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

TL;DR

This paper presents explicit equations for embedding genus 2 Jacobians into a product of projective spaces, providing a more concise model similar to Edwards curves for elliptic curves, and discusses conditions for a universal group law.

Contribution

It introduces a new explicit embedding of genus 2 Jacobians into P^3 x P^3, analogous to Edwards models for elliptic curves, simplifying their description.

Findings

Explicit P^3 x P^3 embedding equations for genus 2 Jacobians

A criterion for universal group law on these abelian surfaces

More succinct representation than standard P^{15} models

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.