[Paper Review] Optimal Curves of Genus 3 over Finite Fields with Discriminant -19

Е. С. Алексеенко, С. И. Алешников+2 more|arXiv (Cornell University)|Feb 11, 2009

Coding theory and cryptography5 references3 citations

TL;DR

This paper provides a complete classification of optimal genus 3 curves over finite fields with discriminant -19, proving they admit explicit equations of the form $ z^2 = ext{quartic in } x + ext{linear in } y $. It constructs a table of maximal and minimal curves over prime fields up to 997, showing that maximal and minimal curves cannot coexist for such fields, and establishes that the Jacobian of any such curve is isogenous to a power of a maximal/minimal elliptic curve over $ ext{F}_q $.

ABSTRACT

In this work we study the properties of maximal and minimal curves of genus 3 over finite fields with discriminant -19. We prove that any such curve can be given by an explicit equation of certain form. Using these equations we obtain a table of maximal and minimal curves over finite fields with discriminant -19 of cardinality up to 997. We also show that existence of a maximal curve implies that there is no minimal curve and vice versa.

Motivation & Objective

• To classify all isomorphism classes of optimal (maximal or minimal) genus 3 curves over finite fields with discriminant -19.
• To derive explicit polynomial equations for such curves using the theory of complex multiplication and Hermitian modules over $ ext{O}_K $, where $ K = ext{Q}( ext{sqrt}(-19)) $.
• To construct a comprehensive table of maximal and minimal curves over prime finite fields $ ext{F}_q $ with $ q \leq 997 $ and discriminant -19.
• To prove that the existence of a maximal curve over such a field implies no minimal curve exists, and vice versa.
• To apply Honda-Tate theory and Deligne’s theorem on isogeny classes to show that the Jacobian of any such curve is isogenous to a power of a maximal/minimal elliptic curve.

Proposed method

• Use the theory of complex multiplication and the endomorphism ring $ ext{O}_K $ for $ K = ext{Q}( ext{sqrt}(-19)) $ to classify isogeny classes of Jacobians of optimal curves.
• Leverage the equivalence between the category of ordinary abelian varieties isogenous to $ E^g $ and the category of $ ext{O}_K $-modules with unimodular Hermitian forms.
• Apply Riemann-Roch and linear system analysis to derive explicit models: for $ ext{dim}(2D) = 2 $, derive equations of the form $ z^2 = ext{quartic in } x + ext{linear in } y $, depending on the divisor type.
• Use valuation arguments at a point $ Q $ to prove linear dependence of monomials in $ x, y, z $, ensuring the minimal polynomial of $ z $ over $ ext{F}_q(x,y) $ has degree 2.
• Apply the strict triangle inequality on valuations to show that certain combinations of functions must vanish, leading to the final form of the curve equation.
• Verify the existence of maximal or minimal curves by checking whether the number of rational points reaches the Hasse-Weil-Serre bound, using the characteristic polynomial $ L(t) = (1 \mp [2\sqrt{q}]t + qt^2)^3 $.

Experimental results

Research questions

• RQ1 What is the complete set of isomorphism classes of optimal genus 3 curves over finite fields with discriminant -19?
• RQ2 Can all such optimal curves be described by explicit polynomial equations, and what form do these equations take?
• RQ3 For which prime fields $ ext{F}_q $ with discriminant -19 do maximal or minimal genus 3 curves exist?
• RQ4 Is it possible for both maximal and minimal curves of genus 3 to exist over the same finite field with discriminant -19?
• RQ5 How does the structure of the endomorphism ring $ ext{O}_K $ for $ K = ext{Q}( ext{sqrt}(-19)) $ constrain the isogeny class of the Jacobian of such curves?

Key findings

• All optimal genus 3 curves over finite fields with discriminant -19 admit an explicit equation of the form $ z^2 = ext{quartic in } x + ext{linear in } y $, as proven in Theorem 5.1.
• For each prime $ q \leq 997 $ with discriminant -19, the paper provides a complete list of maximal and minimal curves, with no overlap between maximal and minimal cases.
• The Jacobian of any such optimal curve is isogenous to $ E^3 $, where $ E $ is a maximal or minimal elliptic curve over $ ext{F}_q $, depending on the case.
• The existence of a maximal curve over $ ext{F}_q $ with discriminant -19 implies that no minimal curve exists over the same field, and vice versa.
• Explicit examples are provided: for $ q = 47 $, a maximal curve is $ y^2 = x^3 + x + 38 $, and a minimal curve is $ z^2 = 5 + 45x + 30x^2 + 10y $.
• For $ q = 997 $, a minimal curve is $ y^2 = x^3 + 500x + 934 $, and $ z^2 = x^2 + 336x + 564 + 196y $, confirming the existence of optimal curves at large prime sizes.