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The Classification of Supersingular Elliptic Curves in Characteristic 3

Alexey Orlov

TL;DR

This work provides an explicit classification of supersingular elliptic curves over the finite field $\\mathbb{F}_{3^d}$ by reducing to canonical forms with $j=0$ and discriminant constraints, then partitioning into Types I, I$^+$, II, and III according to the fourth-power and square status of $-a_4$ and parity of $d$. It derives closed-form point-count formulas for each type using trace sums, Gauss sums, and Artin–Schreier considerations, and it presents an algorithm that counts points efficiently by reducing to these cases and evaluating trace conditions. The results enable precise enumeration of isomorphism classes and provide practical counting procedures for cryptographic and number-theoretic applications in characteristic $3$. The methods blend classical algebraic-geometry transformations with explicit finite-field character sum techniques to yield exact counts and a constructive counting algorithm.

Abstract

We provide an explicit classification of supersingular elliptic curves in characteristic 3 into isomorphism classes, and give explicit formulae for their point counts.

The Classification of Supersingular Elliptic Curves in Characteristic 3

TL;DR

This work provides an explicit classification of supersingular elliptic curves over the finite field by reducing to canonical forms with and discriminant constraints, then partitioning into Types I, I, II, and III according to the fourth-power and square status of and parity of . It derives closed-form point-count formulas for each type using trace sums, Gauss sums, and Artin–Schreier considerations, and it presents an algorithm that counts points efficiently by reducing to these cases and evaluating trace conditions. The results enable precise enumeration of isomorphism classes and provide practical counting procedures for cryptographic and number-theoretic applications in characteristic . The methods blend classical algebraic-geometry transformations with explicit finite-field character sum techniques to yield exact counts and a constructive counting algorithm.

Abstract

We provide an explicit classification of supersingular elliptic curves in characteristic 3 into isomorphism classes, and give explicit formulae for their point counts.
Paper Structure (19 sections, 5 theorems, 69 equations)

This paper contains 19 sections, 5 theorems, 69 equations.

Table of Contents

  1. Canonical Form
  2. Notation
  3. Classification of Supersingular Curve Types
  4. d odd
  5. d even
  6. Type I Curves
  7. Type I+ Curves
  8. Type II Curves
  9. Type III Curves
  10. Type IIIa Curves
  11. Type IIIb Curves
  12. Counting Points
  13. Type I Curves
  14. Type I+ Curves
  15. Type II Curves
  16. ...and 4 more sections

Key Result

Lemma 3.1

Let $q=3^d$. For $a \in \mathbb{F}_3$, define For $d$ odd, For $d$ even,

Theorems & Definitions (6)

  • Lemma 3.1
  • proof
  • Proposition 3.2
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5