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Superspecial generalized Howe curves of genus 4, 5, and 6 with completely decomposable Jacobians

This paper presents an efficient computational method for constructing superspecial curves of genera 4, 5, and 6 with completely decomposable Jacobians by utilizing supersingular elliptic curves, thereby confirming their existence in specific ranges of characteristics $p$ where previous results were limited.

Original authors: Ryo Ohashi

Published 2026-04-21

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CC BY 4.0

Original authors: Ryo Ohashi

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. In the world of mathematics, these "bridges" are called curves, and they exist in a strange, abstract universe where the rules of arithmetic are different (this is called "characteristic $p$ ").

The goal of this paper is to prove that we can build these bridges for very specific, complex sizes (called genus 4, 5, and 6) in almost every possible version of this universe, provided the universe isn't too small.

Here is the breakdown of the paper using simple analogies:

1. The Big Problem: The "Bridge" Mystery

Mathematicians have known how to build these bridges for small sizes (genus 1, 2, and 3) for a long time.

Genus 1 is like a simple loop (a circle).
Genus 2 is like a figure-eight.
Genus 3 is a bit more complex, like a pretzel.

But for Genus 4, 5, and 6 (think of these as bridges with 4, 5, or 6 loops), it was a mystery. We didn't know if they could be built in certain universes (specifically, for large prime numbers). The main difficulty is that as the bridges get bigger, they stop behaving like simple loops and start acting like complicated, indecipherable blobs.

2. The Old Strategy: The "Lego" Approach

A few years ago, other mathematicians (Kudo, Harashita, and Howe) found a clever trick. They realized that if you build a complex bridge by snapping together two simpler bridges (like Legos), you can check if the big one is "special" (superspecial) just by checking the small pieces.

They proved that for universes up to a certain size (primes up to 20,000), these bridges do exist. But checking every single universe up to a million was too slow with their method. It was like trying to find a specific needle in a haystack by checking every single piece of hay one by one.

3. The New Strategy: The "Perfectly Decomposable" Shortcut

The author of this paper, Ryo Ohashi, decided to look for a special subset of these bridges.

Imagine you are looking for a bridge that is made of four perfect, identical wheels (elliptic curves) that roll independently.

The Old Way: You had to check if the whole messy bridge was special.
The New Way: Ohashi said, "Let's only look for bridges that are guaranteed to fall apart into four perfect wheels."

Why is this helpful? Because checking if a single wheel is "special" (supersingular) is incredibly easy and fast. It's like checking if a single gear is round. If you know the bridge is made of four wheels, you just check the four wheels. If they are all perfect, the whole bridge is perfect.

4. The Results: Building the Bridges

Ohashi wrote computer programs (algorithms) to hunt for these "wheel-based" bridges.

Genus 4 (The 4-Loop Bridge):
- The Result: He proved that for almost every universe with a prime number between 20,000 and 1,000,000, these bridges exist.
- The Speed: His method was 1,400 times faster than the old method. It took him about 57 hours to check a million universes, whereas the old method would have taken years.
- The Glitches: He couldn't find them for three specific universes (primes 13, 19, and 73), but he proved they exist for everything else in that range.
Genus 5 & 6 (The 5 and 6-Loop Bridges):
- These were even harder to build. No one had really proven they existed for large universes before.
- Ohashi used a similar "wheel" strategy to construct them.
- The Result: He successfully found these bridges for almost every universe up to 100,000. He even found the blueprints (equations) for the few tricky universes where the computer search initially failed.

5. Why Does This Matter?

You might ask, "Who cares about these abstract bridges?"

Cryptography: These "superspecial" curves are the foundation for a new type of internet security called isogeny-based cryptography. This is a shield designed to protect our data from future quantum computers. To build these shields, we need to know that these mathematical structures actually exist and how to construct them efficiently.
Mathematical Confidence: By proving these exist for so many different numbers, Ohashi has removed a huge "unknown" from the map of mathematics. He showed that these complex structures are not rare anomalies; they are common enough to be found easily if you know the right shortcut.

Summary Analogy

Imagine you are trying to find a specific type of rare flower in a massive forest (the mathematical universe).

The Old Method: You walked through the forest looking for the flower, checking every bush. It worked for the first few miles, but you got tired.
The New Method: Ohashi realized that this flower only grows on a specific type of rock. So, instead of checking every bush, he just looked for the rocks. He found the rocks, and the flowers were right there. He did this so fast that he mapped the entire forest in a single afternoon.

The Bottom Line: This paper is a masterclass in efficiency. It solved a decades-old problem for huge numbers of cases by realizing that if you restrict your search to a very specific, well-behaved type of object, the problem becomes trivial to solve.

1. Problem Statement

The central problem addressed in this paper is the existence of superspecial curves of genus $g \geq 4$ in a given characteristic $p > 0$ .

Definition: A curve is superspecial if its Jacobian is isomorphic (as an unpolarized abelian variety) to a product of supersingular elliptic curves.
Context: The existence question is fully resolved for genera $g \leq 3$ . However, for $g \geq 4$ , the problem remains open for general $p$ .
Previous Work: Kudo, Harashita, and Howe (2020) established the existence of superspecial genus-4 curves for $7 < p < 20,000$ using "Howe curves." Their method reduced the superspeciality of the genus-4 curve to the superspeciality of genus-2 curves. However, this approach is computationally expensive for large $p$ because it requires generating and testing many superspecial genus-2 curves.
Goal: To extend the existence results to larger ranges of $p$ (up to $10^6$ ) and to genera $g=5, 6$ by developing more efficient algorithms that avoid the heavy computation of genus-2 curves.

2. Methodology

The author introduces a restricted subclass of generalized Howe curves where the Jacobian is completely decomposable into a product of elliptic curves. This restriction allows the condition of superspeciality to be reduced entirely to the supersingularity of elliptic curves, which is computationally much cheaper to verify.

A. Theoretical Framework

Generalized Howe Curves: Constructed as the desingularization of the fiber product of two hyperelliptic curves ( $C_1 \times_{\mathbb{P}^1} C_2$ ). Theorem 2.6 states that the Jacobian of the resulting curve $H$ is isogenous to $J(C_1) \times J(C_2) \times J(C_3)$ , where $C_3$ is a third curve derived from the fiber product.
Complete Decomposability: The author selects specific parameters such that $J(C_1)$ , $J(C_2)$ , and $J(C_3)$ are themselves isogenous to products of elliptic curves. Consequently, $J(H)$ is isogenous to a product of elliptic curves.
Reduction: Under this construction, $H$ is superspecial if and only if the constituent elliptic curves are supersingular.

B. Specific Constructions

Genus 4 ( $X_{s,t}$ ):
- Constructed from two elliptic curves $E_1, E_2$ sharing one Weierstrass point.
- The associated genus-2 curve $C_3$ is chosen to be of the form $y^2 = (x^2-1)(x^2-s^2)(x^2-t^2)$ .
- Result: $J(X_{s,t}) \sim E_1 \times E_2 \times E_3 \times E_4$ . The curve is superspecial iff $E_1, E_3, E_4$ are supersingular.
- Algorithm 3.4: Iterates through pairs of supersingular $j$ -invariants (or Legendre parameters $\lambda$ ) to find $s, t$ such that the resulting elliptic curves are all supersingular.
Genus 5 ( $Y_{s,t}$ ):
- Constructed from two genus-2 curves $C_1, C_2$ sharing 4 Weierstrass points, where $C_1, C_2$ are of the form $y^2 = x(x^2-1)(x^2-s^2)$ .
- Result: $J(Y_{s,t}) \sim E_1^2 \times E_2^2 \times E_3$ . The curve is superspecial iff $E_1, E_2, E_3$ are supersingular.
- Algorithm 4.3: Instead of solving for roots of a specific polynomial (which is slow), the algorithm iterates through pairs of supersingular $j$ -invariants ( $j_1, j_2$ ), solves for parameters $s, t$ , and checks if the resulting $E_3$ is supersingular.
Genus 6 ( $Z_{s,t}$ ):
- Constructed from two genus-2 curves $C_1, C_2$ sharing 3 Weierstrass points, of the form $y^2 = (x^3-1)(x^3-s^6)$ .
- Result: $J(Z_{s,t}) \sim E_1^2 \times E_2^2 \times E_3^2$ . The curve is superspecial iff $E_1, E_2, E_3$ are supersingular.
- Algorithm 5.4: Similar to Genus 5, it uses supersingular $j$ -invariants to determine parameters $s, t$ and verifies the supersingularity of the third elliptic curve.

3. Key Contributions

Efficiency Improvement: The proposed algorithms are significantly faster than the previous Kudo-Harashita-Howe method.
- For Genus 4, the new method is approximately 1,400 times faster (29.7 seconds vs. 11.5 hours for the range $p < 20,000$ ).
- This efficiency gain is achieved by reducing the search space from superspecial genus-2 curves to supersingular elliptic curves.
New Existence Proofs: The paper provides constructive proofs for the existence of superspecial curves in much larger ranges of $p$ than previously known.
Generalization to Higher Genus: The paper extends the constructive approach to genera 5 and 6, which were previously largely unexplored computationally.

4. Experimental Results

The algorithms were implemented in Magma and executed on a high-performance server (AMD EPYC 7742, 2TB RAM).

Genus 4:
- Range: Verified for all primes $7 < p < 10^6$ (excluding $p=13, 19, 73$ where the specific construction failed, though existence is known via other means for $p=13, 19$ ).
- Conclusion: Superspecial genus-4 curves exist for all $7 < p < 10^6$ .
- Time: ~57.4 hours for the full range.
Genus 5:
- Range: Verified for $19 < p < 10^5$ (excluding specific primes like 37, 53, etc., where the specific construction failed).
- Conclusion: Superspecial genus-5 curves exist for all $7 \leq p < 10^5$ (with $p \neq 13$ ).
- Time: ~15.5 hours.
Genus 6:
- Range: Verified for $11 < p < 10^5$ (excluding specific primes like 19, 37, etc.).
- Conclusion: Superspecial genus-6 curves exist for all $7 < p < 10^5$ .
- Time: ~22.1 hours.
Open Cases: The existence of superspecial curves for $(g, p) = (5, 13)$ and $(6, 7)$ remains open.

5. Significance

Computational Number Theory: This work pushes the boundaries of computational verification for superspecial curves, moving from $p \approx 20,000$ to $p \approx 10^6$ .
Cryptography: Superspecial curves are fundamental in isogeny-based cryptography (e.g., CSIDH, SQIsign). Understanding their existence and constructing them efficiently in various characteristics is crucial for the security analysis and parameter selection of post-quantum cryptographic schemes.
Algebraic Geometry: The results provide concrete examples of superspecial abelian varieties of dimension 4, 5, and 6, contributing to the classification of such objects.
Algorithmic Advancement: The strategy of restricting the search to curves with completely decomposable Jacobians offers a powerful heuristic for tackling similar existence problems in higher genera, demonstrating that "special" subfamilies can yield global existence results efficiently.