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Rational Points and Zeta Functions of Humbert Surfaces with Square Discriminant

Elira Shaska, Jorge Mello, Sajad Salami, Tony Shaska

TL;DR

This work analyzes the arithmetic of Humbert loci $ abla_n$ parameterizing genus 2 curves with $(n,n)$-split Jacobians, identifying them with Humbert surfaces $ abla ext{H}_{n^2}$ and deriving explicit equations in the weighted projective space $ abla$P$(2,4,6,10)$. It provides concrete point counts over finite fields for $n=2,3,5$, computes zeta functions for $n=2,3$, and develops a framework to construct and detect $(n,n)$-isogenies via explicit Igusa invariants and endomorphism-ring analysis, with applications to post-quantum genus 2 cryptography. The paper also investigates automorphisms, intersections of $ abla_n$, and modern computational methods, including Gröbner-basis approaches and potential machine-learning optimizations, to enhance curve selection, verification, and security assessments across characteristics, notably noting a collapse of $ abla_n$ in characteristic $p=3$. Overall, the results offer a practical pipeline for arithmetic-geometric analysis and cryptographic protocol design using genus 2 curves with controlled split Jacobians, balancing efficiency and security through explicit invariants, zeta data, and endomorphism structure.

Abstract

This paper examines the arithmetic of the loci \(\cL_n\), parameterizing genus 2 curves with \((n, n)\)-split Jacobians over finite fields \(\F_q\). We compute rational points \(|\cL_n(\F_q)|\) over \(\F_3\), \(\F_9\), \(\F_{27}\), \(\F_{81}\), and \(\F_5\), \(\F_{25}\), \(\F_{125}\), derive zeta functions \(Z(\cL_n, t)\) for \(n = 2, 3\). Utilizing these findings, we explore isogeny-based cryptography, introducing an efficient detection method for split Jacobians via explicit equations, enhanced by endomorphism ring analysis and machine learning optimizations. This advances curve selection, security analysis, and protocol design in post-quantum genus 2 systems, addressing efficiency and vulnerabilities across characteristics.

Rational Points and Zeta Functions of Humbert Surfaces with Square Discriminant

TL;DR

This work analyzes the arithmetic of Humbert loci parameterizing genus 2 curves with -split Jacobians, identifying them with Humbert surfaces and deriving explicit equations in the weighted projective space P. It provides concrete point counts over finite fields for , computes zeta functions for , and develops a framework to construct and detect -isogenies via explicit Igusa invariants and endomorphism-ring analysis, with applications to post-quantum genus 2 cryptography. The paper also investigates automorphisms, intersections of , and modern computational methods, including Gröbner-basis approaches and potential machine-learning optimizations, to enhance curve selection, verification, and security assessments across characteristics, notably noting a collapse of in characteristic . Overall, the results offer a practical pipeline for arithmetic-geometric analysis and cryptographic protocol design using genus 2 curves with controlled split Jacobians, balancing efficiency and security through explicit invariants, zeta data, and endomorphism structure.

Abstract

This paper examines the arithmetic of the loci , parameterizing genus 2 curves with \((n, n)\)-split Jacobians over finite fields . We compute rational points \(|\cL_n(\F_q)|\) over , , , , and , , , derive zeta functions \(Z(\cL_n, t)\) for . Utilizing these findings, we explore isogeny-based cryptography, introducing an efficient detection method for split Jacobians via explicit equations, enhanced by endomorphism ring analysis and machine learning optimizations. This advances curve selection, security analysis, and protocol design in post-quantum genus 2 systems, addressing efficiency and vulnerabilities across characteristics.
Paper Structure (61 sections, 8 theorems, 60 equations)

This paper contains 61 sections, 8 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Humbert surfaces
  4. Zeta Function
  5. Good and Bad Reduction of Weighted Hypersurfaces
  6. $\mathbb{F}_q$-Rational Points on Weighted Hypersurfaces
  7. Explicit Equations for $\mathcal{L}_n$
  8. Degree 2
  9. Degree 3
  10. Degree 5
  11. Higher degrees
  12. A few historical remarks
  13. Computing Rational Points and Zeta Function for $\mathcal{L}_2$
  14. Characteristic $p=5$
  15. Computations over $\mathbb{F}_5$:
  16. ...and 46 more sections

Key Result

Proposition 2.1

Let $X=V(F)$ be a weighted hypersurface defined by $F\in S\setminus \{0\}$ of degree $d\geq 1$, $N(F)$ be the set of zeros of $F$ in $\mathbb{F}_q^{n+1}$, and define

Theorems & Definitions (17)

  • Proposition 2.1
  • proof
  • Conjecture 2.2
  • Conjecture 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 3.1
  • Remark 4.1
  • Remark 5.1
  • Example 9.1: $\mathcal{L}_2$ over $\mathbb{F}_5$
  • ...and 7 more