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The dynamics of the Hesse derivative on the $j$ -invariant

This paper investigates the dynamical properties of the Hesse derivative acting on the $j$ -invariant of cubic curves, analyzing orbit structures, establishing conditions for curve periodicity based on $j$ -invariant periodicity, and comparing the orbit sizes of elliptic curves with those of their invariants.

Original authors: Jake Kettinger

Published 2026-03-27

📖 5 min read🧠 Deep dive

CC0 1.0

Original authors: Jake Kettinger

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 have a magical machine in a vast, infinite playground called the Riemann Sphere. This machine doesn't just move things around; it transforms them. Specifically, it takes a special kind of shape called a cubic curve (think of a twisted, three-lobed flower drawn on a piece of paper) and turns it into a new, slightly different flower.

This machine is called the Hesse Derivative.

Now, every flower in this playground has a unique "ID card" called a j-invariant. You can think of the j-invariant as the flower's DNA or its fingerprint. Even if two flowers look slightly different, if they share the same DNA, they are considered "twins" (mathematically isomorphic).

The paper by Jake Kettinger is essentially a study of what happens when you feed a flower's ID card into a special calculator (a rational function called H) that predicts what the ID card of the new flower will be after the machine transforms it.

Here is the breakdown of the paper's journey, explained simply:

1. The Magic Calculator (The Function H)

The author defines a specific formula:
$H(j) = \frac{(6912 - j)^3}{27j^2}$

Think of this as a recipe.

You take a number (the current flower's ID).
You mix it with some ingredients (the numbers 6912, 27, and the operations of cubing and squaring).
You get a new number (the ID of the next flower).

The paper proves that if you take a flower, run it through the Hesse machine, and then check the ID of the result, it will always match the result of plugging the original ID into this recipe.

2. The Dance of Orbits (Dynamics)

The author asks: What happens if we keep feeding the result back into the machine?

Fixed Points: Sometimes, the machine outputs the exact same number you put in. It's like a dancer who spins in a circle and ends up exactly where they started. The paper finds specific IDs (like 1728) where the flower transforms but remains "essentially" the same.
Cycles: Sometimes, the machine outputs a different number, but if you run it again, and again, eventually it loops back to the start.
- Example: Flower A $\to$ Flower B $\to$ Flower C $\to$ Flower A.
- The paper calculates exactly how many of these loops exist for any given length. It's like counting how many different ways you can arrange a group of friends in a circle dance.

3. The Big Surprise: The ID vs. The Flower

Here is the most interesting part of the story.

The author discovers that the ID card (the j-invariant) and the actual flower (the curve) don't always dance to the same beat.

Sometimes, the ID card completes a loop of 3 steps (A $\to$ B $\to$ C $\to$ A).
However, the actual flower might take 6 steps to return to its original shape, even though its ID returned in 3.

The Analogy: Imagine a chameleon changing colors.

The ID is the pattern on its skin. The pattern might repeat every 3 seconds.
The Flower is the chameleon itself. It might take 6 seconds to return to its original pose, even though the pattern looked the same at second 3.

The paper proves that if the ID is periodic (it loops), the flower must also be periodic (it will eventually loop), but the flower's loop might be longer. It's like saying, "If the music repeats, the dancer will eventually repeat their moves, but they might do a few extra spins in between."

4. The Data Tables (The Evidence)

The author ran computer simulations to test this. They created tables showing:

How long the ID card takes to loop.
How long the actual flower takes to loop.

The Finding: There is no simple rule that says "If the ID loops in 3, the flower loops in 3." Sometimes the flower loops in 2, 3, 4, 6, or even 9 times the length of the ID's loop. The relationship is complex and surprising.

5. The Open Questions (What's Next?)

The paper ends by asking questions for future explorers:

The Endomorphism Question: If the flower changes, does its "internal structure" (how it connects to itself) stay the same? The author found examples where the structure changes, even if the ID loops.
The Random Walk Question: If you pick a random number and keep running it through the machine, where does it go? The author has a guess (conjecture) that the numbers will spread out evenly across the negative numbers, the middle numbers, and the large positive numbers, like a crowd of people eventually filling a whole stadium evenly.

Summary

In short, this paper is a detective story about a mathematical machine.

The Clue: There is a formula that predicts how a shape's "fingerprint" changes when the shape is transformed.
The Investigation: The author studied how these fingerprints loop and cycle.
The Twist: The fingerprint's cycle length doesn't always match the shape's cycle length. The shape might take longer to return to its original form than its ID suggests.
The Conclusion: While we can count the loops, the exact relationship between the ID and the shape is a rich, complex puzzle that mathematicians are still trying to solve.

1. Problem Statement

The paper investigates the dynamical system generated by the Hesse derivative acting on the moduli space of elliptic curves, specifically focusing on the $j$ -invariant.

The Hesse Derivative: For a cubic curve $C$ defined by a homogeneous polynomial $f$ , the Hesse derivative $H_C$ is the curve defined by the determinant of the Hessian matrix of $f$ . This operation maps cubic curves to cubic curves.
The Core Question: How does the Hesse derivative affect the $j$ -invariant of an elliptic curve? Specifically, the author studies the rational function $H(j)$ that maps the $j$ -invariant of a curve $C$ to the $j$ -invariant of its Hesse derivative $H_C$ .
Key Inquiry: If the $j$ -invariant of a curve is periodic under the iteration of $H(j)$ , is the curve itself periodic under the iteration of the Hesse derivative? Furthermore, what is the relationship between the size of the orbit of the $j$ -invariant and the size of the orbit of the curve itself?

2. Methodology

The author employs a combination of algebraic geometry, complex dynamics, and computational algebra:

Algebraic Derivation: The paper provides a direct proof deriving the rational function $H(j)$ that relates $j(C)$ to $j(H_C)$ . This involves transforming the Hessian of a Weierstrass form polynomial back into Weierstrass form to compute the new $j$ -invariant.
Complex Dynamics Analysis: The function $H(j) = \frac{(6912 - j)^3}{27j^2}$ is analyzed as a rational map on the Riemann sphere ( $\hat{\mathbb{C}}$ ). The author investigates fixed points, periodic points, multipliers, and the Julia set.
Combinatorial Counting: Using the properties of the degree of the rational function and the Möbius inversion formula, the author derives a closed-form expression for the number of periodic orbits of a given size.
Group Theory and Fibrations: To address the periodicity of the curves (not just their invariants), the paper utilizes the Hesse pencil (a family of cubic curves) and the Hessian group ( $G_{216}$ ). The author analyzes the action of the automorphism group of the pencil on the fibers to determine when $H^n E = E$ .
Computational Verification: Extensive use of computer algebra systems (Macaulay2, Mathematica, Maple) is used to verify specific cases, compute minimal polynomials for periodic points, and generate data tables comparing orbit sizes.

3. Key Contributions and Results

A. The Rational Function $H(j)$

The paper confirms and provides a direct derivation for the function relating the $j$ -invariant of a curve to its Hesse derivative:
$H(j) = \frac{(6912 - j)^3}{27j^2}$

Fixed Points: The function has four fixed points: $j = \infty$ (trilateral), $j = 1728$ (harmonic cubic), and two complex conjugate points.
Dynamics: The function is Post-Critically Finite (PCF). All critical points ($0, 6912, -13824$) are preperiodic. Consequently, the Julia set of $H$ is the entire Riemann sphere ( $J(H) = \hat{\mathbb{C}}$ ). All fixed points are repelling.

B. Counting Orbits (Theorem 1)

The author derives a formula for $B_n$ , the number of distinct orbits of size $n$ for the function $H$ on $\mathbb{C}$ :
$B_n = \frac{1}{n} \sum_{d|n} \mu(d) 3^{n/d}$
where $\mu$ is the Möbius function. This sequence corresponds to OEIS A027376. The derivation relies on the fact that $H$ is a degree 3 rational function with difference-degree 1, ensuring that the number of fixed points of the $n$ -th iterate is exactly $3^n$ .

C. Periodicity of Curves vs. Invariants (Theorem 2)

A central result addresses the relationship between the periodicity of the $j$ -invariant and the curve itself:

Theorem: If an elliptic curve $E$ has a $j$ -invariant that is periodic with period $p$ under $H$ , then the curve $E$ itself is periodic under the Hesse derivative. That is, there exists an $n$ such that $H^n E = E$ .
Bound: The period $n$ of the curve satisfies $n \le 3p$ .
Mechanism: This is proven by analyzing the action of $H$ on the fibers of the Hesse pencil induced by $E$ . The automorphism group of the pencil ( $G_{216}$ ) acts on the set of isomorphic fibers. Since the set of isomorphic fibers is finite (at most 12), the action of $H^p$ on this set must eventually cycle, forcing the curve to return to itself.

D. Discrepancy in Orbit Sizes

The paper demonstrates that the orbit size of the $j$ -invariant does not uniquely determine the orbit size of the curve.

Example: There exist curves where the $j$ -invariant has period 1 (fixed), but the curve has period 2 or 3 under the Hesse derivative.
Data: Tables 1 and 2 provide empirical data for curves with $j$ -invariants of periods 1 through 6, showing various curve orbit sizes (e.g., for a $j$ -invariant of period 1, the curve orbit can be 2 or 3; for period 2, curve orbits can be 4, 6, or 12).

4. Significance and Future Directions

Theoretical Insight: The work bridges the gap between the arithmetic dynamics of rational functions on the moduli space and the geometric dynamics of the curves themselves. It clarifies that while the $j$ -invariant captures the isomorphism class, the Hesse derivative operation depends on the specific embedding and automorphisms of the curve.
Computational Data: The paper provides the first comprehensive dataset comparing orbit sizes of $j$ -invariants versus their corresponding curves, revealing non-trivial relationships between the degree of the minimal polynomial of $j$ and the curve's orbit size.
Open Questions:
1. Algorithmic Determination: How to compute the minimal $n$ such that $H^n E = E$ given the period $p$ of $j(E)$ ?
2. Endomorphism Rings: Can elliptic curves $E$ be classified such that $\text{End}(E) = \text{End}(H_E)$ ? The paper notes that $E$ and $H_E$ are not necessarily isogenous over $\mathbb{Q}$ , even if $j(E)$ is periodic.
3. p-adic Dynamics: For which rational $j$ and prime $q$ does the sequence $H^n(j)$ converge in the $q$ -adic metric?
4. Statistical Distribution: A conjecture is proposed regarding the asymptotic distribution of the signs of the iterates $H^n(j)$ for generic real $j$ , suggesting an equal distribution among negative, small positive, and large positive values.

In summary, Kettinger's paper establishes the Hesse derivative as a well-defined dynamical system on the moduli space of elliptic curves, proves that periodicity of the $j$ -invariant implies periodicity of the curve, and provides a rigorous combinatorial framework for counting these orbits, while highlighting the subtle geometric complexities that arise beyond the $j$ -invariant.