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The Classification of Supersingular Elliptic Curves in Characteristic 3

This paper provides a concrete, implementation-oriented classification of supersingular elliptic curves in characteristic 3 and explicit formulae for their point counts to support the development of the Hecke.jl software package.

Original authors: Alexey Orlov

Published 2026-02-10

📖 3 min read🧠 Deep dive

CC BY 4.0

Original authors: Alexey Orlov

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 a librarian in a massive, infinite library. Most of the books are standard novels, but occasionally, you stumble upon "Magic Books"—special volumes that follow much more complex, mystical rules.

In mathematics, Elliptic Curves are like these magic books. They are beautiful, complex shapes used heavily in modern cryptography (the math that keeps your credit card safe when you shop online). Specifically, this paper focuses on a very rare, "supersingular" breed of these curves in a specific mathematical universe called "Characteristic 3."

Here is the breakdown of what Alexey Orlov did in this paper, using everyday analogies.

1. The Problem: The Messy Closet

Imagine you have a massive closet filled with thousands of different magic books. They all look slightly different—some have gold spines, some have blue, some have different sized covers. If you wanted to count exactly how many "special" pages are in every single book, it would take you a lifetime because you’d be treating every book as a unique mystery.

In math, "counting points" on a curve is like counting the special pages. Doing this for every single curve individually is incredibly slow and computationally expensive.

2. The Solution: The Sorting Hat (Classification)

Orlov’s first big move is to realize that even though the books look different, many of them are actually identical twins or triplets in disguise.

He uses a mathematical "Sorting Hat" to group these curves into Isomorphism Classes. In our library, this is like realizing that a book with a blue cover and a book with a red cover are actually the exact same story, just printed in different colors. Once you know the "story" (the isomorphism class), you don't have to re-read the whole book to understand it.

He sorts them into four main "shelves":

Type I: The standard collection.
Type I+: A slightly different variation.
Type II: The "twins" (curves that are reflections of Type I).
Type III: The "exotics" (curves that follow a different rhythm entirely).

3. The Shortcut: The Cheat Sheet (Point Counting)

Once the books are sorted onto their correct shelves, Orlov provides a "Cheat Sheet" (the formulas in Section 3).

Instead of manually counting every single "special page" (point) in a book, he says: "If your book is on Shelf Type I and has a 'Trace' value of 1, just use this specific math shortcut, and you'll get the answer instantly."

He provides explicit, plug-and-play formulas. This is the difference between counting grains of sand one by one and simply using a scale to weigh them.

4. The Goal: Making Computers Faster

The author mentions he wrote this to help a software tool called Hecke.jl.

Think of Hecke.jl as a high-speed robot librarian. Before this paper, the robot was trying to count pages by hand, which was slow. Orlov gave the robot a Categorization Manual and a Calculator. Now, when the robot encounters a curve, it quickly identifies its "Type," looks up the corresponding formula, and gives the answer in a fraction of a second.

Summary for the Non-Mathematician

The Paper's "TL;DR":

The Subject: Rare, special mathematical shapes (Supersingular Elliptic Curves).
The Method: Instead of treating every shape as a new problem, he proved they actually belong to a few small "families."
The Result: He created a "recipe book" so that computers can calculate complex properties of these shapes instantly, rather than grinding through long, slow calculations.

Technical Summary: The Classification of Supersingular Elliptic Curves in Characteristic 3

Author: Alexey Orlov
Subject Matter: Arithmetic Geometry / Computational Number Theory

1. Problem Statement

The paper addresses the classification of supersingular elliptic curves over finite fields of characteristic 3 ( $\mathbb{F}_q$ , where $q = 3^d$ ) into isomorphism classes. While such classifications exist in literature (notably by Morain), the author identifies a need for a more "self-contained and implementation-oriented" treatment. The primary goal is to provide explicit, concrete formulae for the number of points ( $\#E(\mathbb{F}_q)$ ) on these curves to facilitate efficient point-counting algorithms in computational environments like Hecke.jl.

2. Methodology

The author employs a constructive algebraic approach:

Canonical Reduction: The author first reduces the general Weierstrass equation to a simplified canonical form for supersingular curves in characteristic 3: $E: y^2 = x^3 + a_4x + a_6$ , where $a_4 \neq 0$ .
Isomorphism Analysis: By analyzing the transformation laws for elliptic curves, the author determines the conditions under which two curves are isomorphic over $\mathbb{F}_q$ . This involves solving for isomorphism parameters $(u, r)$ and utilizing the Artin–Schreier equation ( $r^3 - r = c$ ) and Hilbert’s Theorem 90 to determine the existence of solutions in $\mathbb{F}_q$ .
Character Sums: To derive point-counting formulae, the author uses the quadratic character $\chi$ and evaluates sums of the form $\sum \chi(f(x))$ . The derivation relies on Gauss sums and the properties of the Trace function ( $\text{Tr}: \mathbb{F}_q \to \mathbb{F}_3$ ).

3. Key Contributions and Results

The paper categorizes supersingular curves into distinct "Types" based on the properties of the coefficient $a_4$ and the parity of $d$ .

A. Classification of Curve Types:

Type I ( $d$ odd/even): Curves where $-a_4$ $- a_{4}$ is a fourth power. These are reduced to the form $y^2 = x^3 - x + b$ $y^{2} = x^{3} - x + b$ .
- If $d$ is odd, there are 3 isomorphism classes based on $\text{Tr}(b)$ .
- If $d$ is even, there are 2 isomorphism classes.
Type I+ ( $d$ odd): Curves where $a_4$ is a fourth power. All such curves are isomorphic to $y^2 = x^3 + x$ .
Type II ( $d$ even): Curves where $-a_4$ is a square but not a fourth power. These are quadratic twists of Type I curves.
Type III (a and b): Curves where $-a_4$ is not a square. These are categorized by whether $a_4$ belongs to specific cosets of the fourth powers, resulting in curves isomorphic to $y^2 = x^3 - \beta x$ or $y^2 = x^3 - \beta^3 x$ (where $\beta$ is a primitive root).

B. Point Counting Formulae:
The author provides explicit, closed-form expressions for $\#E(\mathbb{F}_q)$ for every type.

Type I/II: The point count depends on the trace of a value derived from $a_4$ and $a_6$ . The results involve terms like $\pm \sqrt{3q}$ or $\pm 2\sqrt{q}$ depending on the parity of $d$ and the value of the trace.
Type I+/III: These curves are "simpler" in terms of point counting, often resulting in $\#E(\mathbb{F}_q) = q + 1$ .

C. Algorithmic Implementation:
The paper concludes with a comprehensive, decision-tree style algorithm (Section 4) that takes the coefficients $(a_4, a_6)$ and the dimension $d$ as input and outputs the exact number of points.

4. Significance

The significance of this work is twofold:

Theoretical: It provides a pedagogical and rigorous derivation of the classification of supersingular curves in characteristic 3, bridging the gap between abstract theory and concrete coefficients.
Computational: By providing "elementary and concrete" reasoning and explicit formulae, the paper serves as a direct blueprint for implementing high-speed point-counting routines in mathematical software. This is critical for cryptography and arithmetic geometry applications where the efficiency of working with supersingular curves is paramount.