[Paper Review] Supersingular genus-two curves over fields of characteristic three

Everett W. Howe|arXiv (Cornell University)|Apr 19, 2006

Algebraic Geometry and Number Theory13 references3 citations

TL;DR

This paper classifies the Weil polynomials of supersingular genus-2 curves over finite fields of characteristic 3 by analyzing degree-3 maps from such curves to the supersingular elliptic curve with j-invariant 0. Using moduli space techniques and explicit models, it shows that, except for the curve $ y^2 = x^5 + 1 $, all supersingular genus-2 curves over an algebraic closure of $ \mathbb{F}_3 $ admit exactly 20 non-isomorphic degree-3 maps to this elliptic curve, leading to a complete determination of the occurring Weil polynomials for all finite fields $ \mathbb{F}_q $ with $ q = 3^d $.

ABSTRACT

Let C be a supersingular genus-2 curve over an algebraically closed field of characteristic 3. We show that if C is not isomorphic to the curve y^2 = x^5 + 1 then up to isomorphism there are exactly 20 degree-3 maps phi from C to the elliptic curve E with j-invariant 0. We study the coarse moduli space of triples (C,E,phi), paying particular attention to questions of rationality. The results we obtain allow us to determine, for every finite field k of characteristic 3, the polynomials that occur as Weil polynomials of supersingular genus-2 curves over k.

Motivation & Objective

• Address the gap in classifying isogeny classes of abelian surfaces over finite fields of characteristic 3 that contain Jacobians, particularly for supersingular isogeny classes.
• Complete the classification of Weil polynomials for supersingular genus-2 curves over $ \mathbb{F}_q $ with $ q = 3^d $, which was previously incomplete for $ d \leq 1 $.
• Establish a moduli-theoretic framework to study degree-3 maps from supersingular genus-2 curves to the supersingular elliptic curve with $ j = 0 $ in characteristic 3.
• Provide explicit equations and rationality results for the coarse moduli space of such maps, enabling the classification of occurring Weil polynomials.
• Resolve the case of supersingular isogeny classes in characteristic 3, completing prior work that assumed characteristic $ > 3 $.

Proposed method

• The paper constructs the coarse moduli space $ \mathcal{A} $ of triples $ (C, E, \varphi) $, where $ C $ is a supersingular genus-2 curve, $ E $ is the supersingular elliptic curve with $ j = 0 $, and $ \varphi: C \to E $ is a degree-3 map.
• It establishes an isomorphism between $ \mathcal{A} $ and the coarse moduli space $ \mathcal{B} $ of pairs $ (C, G) $, where $ G $ is a non-isotropic order-4 subgroup of $ (\operatorname{Jac} C)[2] $, using the structure of the 2-torsion and Weil pairing.
• Explicit models are given for $ \mathcal{A} $ and $ \mathcal{B} $, showing that $ \mathcal{A} $ is isomorphic to the affine line with one point removed, and the map $ \mathcal{A} \to \mathcal{S} $ (to the moduli space of supersingular genus-2 curves) has degree 20.
• Using the Igusa invariants $ [J_2 : J_4 : J_6 : J_8 : J_{10}] $, the paper characterizes supersingular genus-2 curves in characteristic 3 by the condition $ J_2 = J_4 = J_8 = 0 $, and shows $ \mathcal{S} \cong \mathbb{A}^1 $.
• Rationality questions are addressed by analyzing the action of Galois groups on roots of associated rational functions, particularly using the function $ F(z) = \frac{(z^9 - z)^5}{(z^9 - z)^{36}} $, and proving splitting behavior over finite fields.
• Key lemmas on traces and splitting of polynomials over $ \mathbb{F}_q $, such as those involving $ z^r - z - c $ and $ (z^{r-1} + 1)^{(r+1)/2} - e z^{r(r-1)/2} $, are used to establish rationality and existence of solutions.

Experimental results

Research questions

• RQ1 What are the Weil polynomials that occur for supersingular genus-2 curves over finite fields of characteristic 3?
• RQ2 How many non-isomorphic degree-3 maps exist from a supersingular genus-2 curve (other than $ y^2 = x^5 + 1 $) to the supersingular elliptic curve with $ j $-invariant 0 in characteristic 3?
• RQ3 What is the structure of the coarse moduli space of such degree-3 maps, and how does it relate to the moduli space of supersingular genus-2 curves?
• RQ4 How can rationality conditions over finite fields be used to determine which isogeny classes of abelian surfaces contain Jacobians?
• RQ5 What is the role of the 2-torsion subgroup and the Weil pairing in classifying these maps and curves?

Key findings

• For every finite field $ \mathbb{F}_q $ with $ q = 3^d $, the paper provides a complete list of the Weil polynomials that occur for supersingular genus-2 curves, depending on whether $ d $ is odd or even.
• When $ d $ is odd, the Weil polynomials are: $ (x^2 + q)(x^2 - s x + q) $ for $ s = \pm \sqrt{3q} $, $ (x^2 + q)^2 $ (if $ q > 3 $), $ x^4 + q^2 $, $ x^4 + q x^2 + q^2 $, and $ x^4 - 2q x^2 + q^2 $ (if $ q > 3 $).
• When $ d $ is even, the Weil polynomials are: $ (x^2 - 2s x + q)(x^2 + s x + q) $ for $ s = \pm \sqrt{q} $, $ (x^2 - s x + q)^2 $ for $ s = 0, \pm \sqrt{q} $, $ (x^2 - 2s x + q)^2 $ for $ s = \pm \sqrt{q} $ (if $ q > 9 $), $ x^4 + q^2 $, and $ x^4 - s x^3 + q x^2 - s q x + q^2 $ for $ s = \pm \sqrt{q} $.
• Except for the curve $ y^2 = x^5 + 1 $, every supersingular genus-2 curve over an algebraic closure of $ \mathbb{F}_3 $ admits exactly 20 non-isomorphic degree-3 maps to the supersingular elliptic curve with $ j = 0 $.
• The coarse moduli space $ \mathcal{A} $ of such maps is isomorphic to $ \mathbb{A}^1 \setminus \{0\} $, and the natural map $ \mathcal{A} \to \mathcal{S} $ (to the moduli space of supersingular genus-2 curves) has degree 20.
• The moduli space $ \mathcal{S} $ of supersingular genus-2 curves in characteristic 3 is isomorphic to the affine line $ \mathbb{A}^1 $, and the supersingularity condition is characterized by $ J_2 = J_4 = J_8 = 0 $ in Igusa invariants.