point).

Here is a breakdown of what this paper achieves, using the analogy of a Grand Architectural Survey.

1. The "One True Shortcut" (The Morphism Problem)

Imagine you are looking at a giant, complex castle ( $X_0(p)$ ). You want to know if there are any simpler, smaller buildings ( $Y$ ) that this castle can be "squashed" into without losing its essential shape. In math, this "squashing" is called a morphism.

The authors investigated whether these giant castles have any secret, unusual ways of being squashed into smaller, simpler buildings (curves with a "genus" or complexity of 2 or more).

The Discovery: For almost every castle they checked (up to a massive size of $p < 3000$ ), they found that there is really only one standard way to squash the castle: by using a specific "mirror" called the Atkin-Lehner involution. It’s like discovering that every complex castle in a kingdom, no matter how big, only has one specific, predictable secret door.

2. The "Six-Step" Rule (Degree 6 Points)

Now, imagine you are a traveler. You want to know: "Can I find an infinite number of ways to visit this castle if I am only allowed to use 6 steps at a time?"

In math, "infinitely many points of degree 6" means there are endless ways to navigate the curve using a specific level of complexity.

The Discovery: The authors created a master list. They categorized every single "castle" (prime level $p$ ) and told us exactly which ones allow for infinite 6-step journeys and which ones are "dead ends" where you can only find a finite number of such paths.

The Winners: They identified a specific group of primes (like $p \leq 151$ or certain others like $269$) where the paths are endless.
The Losers: For almost all other large castles, the 6-step paths eventually run out.

3. The "Mystery of 193" (The Unsolved Case)

Every good detective story has a "cold case." In this paper, that case is the number 193.

The authors found that the castle at level 193 is a strange, stubborn outlier. It doesn't fit the easy patterns they found for the others. They can't prove if it has infinite 6-step paths or not. It sits there in the middle of their map, a blurry, mysterious island that refuses to be fully surveyed.

Summary for the Non-Mathematician

If the universe of numbers were a collection of vast, intricate landscapes, this paper is a high-resolution GPS map.

It tells us:

How the landscapes are connected: Most of them only connect to other shapes in one very specific, predictable way.
Where the infinite roads are: It tells us exactly which landscapes allow for endless travel using a "6-step" movement pattern.
Where the fog is: It points out the one specific location (193) where our current maps fail and we need more light.

This paper, titled "Morphisms on the Modular Curve $X_0(p)$ and Degree 6 Points" by Maarten Derickx and Petar Orlić, investigates the existence of non-constant morphisms from the modular curve $X_0(p)$ (where $p$ is prime) to curves of higher genus and uses these findings to classify modular curves with infinitely many points of degree 6.

1. The Problem

The authors address two interconnected problems in arithmetic geometry:

Morphism Classification: Determining the possible non-constant rational morphisms $f: X_0(p) \to Y$ , where $Y$ is a curve of genus $g(Y) \geq 2$ .
Degree $d$ Points: Identifying which modular curves $X_0(p)$ possess infinitely many points of a specific degree $d$ over $\mathbb{Q}$ . Specifically, the paper focuses on the case $d=6$ .

The study of these points is significant because, by Faltings' Theorem, a curve has infinitely many rational points only if its genus is 0 or 1. For higher degrees, the existence of infinitely many points is tied to the curve's gonality and its ability to map to curves of lower genus (like $\mathbb{P}^1$ or elliptic curves) or to admit specific geometric structures (like Debarre-Fahlaoui curves).

2. Methodology

The authors employ a combination of theoretical arithmetic geometry and heavy computational verification:

Jacobian Decomposition: They utilize the fact that for prime levels $p$ , the Jacobian $J_0(p)$ is isogenous to a product of simple abelian varieties that occur with multiplicity 1. This allows them to enumerate all abelian subvarieties of $J_0(p)$ using SageMath.
Riemann-Hurwitz Formula & Polarization: They use the properties of the induced polarization on abelian subvarieties and the Riemann-Hurwitz formula to bound the possible degrees of morphisms and the genera of the target curves.
Computational Enumeration: For $p < 3000$ , they exhaustively check all abelian subvarieties to classify morphisms. For larger $p$ , they use analytic bounds on class numbers and genus formulas to prove that certain morphisms cannot exist.
Symmetric Powers and Brill-Noether Theory: To handle the case of degree 6 points, they analyze the image of the $d$ -th symmetric power of the curve in its Jacobian (the $W_d^0$ variety) and use results relating the existence of infinitely many points to the presence of positive-rank abelian subvarieties within these images.

3. Key Contributions and Results

A. Classification of Morphisms (The "Main Conjecture" Support)

Theorem 1.3: For $p < 3000$ , any non-constant morphism $f: X_0(p) \to Y$ (with $g(Y) \geq 2$ and $\text{deg}(f) > 1$ ) is isomorphic to the standard quotient map $X_0(p) \to X_0^+(p)$ (the map induced by the Atkin-Lehner involution $w_p$ ).
Theorem 1.4: They provide a theoretical condition under which this result holds for all primes $p$ , based on the dimension of the abelian subvarieties of $J_0(p)$ . They show this condition applies to $\approx 98.9\%$ of primes below $1,000,000$.
Conjecture 1.5: Based on their evidence, they conjecture that for any prime $p$ , the only such morphism is the Atkin-Lehner quotient map.

B. Classification of Degree 6 Points

The paper provides a complete classification of prime-level modular curves $X_0(p)$ with infinitely many points of degree 6 over $\mathbb{Q}$ :

Theorem 1.9: $X_0(p)$ has infinitely many degree 6 points if and only if $p \leq 151$ or $p \in \{163, 167, 179, 181, 191, 227, 239, 269\}$ .
The case $p = 193$ is identified as the sole remaining exception where the status is currently unknown.

4. Significance

This work advances the program of classifying "low degree points" on modular curves, a field that connects the geometry of curves to the arithmetic of elliptic curves and modular forms.

Completeness: It extends previous work (which covered degrees $d \leq 5$ ) to the case $d=6$ , providing a much more complete picture of the arithmetic of $X_0(p)$ .
Algorithmic Utility: The methods used to bound the genus and analyze the Jacobian decomposition provide a roadmap for studying higher degrees ( $d \geq 7$ ), although the authors note that increasing complexity makes such computations significantly harder.
Theoretical Insight: The results reinforce the idea that the Atkin-Lehner involution is the "dominant" geometric feature of these curves when considering maps to higher-genus targets.