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Superspecial plane quintics with large automorphism groups

This paper investigates superspecial plane quintic curves with large automorphism groups, explicitly characterizing their superspeciality via truncated Gaussian hypergeometric series to determine the exact count of isomorphism classes for cyclic groups of order 10 and to provide an efficient enumeration algorithm for those of order 8.

Original authors: Ryo Ohashi

Published 2026-05-29

📖 5 min read🧠 Deep dive

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 the mathematical world as a vast landscape filled with intricate shapes called "curves." In this paper, the author, Ryo Ohashi, is exploring a specific neighborhood of this landscape: plane quintic curves.

To understand what these are, think of them as complex, five-sided geometric flowers drawn on a flat sheet of paper (a plane). While some flowers are simple, these specific ones are "non-hyperelliptic," meaning they have a very specific, twisted complexity that prevents them from being flattened into a simpler shape. They are defined by equations involving powers of five (like $X^5 + Y^5 + Z^5 = 0$ ).

The author is particularly interested in two things about these flowers:

Symmetry (Automorphisms): How many ways can you rotate or flip the flower so it looks exactly the same? Some flowers have very little symmetry, while others are highly symmetrical.
Superspeciality: This is a special "superpower" property. In the world of these curves, being "superspecial" is like a flower having a perfect, ultra-efficient internal structure that makes it behave in a very specific, rare way in certain mathematical environments (specifically, in fields with a specific type of "clock" called characteristic $p$ ).

The Mission: Finding the "Super-Symmetrical" Flowers

The paper focuses on finding these "super-special" flowers that also have large groups of symmetry. The author looks at two main groups of symmetry:

The Order 10 Group: Flowers that can be rotated in 10 distinct ways to look the same.
The Order 8 Group: Flowers that can be rotated in 8 distinct ways.

The author treats these flowers as members of families. Some families are fixed (like a single, unique flower), while others are "one-parameter families," which is like a garden where you can tweak a single dial (a variable called $r$ ) to grow slightly different versions of the same flower.

The Magic Tool: Gaussian Hypergeometric Series

How does the author tell if a flower is "super-special"? He doesn't just look at it; he uses a sophisticated mathematical recipe called a Gaussian hypergeometric series.

Think of this series as a magic spell or a complex recipe.

For the Order 10 flowers, the recipe is surprisingly simple. It's like a single line of code. If you plug in the right numbers, the spell tells you exactly how many super-special flowers exist.
For the Order 8 flowers, the recipe is much more complicated. It's like a spell with multiple ingredients that all have to be zero at the same time. Because it's so complex, you can't just write a simple formula to count them. Instead, the author had to build a computer program (an algorithm) to check each possibility one by one.

The Discoveries

1. The Order 10 Discovery (The Predictable Garden)
The author found a beautiful, clean rule for the Order 10 flowers.

If the "clock" of the mathematical world ( $p$ ) ticks in a specific rhythm (specifically, if $p$ leaves a remainder of 9 or 19 when divided by 20), then super-special flowers exist.
The number of these flowers follows a neat formula: roughly $3p/20$ .
If the clock ticks in any other rhythm, there are zero such flowers.
Analogy: It's like finding that a specific type of rare bird only nests in trees that are exactly 9 or 19 feet tall. If the tree is any other height, the bird isn't there.

2. The Order 8 Discovery (The Chaotic Garden)
The Order 8 flowers were much trickier.

The author ran his computer algorithm on thousands of different "clocks" (prime numbers between 13 and 10,000).
The Big Finding: If the clock ticks in a rhythm where the remainder is 1, 3, or 5 when divided by 8, no super-special flowers exist at all.
However, if the clock ticks with a remainder of 7 (e.g., 23, 31, 47...), the flowers do exist, but their numbers are irregular. There isn't a simple formula like the Order 10 case. The count jumps up and down unpredictably.
Analogy: Imagine a garden where a specific flower only blooms if the weather is "7 degrees" (in a weird mathematical sense). If it's 1, 3, or 5 degrees, the garden is empty. But if it is 7 degrees, sometimes you find 1 flower, sometimes 5, and sometimes 0, with no obvious pattern to the count.

The Result

The paper provides:

A perfect formula for counting the Order 10 super-special flowers.
A fast computer program that successfully counted the Order 8 flowers for thousands of cases, revealing that they only appear when the mathematical "clock" ticks in a specific way (remainder 7).
A list of these counts (Table 3 in the paper), which serves as a catalog for mathematicians.

The author concludes that while the Order 10 case is a neat, solved puzzle, the Order 8 case is a bit more chaotic, and proving why they don't exist for other clock rhythms remains a mystery for future mathematicians to solve. The paper essentially maps out where these rare, super-symmetrical mathematical flowers can be found and how many of them there are.

Technical Summary: Superspecial Plane Quintics with Large Automorphism Groups

Problem Statement
This paper investigates the existence and enumeration of superspecial plane quintic curves (non-singular projective curves of genus 6) defined over an algebraically closed field of characteristic $p > 13$ . While the superspeciality of plane quartic curves (genus 3) has been extensively studied, the classification for plane quintics remains less complete. The specific focus is on curves possessing large automorphism groups: those with order greater than 10, as well as those with cyclic automorphism groups of order 8 and 10. The goal is to determine necessary and sufficient conditions on the characteristic $p$ for these curves to be superspecial and to enumerate the number of their isomorphism classes.

Methodology
The authors employ a combination of algebraic geometry and computational number theory, leveraging the classification of plane quintic curves by Badr and Bars. The methodology proceeds as follows:

Reduction to Superelliptic Curves: The plane quintic curves under consideration are transformed into superelliptic curves of the form $y^n = f(x)$ . This allows the application of known criteria for superspeciality.
Hypergeometric Characterization: Utilizing Theorem 2.2 (based on the vanishing of specific coefficients in the expansion of $f(x)^{(ip-j)/n}$ ), the authors translate the condition of superspeciality into the vanishing of truncated Gaussian hypergeometric series, denoted as $G(d)(a, b, c | z)$ .
Case Analysis by Automorphism Group:
- 0-Dimensional Families: For curves with fixed defining equations (Types 1–5 in Table 1), the authors derive explicit congruence conditions on $p$ using the binomial theorem and properties of the hypergeometric series.
- 1-Dimensional Families (Order 10): For the family $X^5 + Y^5 + XZ^4 + rX^3Z^2 = 0$ , the authors establish a correspondence between the parameter $r$ and a parameter $\lambda$ in a superelliptic model $y^5 = x(x^2-1)(x^2-\lambda)$ . They prove that superspeciality corresponds to the vanishing of a single specific hypergeometric series $G(\lambda)$ .
- 1-Dimensional Families (Order 8): For the family $X^5 + Y^4Z + XZ^4 + rX^3Z^2 = 0$ , the superspeciality condition is more complex, requiring the simultaneous vanishing of multiple hypergeometric series. The authors define $G(\lambda)$ as the greatest common divisor (gcd) of these polynomials.
Algorithmic Enumeration: For the order 8 case, where a simple closed-form formula for the number of solutions is not immediately apparent, the authors develop an efficient algorithm to compute the degree of the polynomial $G(\lambda)$ and count the valid isomorphism classes. This algorithm is implemented in Magma.

Key Contributions and Results

Classification of 0-Dimensional Families: The paper provides explicit necessary and sufficient conditions on $p$ for the superspeciality of the Fermat quintic, Hurwitz quintic, and other specific forms with automorphism groups of order 150, 39, 30, 20, and 16. For instance, the Fermat quintic is superspecial if and only if $p \equiv 4 \pmod 5$ .
Enumeration for Order 10 (Theorem 1.1): The authors derive a precise formula for the number of isomorphism classes of superspecial plane quintics with a cyclic automorphism group of order 10. The count depends on $p \pmod{20}$ $p (mod 20)$ :
- If $p \equiv 9 \pmod{20}$ , the number is $(3p - 27)/20$ .
- If $p \equiv 19 \pmod{20}$ , the number is $(3p - 37)/20$ .
- Otherwise, the number is 0.
  This result is achieved by analyzing the roots of a single hypergeometric series and excluding cases where the automorphism group is larger (order 150 or 20).
Enumeration for Order 8 (Theorem 1.2): For curves with cyclic automorphism groups of order 8, the authors prove that no such superspecial curves exist if $p \equiv 1, 3, 5 \pmod 8$ $p \equiv 1, 3, 5 (mod 8)$ . For $p \equiv 7 \pmod 8$ $p \equiv 7 (mod 8)$ , they provide an algorithm to compute the number of isomorphism classes.
- Computational results for $13 < p < 10000$ are presented in Table 3, showing that the number of classes varies irregularly with $p$ and does not follow a simple linear formula like the order 10 case.
- The computation for the range up to 10,000 was completed in approximately 36.2 seconds.

Significance and Claims
The paper claims to provide the first explicit characterization of superspeciality for plane quintic curves with large automorphism groups in terms of Gaussian hypergeometric series.

Theoretical Insight: It demonstrates that while the order 10 case yields a clean, explicit formula for the number of isomorphism classes, the order 8 case is significantly more intricate, requiring the intersection of multiple hypergeometric conditions.
Computational Efficiency: The proposed algorithm for the order 8 case is shown to be efficient enough to handle characteristics up to 10,000, suggesting that the upper bound for computational verification can be extended.
Future Work: The authors modestly note that a theoretical proof for the non-existence of order 8 superspecial curves when $p \not\equiv 7 \pmod 8$ (observed computationally) remains an open problem. They also suggest that investigating other automorphism groups may require multivariable hypergeometric series.

The work is supported by JSPS Grant-in-Aid for Young Scientists 25K17225, and the implementation code is made publicly available.

The Mission: Finding the "Super-Symmetrical" Flowers

The Magic Tool: Gaussian Hypergeometric Series

The Discoveries

The Result

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