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On superspecial hyperelliptic curves of genus 5 whose automorphism groups contain $(\mathbb{Z}/2\mathbb{Z})^3$

This paper presents and implements an algorithm in Magma to successfully enumerate superspecial hyperelliptic curves of genus 5 with automorphism groups containing $(\mathbb{Z}/2\mathbb{Z})^3$ across all characteristics between 11 and 1000, addressing the limited understanding of such curves in higher genera.

Original authors: Ryo Ohashi, Momonari Kudo

Published 2026-03-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ryo Ohashi, Momonari Kudo

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 trying to build a very specific type of bridge. But instead of steel and concrete, you are building with mathematical shapes called "curves."

This paper is about finding a specific kind of bridge that is not only beautiful but also "super-strong" (mathematically known as superspecial) and has a very specific, symmetrical design (containing a group of symmetries called $(\mathbb{Z}/2\mathbb{Z})^3$ ).

Here is the story of how the authors, Ryo Ohashi and Momonari Kudo, went on a treasure hunt to find these bridges.

1. The Setting: A World of Mathematical Bridges

In the world of mathematics, curves are like roads. Some roads are simple loops, others are complex, twisting paths.

Genus: This is a measure of how "twisty" or "hole-filled" the road is. A genus-1 curve is a simple loop (like a donut). A genus-5 curve is a complex road with 5 holes.
Superspecial: This is the "Gold Standard" of these roads. A superspecial curve is incredibly special because its underlying structure is made entirely of "super-strong" building blocks (supersingular elliptic curves). These are crucial for a new type of future-proof internet security called Post-Quantum Cryptography.

2. The Problem: The Missing Blueprints

For a long time, mathematicians knew how to find these super-strong bridges if they were small (Genus 1, 2, or 3). They even figured out how to find some Genus 4 bridges.
But Genus 5? That was the "Wild West." No one had a map. It was too big, too complex, and too hard to count.

The authors decided to tackle Genus 5, but with a twist. They only looked for bridges that had a specific symmetry: they could be flipped or rotated in three different ways without changing their shape. Think of it like a snowflake that looks the same if you flip it over three different axes. This symmetry made the problem solvable.

3. The Strategy: The "Lego" Trick

The authors realized something brilliant. Instead of trying to build the massive Genus 5 bridge from scratch, they could deconstruct it.

They discovered that any Genus 5 bridge with this specific symmetry is actually just a bundle of smaller, simpler bridges tied together.

Imagine taking a complex 5-hole road and realizing it's actually just three simple loops and one 2-hole road glued together.
If you know how to build the small pieces (the Genus 1 and Genus 2 roads), you can figure out if the big Genus 5 road is "superspecial" just by checking if the small pieces are "superspecial."

This is like checking if a giant, complex machine works by checking if its tiny gears are perfect. If the gears are good, the machine is good.

4. The Expedition: The Computer Search

With this "Lego trick" in hand, the authors wrote a computer program (using a tool called Magma) to go on a massive search.

The Terrain: They searched through different "worlds" defined by numbers called characteristics (denoted by $p$ ). Think of $p$ as the rules of physics for that specific world.
The Range: They checked every world where the rule number $p$ was between 11 and 1000.

They ran their algorithm, which essentially said:

Find all the perfect small bridges (Genus 2).
See if they can be glued together to make a perfect Genus 5 bridge.
Check if the resulting bridge has the required symmetry.
Count how many unique designs exist.

5. The Results: A Map of Success and Failure

The results were fascinating and revealed a surprising pattern:

Success: In most worlds (like $p=23, 31, 71$ , etc.), they found many beautiful, superspecial bridges. They even counted exactly how many unique designs existed for each world.
Failure: However, in some worlds (like $p=37, 41, 43$ , etc.), no such bridge existed at all. It was as if the laws of physics in those worlds simply didn't allow for this specific type of symmetry to exist.

This was a big deal. For Genus 4 bridges, mathematicians thought they existed in every world. But for Genus 5, the authors proved that there are "dead zones" where these bridges cannot be built.

6. Why Does This Matter?

You might ask, "Who cares about counting math bridges?"

Cryptography: These "superspecial" curves are the foundation for the next generation of encryption. As quantum computers get stronger, they will break current internet security. These curves are the proposed shield against them.
Understanding the Universe: By mapping out where these shapes exist and where they don't, mathematicians are learning the fundamental rules of the mathematical universe. It's like discovering that certain types of crystals can only form in specific temperatures.

The Takeaway

Ohashi and Kudo didn't just find a few bridges; they built a feasible algorithm (a recipe) that anyone can use to find these shapes. They successfully mapped out the territory for numbers up to 1000, proving that while these shapes are common, they are not everywhere.

It's a bit like saying, "We found a recipe for a perfect cake that works in 90% of kitchens, but in these specific 27 kitchens, the ingredients just don't mix right." This knowledge helps future cryptographers know exactly which "kitchens" (mathematical fields) are safe to build their security systems in.

1. Problem Statement

The enumeration of superspecial curves (curves whose Jacobian is isomorphic to a product of supersingular elliptic curves) is a fundamental problem in arithmetic geometry and has significant applications in post-quantum cryptography (specifically isogeny-based cryptography).

Context: While the number of superspecial curves is well-understood for genus $g \le 3$ , and computational methods exist for genus $g=4$ , the case for genus $g=5$ remains largely unexplored.
Specific Challenge: The authors focus on hyperelliptic curves of genus 5 defined over finite fields $\mathbb{F}_{p^2}$ (where $p$ is a prime) whose automorphism groups contain the subgroup $(\mathbb{Z}/2\mathbb{Z})^3$ .
Goal: To construct a feasible algorithm to enumerate these curves for all primes $p$ in the range $11 < p < 1000$ and determine their existence and isomorphism classes.

2. Methodology

The authors employ a decomposition strategy that reduces the complexity of genus 5 curves to lower-genus components.

A. Curve Classification and Decomposition

The paper focuses on hyperelliptic curves defined by the equation:
$H_{a,b,c}: y^2 = (x^4 + ax^2 + 1)(x^4 + bx^2 + 1)(x^4 + cx^2 + 1)$
where $a, b, c \in \mathbb{F}_{p^2}$ are distinct and not equal to $\pm 2$ .

Automorphism Group: These curves possess three order-2 automorphisms ( $\sigma: x \to -x$ , $\tau: x \to 1/x$ , and the hyperelliptic involution), ensuring the automorphism group contains $(\mathbb{Z}/2\mathbb{Z})^3$ .
Jacobian Decomposition (Theorem 3.2): The authors prove that the Jacobian of $H_{a,b,c}$ is isogenous to a product of three elliptic curves ( $E_1, E_2, E_3$ ) and a genus-2 curve ( $C$ ):
$\text{Jac}(H_{a,b,c}) \sim E_1 \times E_2 \times E_3 \times \text{Jac}(C)$
The parameters of these components are derived from $a, b, c$ via specific Möbius transformations.
Superspeciality Criterion (Corollary 3.3): $H_{a,b,c}$ is superspecial if and only if $E_1, E_2, E_3$ are supersingular and $C$ is superspecial. This allows the problem to be reduced to checking properties of genus $\le 2$ objects.

B. Handling Special Cases

The authors analyze sub-cases where the automorphism group is larger than $(\mathbb{Z}/2\mathbb{Z})^3$ (Types 7, 9, 10, 11, 12, 15 in their classification).

For these "special" types, the Jacobian decomposes completely into a product of five elliptic curves.
They derive specific conditions for superspeciality based on the $j$ -invariants of these elliptic curves or truncated Gaussian hypergeometric series.

C. Isomorphism Testing

To avoid counting isomorphic curves multiple times, the authors establish Theorem 4.5, which provides a complete set of conditions under which two curves $H_{a,b,c}$ and $H_{a',b',c'}$ are isomorphic. This involves checking if the sets of parameters $\{a,b,c\}$ and $\{a',b',c'\}$ are related by specific transformations (permutations, sign changes, and fractional linear transformations).

D. Algorithm Design

The authors propose four distinct algorithms based on the curve type:

Algorithm 1 (Generic Case): Iterates through known superspecial genus-2 curves (in Rosenhain form), checks if the associated elliptic curves are supersingular, and reconstructs the genus-5 curve parameters $(a,b,c)$ $(a, b, c)$ .
- Complexity: $\tilde{O}(p^3)$ .
Algorithms 2, 3, 4 (Special Cases): These algorithms directly search for parameters ( $A$ $A$ or $\delta$ $δ$ ) that satisfy specific supersingularity conditions (e.g., $j$ $j$ -invariants in the set of supersingular $j$ $j$ -invariants or roots of hypergeometric polynomials).
- Complexity: Expected $\tilde{O}(p)$ due to heuristics regarding the number of solutions.

3. Key Contributions

First Systematic Enumeration: This is the first work to systematically enumerate superspecial hyperelliptic curves of genus 5 with a specific large automorphism group.
Theoretical Reduction: The proof that the superspeciality of these specific genus-5 curves can be reduced to the superspeciality of genus-2 curves and elliptic curves, significantly lowering the computational barrier.
Isomorphism Criteria: The derivation of explicit necessary and sufficient conditions for the isomorphism of curves in the generic case (Theorem 4.5), which is crucial for efficient enumeration.
Complexity Analysis: Theoretical proof that the generic algorithm runs in $\tilde{O}(p^3)$ arithmetic operations over $\mathbb{F}_{p^2}$ , and heuristic analysis suggesting the special case algorithms are much faster ( $\tilde{O}(p)$ ).

4. Experimental Results

The authors implemented the algorithms in Magma and executed them for all primes $p$ where $11 < p < 1000$ .

Main Result (Theorem 1.1): Superspecial hyperelliptic curves of genus 5 with the required automorphism group exist for almost all primes in the tested range ( $23 \le p < 1000$ ).
Exceptions: The curves do not exist for the following primes:
$37, 41, 43, 53, 61, 67, 73, 97, 181, 193, 197, 233, 241, 277, 313, 331, 373, 421, 541, 571, 613, 643, 673, 709, 769, 877, 977$
Data: The paper provides comprehensive tables (Tables 2 and 3) listing the number of isomorphism classes for each prime $p$ , broken down by the specific automorphism group type (e.g., Type 4-1, 7, 9, etc.).
Observation: Unlike genus 4, where superspecial curves seem to exist for all sufficiently large $p$ , the genus 5 case exhibits "gaps" where no such curves exist for certain primes.

5. Significance

Cryptographic Relevance: Superspecial curves are the foundation for isogeny-based cryptographic schemes (like CSIDH). Understanding the distribution and existence of these curves in higher genera (genus 5) is vital for assessing the security landscape and potential new cryptographic primitives.
Algorithmic Advancement: The paper demonstrates that enumerating high-genus superspecial curves is feasible by leveraging Jacobian decomposition, offering a blueprint for future research into genus $g > 5$ or non-hyperelliptic cases.
Mathematical Insight: The discovery of specific primes where these curves do not exist suggests a deeper arithmetic obstruction that warrants further theoretical investigation, contrasting with the behavior observed in lower genera.