[Paper Review] A census of all genus 4 curves over the field with 2 elements

Xavier Xarles|arXiv (Cornell University)|Jul 15, 2020

Algebraic Geometry and Number Theory6 references4 citations

TL;DR

This paper presents a complete census of all genus 4 curves over the finite field $\mathbb{F}_2$ up to isomorphism, computing both hyperelliptic and trigonal cases via group actions on polynomials and algebraic geometry techniques. The key contribution is the explicit enumeration of 16 isomorphism classes of genus 4 curves over $\mathbb{F}_2$, including detailed equations and invariants such as point counts and Weil polynomials, with several curves achieving maximal point counts over extension fields.

ABSTRACT

We explain how we computed equations for all genus 4 curves defined of the field with 2 elements, up-to-isomorphism, and some of the data we obtained. We give descriptions also of nice models for genus 4 curves over characteristic 2 fields, in both the hyperelliptic case and the trigonal case.

Motivation & Objective

• To compute all isomorphism classes of genus 4 curves defined over $\mathbb{F}_2$.
• To classify both hyperelliptic and trigonal curves over $\mathbb{F}_2$ using algebraic and geometric invariants.
• To provide explicit equations and invariants (e.g., Weil polynomials, point counts) for each isomorphism class.
• To identify curves achieving maximal numbers of rational points over $\mathbb{F}_{2^k}$ for $k=1,\dots,5$, and determine their properties.

Proposed method

• Use of standard models for hyperelliptic curves: $y^2 + q(x)y = p(x)$ with $q(x)$ monic and degree constraints.
• Application of the $\operatorname{PGL}(2,\mathbb{F}_2)$-action on polynomials via $\psi_n(A)(q) = (cx+d)^n q\left(\frac{ax+b}{cx+d}\right)$ to classify isomorphism classes.
• Computation of representatives for $\overline{\mathbb{F}_2[x]_5}$ via orbit decomposition of degree-5 polynomials under $\operatorname{PGL}(2,\mathbb{F}_2)$.
• Enumeration of trigonal curves as complete intersections of quadrics and cubics in $\mathbb{P}^3$, using geometric invariants.
• Use of Weil polynomials and point counting over $\mathbb{F}_{2^k}$ to distinguish isogeny classes and detect maximal curves.
• Verification of isomorphism non-equivalence via degree and coefficient analysis of $p(x)$ and $q(x)$ in hyperelliptic models.

Experimental results

Research questions

• RQ1 How many isomorphism classes of genus 4 curves exist over $\mathbb{F}_2$, and what are their defining equations?
• RQ2 Which genus 4 curves over $\mathbb{F}_2$ achieve the maximal number of rational points over $\mathbb{F}_{2^k}$ for $k=1,\dots,5$?
• RQ3 Can curves with isogenous Jacobians have different defining equations, and if so, how many such curves exist?
• RQ4 What is the structure of the 2-torsion subgroup of the Jacobian for curves with maximal point counts?
• RQ5 Are there multiple non-isomorphic curves with the same Weil polynomial, and if so, how are they related geometrically?

Key findings

• There are exactly 16 isomorphism classes of genus 4 curves over $\mathbb{F}_2$, comprising 4 hyperelliptic and 12 trigonal curves.
• Four hyperelliptic curves share the same Weil polynomial and thus have isogenous Jacobians, with equations differing only in the constant term of $p(x)$.
• One curve achieves the maximal number of $71$ rational points over $\mathbb{F}_{2^5}$, and is the only such curve defined over $\mathbb{F}_2$ with $a_5 = 14$.
• The curve with $|C(\mathbb{F}_2)| = 1$ and $|C(\mathbb{F}_{2^4})| = 45$ is the unique genus 4 curve over $\mathbb{F}_2$ achieving the maximum possible number of points over $\mathbb{F}_{2^4}$.
• Two distinct trigonal curves with different quadric components have isogenous Jacobians, and the Jacobian of a hyperelliptic curve is isogenous to four distinct trigonal curves.
• The 2-torsion subgroup of the Jacobian for the curve with $a_1 = 3$, $a_2 = 3$, $a_3 = 2$, $a_4 = 3$ is isomorphic to $\mathbb{Z}/30\mathbb{Z}$.