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An analog of the Edwards model for Jacobians of genus 2 curves

E. Victor Flynn, Kamal Khuri-Makdisi

TL;DR

This work constructs an explicit ${\bf P}^3 \times {\bf P}^3$ embedding of the Jacobian of a genus $2$ curve, using a pair of Kummer coordinates for $D$ and $D+D_1$ with a fixed order $4$ point $D_1$, to obtain a concise model that surpasses the standard ${\bf P}^{15}$ representation. It develops a robust framework based on algebraic theta functions and Kempf's theta group to derive a complete bigraded ideal of relations, with precise generators and dimensions (Theorems generatorsInEachBidegree and PQAddition), and provides explicit equations for the Jacobian and its group law (Theorem defeqns and Theorem GroupLawGenus2), including a twisted version. The Edwards-curve analogy is extended to genus $2$, illustrating how a universal group law can be achieved in an elliptic setting and mirrored in the genus $2$ construction; the approach yields explicit addition formulas via matrices $A_{ij}$ and $J_{ij}$. Finally, the paper gives a concrete nondegeneracy criterion for the group law: if $c$, $cd$, and $g(g-b^2(c-1))$ are nonsquares in the ground field $K$, then the induced group law is universal on $K$-rational points, enabling robust arithmetic on genus $2$ Jacobians and potential cryptographic applications.

Abstract

We give the explicit equations for a P^3 x P^3 embedding of the Jacobian of a curve of genus 2, which gives a natural analog for abelian surfaces of the Edwards curve model of elliptic curves. This gives a much more succinct description of the Jacobian variety than the standard version in P^{15}. We also give a condition under which, as for the Edwards curve, the abelian surfaces have a universal group law.

An analog of the Edwards model for Jacobians of genus 2 curves

TL;DR

This work constructs an explicit embedding of the Jacobian of a genus curve, using a pair of Kummer coordinates for and with a fixed order point , to obtain a concise model that surpasses the standard representation. It develops a robust framework based on algebraic theta functions and Kempf's theta group to derive a complete bigraded ideal of relations, with precise generators and dimensions (Theorems generatorsInEachBidegree and PQAddition), and provides explicit equations for the Jacobian and its group law (Theorem defeqns and Theorem GroupLawGenus2), including a twisted version. The Edwards-curve analogy is extended to genus , illustrating how a universal group law can be achieved in an elliptic setting and mirrored in the genus construction; the approach yields explicit addition formulas via matrices and . Finally, the paper gives a concrete nondegeneracy criterion for the group law: if , , and are nonsquares in the ground field , then the induced group law is universal on -rational points, enabling robust arithmetic on genus Jacobians and potential cryptographic applications.

Abstract

We give the explicit equations for a P^3 x P^3 embedding of the Jacobian of a curve of genus 2, which gives a natural analog for abelian surfaces of the Edwards curve model of elliptic curves. This gives a much more succinct description of the Jacobian variety than the standard version in P^{15}. We also give a condition under which, as for the Edwards curve, the abelian surfaces have a universal group law.
Paper Structure (5 sections, 26 theorems, 116 equations)

This paper contains 5 sections, 26 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Generators of the ideal of relations
  3. An approach for an explicit derivation of the Edwards curve
  4. An analog for Jacobians of genus 2 curves
  5. Conditions for non-degeneracy of the group law

Key Result

Theorem 2.2

Suppose that $k$ is both even and invertible in $\overline{K}$. Choose a symplectic decomposition $\mathfrak{J}[2k] = A_{2k} \oplus B_{2k}$; this also determines a decomposition $\mathfrak{J}[k] = A_k \oplus B_k$, with $B_k = [2]B_{2k} = B_{2k} \cap \mathfrak{J}[k]$, and an analogous $A_k$. Then the Here $b_1, b_2 \in B_k$, and we choose $d_1, d_2 \in B_{2k}$ satisfying $2d_1 = b_1, 2d_2 = b_2$. (

Theorems & Definitions (57)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Corollary 2.6
  • proof
  • Lemma 2.7
  • ...and 47 more