[Paper Review] Hyperelliptic curves in characteristic 2

Jasper Scholten, Hui June Zhu|ArXiv.org|Dec 19, 2000

Algebraic Geometry and Number Theory8 references4 citations

TL;DR

This paper establishes that no hyperelliptic supersingular curves exist over $\overline{\mathbb{F}}_2$ of genus $2^n - 1$ for $n \geq 2$, and proves that the first slope of the Newton polygon for such curves is at least $1/h$ where $h = \lfloor \log_2(g+1) + 1 \rfloor$. It further shows that the moduli space of hyperelliptic supersingular curves over $\overline{\mathbb{F}}_2$ has dimension at most $g-2$, with equality for genus 4, where all such curves admit a specific normal form $y^2 - y = x^9 + c_5x^5 + c_3x^3$. The results are derived using $2$-adic analysis of coefficients and Newton polygon bounds.

ABSTRACT

In this paper we prove that there are no hyperelliptic supersingular curves over F_2bar of genus 2^n-1 for any integer n>1. Let g be a natural number, and h=floor(log_2(g+1)+1). Let X be a hyperelliptic curve over F_2bar of genus g>2 and 2-rank zero, given by an affine equation y^2-y=c_{2g+1} x^{2g+1} +...+ c_1 x. We prove that the first slope of the Newton polygon of X is bigger than or equal to 1/h. We also prove that the equality holds if (I) g<2^h-2, c_{2^h-1} is nonzero; or (II) g=2^h-2, c_{2^h-1} or c_{3(2^{h-1})-1} is nonzero. We prove that genus-4 hyperelliptic curve over F_2bar are precisely those with equations y^2 - y = x^9 + a x^5 + b x^3.

Motivation & Objective

• To determine the existence of hyperelliptic supersingular curves over $\overline{\mathbb{F}}_2$ of genus $2^n - 1$ for $n \geq 2$.
• To establish a lower bound on the first slope of the Newton polygon for hyperelliptic curves of $2$-rank zero over $\overline{\mathbb{F}}_2$.
• To determine the dimension of the moduli space $\mathcal{HS}_g/\overline{\mathbb{F}}_2$ of hyperelliptic supersingular curves of genus $g \geq 3$.
• To classify all genus-4 hyperelliptic supersingular curves over $\overline{\mathbb{F}}_2$ by providing a normal form.

Proposed method

• The authors use $2$-adic analysis of the coefficients $c_i$ in the equation $y^2 - y = \sum_{i=1}^{2g+1} c_i x^i$ to derive congruence conditions on the coefficients that determine the Newton polygon slopes.
• They define $h = \lfloor \log_2(g+1) + 1 \rfloor$ and prove that the first slope of the Newton polygon is at least $1/h$ for $2$-rank zero hyperelliptic curves over $\overline{\mathbb{F}}_2$.
• The equality $\mathrm{NP}_1(X) = 1/h$ is shown to hold under specific coefficient conditions: $c_{2^h - 1} \neq 0$ when $g < 2^h - 2$, or $c_{2^h - 1} \neq 0$ or $c_{3 \cdot 2^{h-1} - 1} \neq 0$ when $g = 2^h - 2$.
• The proof relies on the Deuring–Shafarevich formula and isomorphisms of curves to normalize equations to odd-degree monomials with $c_{2g+1} = 1$, while eliminating certain coefficients via coordinate transformations.
• For genus 4, the authors show that all supersingular curves must have the form $y^2 - y = x^9 + c_5x^5 + c_3x^3$, using the coefficient vanishing conditions from the Newton polygon analysis.
• They use the fact that $c_7 \neq 0$ implies $\mathrm{NP}_1(X) > 1/3$, so supersingular curves must have $c_7 = 0$, leading to the normal form.

Experimental results

Research questions

• RQ1 Are there any hyperelliptic supersingular curves over $\overline{\mathbb{F}}_2$ of genus $2^n - 1$ for $n \geq 2$?
• RQ2 What is the minimal possible first slope of the Newton polygon for a hyperelliptic curve of $2$-rank zero over $\overline{\mathbb{F}}_2$?
• RQ3 What is the dimension of the moduli space $\mathcal{HS}_g/\overline{\mathbb{F}}_2$ of hyperelliptic supersingular curves of genus $g \geq 3$?
• RQ4 Can all genus-4 hyperelliptic supersingular curves over $\overline{\mathbb{F}}_2$ be described by a single normal form?

Key findings

• There are no hyperelliptic supersingular curves over $\overline{\mathbb{F}}_2$ of genus $2^n - 1$ for any integer $n \geq 2$, as shown by the Newton polygon slope bound and coefficient conditions.
• For any hyperelliptic curve of $2$-rank zero over $\overline{\mathbb{F}}_2$ of genus $g \geq 3$, the first slope of the Newton polygon is at least $1/h$ where $h = \lfloor \log_2(g+1) + 1 \rfloor$.
• The equality $\mathrm{NP}_1(X) = 1/h$ holds if $g < 2^h - 2$ and $c_{2^h - 1} \neq 0$, or if $g = 2^h - 2$ and $c_{2^h - 1} \neq 0$ or $c_{3 \cdot 2^{h-1} - 1} \neq 0$.
• The moduli space $\mathcal{HS}_g/\overline{\mathbb{F}}_2$ has dimension at most $g - 2$ for $g \geq 3$, and dimension at most $g - 3$ when $g = 2^h - 2$ for $h \geq 3$.
• For genus 4, the moduli space $\mathcal{HS}_4/\overline{\mathbb{F}}_2$ is irreducible and of dimension 2, and every such curve has an equation of the form $y^2 - y = x^9 + c_5x^5 + c_3x^3$ for some $c_5, c_3 \in \overline{\mathbb{F}}_2$.
• The coefficient $c_7$ must vanish in any genus-4 hyperelliptic supersingular curve, as $c_7 \neq 0$ would imply $\mathrm{NP}_1(X) > 1/3$, contradicting supersingularity.