🔢 mathematics

Automorphism groups of hyperelliptic curves of $2$-rank zero

This paper determines the reduced automorphism groups of low-genus hyperelliptic curves of 2-rank zero in characteristic 2 by establishing their semidirect-product structures and deriving specific group structures via Magma computations, ultimately formulating two conjectures analogous to the Oort conjecture.

Original authors: Kohtaro Yamaguchi, Shushi Harashita

Published 2026-04-21

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Kohtaro Yamaguchi, Shushi Harashita

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 an architect designing a very specific, intricate type of building called a Hyperelliptic Curve. In the world of mathematics, these aren't physical buildings, but complex shapes defined by equations.

This paper is like a detailed inspection report on a specific, rare subset of these buildings: those built in a world where the rules of math are slightly different (called "Characteristic 2") and which have a special property called "2-rank zero."

Here is the breakdown of what the authors, Kohtaro Yamaguchi and Shushi Harashita, discovered, explained in simple terms:

1. The Building Blocks: The "Artin-Schreier" House

Think of these curves as houses built with a specific blueprint. In this special math world, every house of this type follows a formula that looks like this:
$y^2 - y = f(x)$
Where $f(x)$ is a polynomial (a fancy algebraic expression). The authors focus on a specific version of this formula where the shape of the house is determined by a number $n$ (which relates to the "genus" or complexity of the curve).

2. The Symmetry Test: "Automorphisms"

The main question the authors asked is: "How symmetrical are these houses?"

In math, an automorphism is a way to rotate, flip, or shift a shape so that it looks exactly the same as it did before.

Imagine a square: You can rotate it 90 degrees, and it looks the same. It has high symmetry.
Imagine a lopsided rock: You can barely move it without it looking different. It has low symmetry.

The authors wanted to count exactly how many ways you can "move" these specific curves without changing their appearance. They call this collection of moves the Automorphism Group.

3. The "Reduced" Group: Ignoring the Trivial Flip

Every one of these curves has one very obvious, boring symmetry: a simple flip that swaps the top and bottom halves. The authors decided to ignore this "boring flip" to see the real interesting symmetries. They call this the Reduced Automorphism Group.

Think of it like looking at a person. Everyone has a reflection in a mirror. That's the "boring flip." The authors wanted to know: "If we ignore the mirror image, how many other ways can this person stand and still look like themselves?"

4. The Big Discovery: The "Oort Conjecture" Analogy

There is a famous idea in math called the Oort Conjecture. It basically says: "If you pick a random, generic supersingular abelian variety (a very complex math object), it will have almost no symmetry. It will be unique and rigid."

The authors asked: "Does this rule apply to our specific Hyperelliptic curves?"

They ran massive computer simulations (using a tool called Magma, which is like a super-calculator for algebra) to check curves with small complexities (genus 1 through 9).

The Results:

For most cases (Genus 3 to 6): The answer was YES. The generic curves were very rigid. They had almost no symmetry (just the trivial flip). This supports the idea that "generic" curves are unique.
The Surprise (Genus 2, 4, 8): The answer was NO. When the complexity of the curve was a power of 2 (like 2, 4, or 8), the curves suddenly became very symmetrical. They had huge groups of symmetries.

5. The New Theory: "The Power of Two" Rule

Because of this surprise, the authors formulated a new guess (a conjecture):

"If you have a generic supersingular hyperelliptic curve in this math world, it will have almost no symmetry UNLESS its complexity is a power of 2 (2, 4, 8, 16...). If it is a power of 2, it will be highly symmetrical."

Why is this important?

Classification: Just like biologists classify animals by their traits, mathematicians classify curves by their symmetries. This paper adds a new chapter to that classification book.
Counter-Examples: They found that the old "Oort Conjecture" isn't a universal law for these specific curves; it has a "loophole" involving powers of 2.
Computational Power: They used computers to solve equations that are too messy for humans to do by hand, proving that sometimes you need a digital assistant to see the pattern.

Summary Analogy

Imagine a gallery of sculptures.

The Oort Conjecture predicted that if you pick a random sculpture, it will be a unique, asymmetrical rock.
Yamaguchi and Harashita looked at a specific type of sculpture (the "2-rank zero" ones).
They found that for most sizes, the sculptures were indeed unique rocks.
However, they discovered that if the sculpture was built in sizes of 2, 4, or 8 units, they were actually perfect, highly symmetrical spheres that could be spun in many ways.

They concluded: "The rule about uniqueness holds true, except when the size is a power of two."

1. Problem Statement and Motivation

The paper investigates the automorphism groups of hyperelliptic curves defined over an algebraically closed field $K$ of characteristic 2, specifically focusing on curves with 2-rank 0 (which includes supersingular curves).

Context: The research is motivated by Oort's Conjecture, which posits that the automorphism group of a generic principally polarized supersingular abelian variety of dimension $g \ge 2$ is isomorphic to $\mathbb{Z}_2$ (the group $\{\pm 1\}$ ).
Specific Question: The authors ask if an analogous statement holds for hyperelliptic curves of genus $g \ge 2$ with 2-rank 0 in characteristic 2. Specifically, is the automorphism group of a generic such curve simply $\mathbb{Z}_2$ (generated by the hyperelliptic involution)?
Gap in Knowledge: While Oort's conjecture has been resolved for general cases (excluding small genera and specific characteristics), the behavior of the specific subclass of 2-rank 0 hyperelliptic curves in characteristic 2 requires distinct analysis. Previous classifications (e.g., Igusa for genus 2) did not restrict to the 2-rank 0 condition in characteristic 2.

2. Mathematical Framework

The study focuses on Artin-Schreier curves defined by the equation:
$C: y^2 - y = f(x)$
where $f(x) = x(x^{2n} + a_{n-1}x^{2(n-1)} + \dots + a_1x^2 + a_0)$ .

Genus: The genus of such a curve is $g = n$ .
2-Rank 0: It is established that any hyperelliptic curve of 2-rank 0 in characteristic 2 can be transformed into this specific form.
Automorphisms: An automorphism $\sigma$ of $C$ is induced by a coordinate transformation:
$x \mapsto \alpha x + \beta, \quad y \mapsto y + \gamma_n x^n + \dots + \gamma_0$
where $\alpha, \beta, \gamma_i \in K$ and $\alpha \neq 0$ .
Reduced Automorphism Group ($RA(C)$): The full automorphism group $Aut(C)$ contains the hyperelliptic involution $\sigma: (x, y) \mapsto (x, y+1)$ . The quotient group $RA(C) = Aut(C) / \langle \sigma \rangle$ is the reduced automorphism group. Determining $RA(C)$ is central to the paper.

3. Methodology

The authors employ a combination of theoretical algebraic geometry and computational algebra:

Theoretical Structure Analysis:
- They define homomorphisms $\rho_n: Aut(C) \to \mu_{2n+1}$ (mapping to the scaling factor $\alpha$ ) and $\tau_n: Aut(C) \to M_n$ (mapping to the affine transformation $(\alpha, \beta)$ ).
- They prove that $Aut(C)$ has a semidirect product structure involving the kernel of these maps and the image groups.
- Specifically, they establish that $RA(C) \cong U_n \rtimes \mathbb{Z}_{\#Im\rho_n}$ , where $U_n$ is an elementary abelian 2-group (related to translations $\beta$ ).
Computational Algebra (Magma):
- To determine the specific group structures for small genera ( $n \le 6$ ), the authors use the computer algebra system Magma.
- Algorithm: They construct an ideal generated by the polynomial conditions required for the coordinate transformation to be an automorphism (derived by equating coefficients of $Y^2 - Y = f(X)$ ).
- Gröbner Bases: They compute Gröbner bases of these ideals using lexicographic monomial orders (specifically $\beta \succ \alpha$ ) to solve for the number of valid solutions for $\alpha$ and $\beta$ under various conditions on the coefficients $a_i$ .
- This allows them to count the order of $Im\rho_n$ and $U_n$ and determine the group structure based on the vanishing or non-vanishing of specific polynomial expressions in the coefficients $a_i$ .
Extension to Scholten-Zhu Curves:
- They apply similar methods to specific supersingular curves of genus $g \le 9$ defined by Scholten and Zhu, which have specific sparse polynomial forms.

4. Key Contributions and Results

A. Theoretical Structure (Theorem 1)

The authors prove that for any $n$ :

$Ker(\tau_n) = \langle \sigma \rangle$ , so $RA(C) \cong Im(\tau_n)$ .
$RA(C) \cong U_n \rtimes \mathbb{Z}_{\#Im\rho_n}$ , where $U_n$ is an elementary abelian 2-group ( $\mathbb{Z}_2^\ell$ ).
If $n$ is odd and $n \ge 3$ , then $U_n$ is trivial ( $\{e\}$ ), meaning $RA(C)$ is cyclic.

B. Detailed Classification for Small Genus (Theorem 2)

For $n = 1, \dots, 6$ , the authors provide a complete classification of $RA(C)$ and $|Aut(C)|$ based on the coefficients $a_i$ .

Generic Case: For "generic" coefficients (where specific polynomial conditions are non-zero), the reduced automorphism group is often trivial ( $Z_1$ ), implying $Aut(C) \cong \mathbb{Z}_2$ .
Special Cases: When specific coefficients vanish (e.g., $a_1=0$ for $n=2$ , or all lower coefficients vanish for higher $n$ ), the group size increases significantly, often involving cyclic groups of order $2n+1$ or larger 2-groups.
Example: For $n=2$ (Genus 2), if $a_1=0$ , $RA(C) \cong \mathbb{Z}_2^4 \rtimes \mathbb{Z}_5$ , leading to $|Aut(C)| = 160$.

C. Scholten-Zhu Curves (Theorem 3)

The authors analyze specific supersingular curves of genus $g \in \{1, \dots, 9\}$ .

They find that for genera $g=2, 4, 8$ (powers of 2), the automorphism groups can be large even in the "generic" supersingular case (depending on parameters $c_i$ ).
For $g=9$ (not a power of 2), the generic case yields a small group.

D. New Conjectures

Based on the computational results, the authors formulate two refined conjectures:

Conjecture 1 (2-rank 0): The automorphism group of a generic hyperelliptic curve of genus $g > 2$ and 2-rank 0 in characteristic 2 is $\{\pm 1\}$ (isomorphic to $\mathbb{Z}_2$ ).
Conjecture 2 (Supersingular): The automorphism group of a generic supersingular hyperelliptic curve of genus $g$ $g$ in characteristic 2 is $\{\pm 1\}$ ${\pm 1}$ if and only if $g$ $g$ is not a power of 2.
- Significance: This suggests that supersingular curves of genus $2^k$ possess "exceptional" symmetries that generic curves of other genera do not.

5. Significance and Future Work

Counter-Example Refinement: The results clarify that while Oort's conjecture holds for generic cases in many dimensions, the restriction to hyperelliptic curves of 2-rank 0 in characteristic 2 reveals a dependence on the genus being a power of 2.
Computational Limits: The paper highlights the difficulty of extending these results to higher genera ( $n > 6$ ) due to the complexity of the Gröbner basis computations.
Future Directions: The authors propose a theoretical approach to generalize the results for arbitrary $n$ $n$ , particularly focusing on:
- Classifying the image of $\rho_n$ for odd $n$ .
- Determining the kernel structure for even $n$ .
- Investigating the pattern where $U_n$ appears to be non-trivial only when $n$ is a power of 2.

In summary, this paper provides a rigorous computational and theoretical classification of automorphism groups for a specific class of hyperelliptic curves in characteristic 2, challenging and refining existing conjectures regarding the symmetry of supersingular abelian varieties.