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Automorphisms of Plane Curves defined from Chebychev polynomials

This paper determines the full automorphism groups of algebraic curves defined by $y^d = \varphi_d(x)$ , where $\varphi_d(x)$ is the Chebyshev polynomial, for specific cases including $d=4$ and conditions where $2d$ or $4d$ equals $p^r+1$ , while also proposing a general expectation for other degrees and applying these results to distinguish non-isomorphic maximal curves of the same genus.

Original authors: Saeed Tafazolian, Jaap Top

Published 2026-03-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Saeed Tafazolian, Jaap Top

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 have a magical piece of paper with a drawing on it. This drawing isn't just any picture; it's a mathematical shape called a curve, defined by a specific recipe involving numbers and variables.

The authors of this paper, Saeed and Jaap, are like master detectives trying to figure out the symmetry of these drawings. In math, "symmetry" means: If I twist, flip, or rotate this shape, can I make it look exactly the same as before? The collection of all these possible moves is called the Automorphism Group.

Here is the story of their investigation, broken down into simple concepts.

1. The Special Ingredients: Chebyshev Polynomials

The curves they are studying are made using a special ingredient called a Chebyshev polynomial (let's call it $\phi_d$ ).

The Analogy: Think of a Chebyshev polynomial as a very strict, rhythmic drumbeat. It has a unique pattern that repeats in a specific way.
The Curve: The shape they are studying is defined by the equation $y^d = \phi_d(x)$ . Imagine this as a 3D landscape where the height ( $y$ ) depends on the rhythm ( $\phi_d$ ) of the ground ( $x$ ).

2. The Goal: Counting the "Twists and Turns"

The main question is: How many ways can we move this shape so it looks unchanged?

If the shape is a perfect circle, you can spin it any amount, and it looks the same. It has infinite symmetry.
If the shape is a square, you can only spin it 90 degrees, 180, 270, or flip it. It has a limited number of symmetries.
The authors want to count exactly how many "moves" are possible for these specific Chebyshev curves.

3. The Clue: The "Touching Points" (Inflection Points)

To solve the mystery, the authors look for special spots on the curve called Total Inflection Points.

The Analogy: Imagine the curve is a rollercoaster track. A "total inflection point" is a spot where the track is so flat that if you laid a ruler on it, the ruler would touch the track at every single point of the curve (in a mathematical sense).
The Discovery: The authors found that the number of these special "flat spots" tells them everything about the symmetry.
- Scenario A (The Normal Case): Usually, there are exactly $d$ of these flat spots, and they all sit on a single straight line (like beads on a string).
- Scenario B (The Special Case): Sometimes, if the numbers work out perfectly (specifically when $2d$ or $4d$ relates to the size of the number system being used), there are $3d$ flat spots. These are arranged in a triangle of three lines.

4. The Results: What the Symmetry Looks Like

Based on how many "flat spots" they found, the authors determined the symmetry group:

The "Fermat" Surprise: When there are $3d$ flat spots (the special case), the curve turns out to be mathematically identical to a famous shape called the Fermat Curve. This shape is like a perfect, multi-sided star. Its symmetry group is huge and complex, involving a mix of rotations and flips (mathematically described as $(Z/dZ \times Z/dZ) \rtimes S_3$ ).
- Simple translation: "If the curve has this specific special pattern, it's actually a disguised version of a famous, highly symmetrical star."
The "Normal" Case (Most $d$ values): When there are only $d$ flat spots, the symmetry is much more modest.
- For $d=4$ $d = 4$ (a specific size), the symmetry depends on the "environment" (the characteristic of the field, which is like the rules of the number system).
  - In some environments, it's a small group of 16 moves.
  - In others (like when the number system is based on the number 5), it's a larger group of 48 moves.
- For most other sizes ( $d > 4$ ), the symmetry is usually just a simple rotation and a flip (like a rectangle).

5. The "Gotcha": Two Curves That Look Alike But Aren't

The paper ends with a cool trick.

The Setup: The authors found two different curves that have the exact same number of holes (genus) and the same number of points on a finite grid. In the world of coding theory and cryptography, these are called "Maximal Curves."
The Twist: Even though they look identical in terms of size and point-count, they are not the same shape.
The Proof: How do they know? Because their symmetry groups are different! One curve allows for a "spin of 120 degrees" (an order 3 symmetry), while the other does not.
- Analogy: Imagine two identical-looking boxes. One has a secret latch that opens with a specific twist, and the other doesn't. They look the same from the outside, but their internal "keys" (symmetries) are different.

Summary

This paper is a detective story about mathematical shapes.

The Clue: They counted the "flat spots" on the curve.
The Deduction: The number of flat spots revealed the curve's true identity.
The Surprise: Sometimes these curves are actually famous "Fermat" stars in disguise.
The Application: They used this knowledge to prove that two curves, which seemed identical, were actually different because their "keys to unlock them" (symmetries) didn't match.

This is important for mathematicians and computer scientists because understanding these symmetries helps in designing better error-correcting codes for sending data across the internet or space.

1. Problem Statement

The paper investigates the automorphism groups of a specific family of smooth plane algebraic curves $C_d$ defined over a field $k$ (with characteristic $p$ such that $p \nmid 2d$ ). The curves are given by the affine equation:
$y^d = \phi_d(x)$
where $\phi_d(x)$ is the $d$ -th Chebyshev polynomial of the first kind.

The primary objectives are:

To determine the full automorphism group $\text{Aut}(C_d)$ for various values of $d$ and characteristics $p$ .
To classify the total inflection points of these curves, as these points are invariant under automorphisms and provide a structural constraint.
To apply these classification results to distinguish between maximal curves of the same genus over finite fields, specifically showing that certain curves defined by Chebyshev polynomials are not isomorphic to Fermat curves or other standard maximal curves, despite sharing the same genus.

2. Methodology

The authors employ a combination of algebraic geometry, group theory, and explicit polynomial computations:

Inflection Point Analysis: The authors first characterize the "total inflection points" of $C_d$ (points where the tangent line intersects the curve with multiplicity $d$ ). They prove that for most cases, these points lie on the line $y=0$ . However, in specific characteristic-dependent cases (where $2d = p^r + 1$ ), the number of inflection points triples to $3d$ , lying on the union of lines $y=0$ and $x=\pm 2$ .
Linearization of Automorphisms: Relying on a classical result by B. Segre, the authors utilize the fact that automorphisms of smooth plane curves extend to linear automorphisms of the projective plane $\mathbb{P}^2$ . This allows them to study $\text{Aut}(C_d)$ as a subgroup of $\text{PGL}(3, k)$ .
Reduction to $\text{PGL}(2, k)$ : By analyzing the action of automorphisms on the set of total inflection points (which lie on a line isomorphic to $\mathbb{P}^1$ ), the authors construct a homomorphism from $\text{Aut}(C_d)$ to $\text{Aut}(\mathbb{P}^1) \cong \text{PGL}(2, k)$ . The kernel of this map is identified as a cyclic group of order $d$ (scaling $y$ by roots of unity).
Classification of Subgroups: The image of the homomorphism is analyzed using the classification of finite subgroups of $\text{PGL}(2, k)$ (specifically referencing Dickson's classification). The authors determine which subgroups can permute the zeros of the Chebyshev polynomial $\phi_d(x)$ .
Explicit Isomorphisms and Counter-examples: For specific cases (e.g., $2d = q+1$ ), the authors construct explicit birational maps to Fermat curves. For non-isomorphic cases, they compare the structure of the automorphism groups (e.g., existence of elements of order 3) to prove non-isomorphism.

3. Key Contributions and Results

A. Classification of Total Inflection Points (Proposition 2.1)

The paper establishes a precise count of total inflection points for $C_d$ :

Generic Case: There are exactly $d$ inflection points, all located on the line $y=0$ (the zeros of $\phi_d(x)$ ).
Exceptional Case: If $\text{char}(k) = p \geq 3$ and $d = (p^m + 1)/2$ for some $m \geq 1$ , there are $3d$ inflection points. These are the intersection of $C_d$ with the lines $y=0$ , $x=2$ , and $x=-2$ .

B. Automorphism Groups in Specific Cases

The authors determine $\text{Aut}(C_d)$ for several significant scenarios:

Case $2d = q + 1$ (where $q$ is a power of $p$ ):
- The curve $C_d$ is isomorphic to the Fermat curve $v^d = u^d + 1$ .
- The automorphism group is $\text{Aut}(C_d) \cong (\mathbb{Z}/d\mathbb{Z} \times \mathbb{Z}/d\mathbb{Z}) \rtimes S_3$ .
- This group has order $6d^2$ .
Case $d = 4$ :
- If $p = 7$ : $2d = 8 = 7+1$ , so it falls into the Fermat case. $\text{Aut}(C_4) \cong (\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}) \rtimes S_3$ (Order 96).
- If $p = 5$ : The group is $\mathbb{Z}/4\mathbb{Z} \rtimes A_4$ (Order 48).
- If $p \neq 2, 5, 7$ : The group is $\mathbb{Z}/4\mathbb{Z} \rtimes (\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z})$ (Order 16).
Case $4d = q + 1$ (with $d > 4$ ):
- The curve is not isomorphic to a Fermat curve.
- The automorphism group is $\mathbb{Z}/d\mathbb{Z} \rtimes S_3$ (isomorphic to $\mathbb{Z}/d\mathbb{Z} \rtimes D_3$ ).
- The order is $6d$ . The quotient group $G/N$ is isomorphic to the dihedral group $D_3$ .
General Expectation (Expectation 3.19):
- For $d > 4$ where neither $2d-1$ nor $4d-1$ is a power of $p$ , the authors conjecture that $\text{Aut}(C_d) \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/d\mathbb{Z}$ . This implies the only non-trivial automorphisms are scaling $y$ and the involution $x \mapsto -x$ .

C. Applications to Maximal Curves

The paper addresses the problem of distinguishing maximal curves over finite fields $\mathbb{F}_{q^2}$ .

Context: A curve is maximal over $\mathbb{F}_{q^2}$ if it attains the Hasse-Weil upper bound for the number of rational points.
Result: The authors prove that for certain parameters, the curve $C_m: y^m = \phi_n(x)$ and the curve $D: y^m = x^n + 1$ (or Fermat curves) have the same genus and are both maximal, yet they are not isomorphic.
Proof Mechanism: The non-isomorphism is proven by comparing their automorphism groups. For instance, in the case $4n = q+1$ , the Chebyshev curve admits an automorphism of order 3 (derived from the specific structure of $\phi_n$ ), whereas the corresponding Fermat-type curve does not.

4. Significance

Refinement of Existing Literature: The paper provides a transparent, elementary proof for the isomorphism between Chebyshev curves and Fermat curves in the $2d=q+1$ case, avoiding the heavy machinery used in previous works (e.g., [5]).
New Classification: It offers a complete classification of automorphism groups for these curves in cases where $d=4$ and $4d=q+1$ , filling gaps in the literature.
Counter-examples in Coding Theory: By demonstrating the existence of non-isomorphic maximal curves with the same genus, the paper refines the understanding of the moduli space of maximal curves. This is crucial for algebraic geometry codes, where the specific structure of the curve (and its automorphisms) affects code parameters.
Conjecture: The paper formulates a clear expectation (Expectation 3.19) for the generic case ( $d > 4$ ), providing a target for future research.

In summary, the paper successfully links the arithmetic properties of Chebyshev polynomials to the geometric structure of plane curves, utilizing inflection points and group theory to classify automorphisms and distinguish between seemingly similar maximal curves.