Explicit Rational Group Law on Hyperelliptic Jacobians of any Genus

TL;DR

This paper provides explicit polynomial and rational functions describing the group law on hyperelliptic Jacobians of any genus, extending classical models and enabling more concrete computations in higher dimensions.

Contribution

It introduces explicit equations for the group law on hyperelliptic Jacobians, generalizing the elliptic curve chord method to higher genus cases.

Findings

Explicit equations for hyperelliptic Jacobian group law derived

Extension of Mumford's models to higher genus

Facilitates computational approaches in algebraic geometry

Abstract

It is well-known that abelian varieties are projective, and so that there exist explicit polynomial and rational functions which define both the variety and its group law. It is however difficult to find any explicit polynomial and rational functions describing these varieties or their group laws in dimensions greater than two. One exception can be found in Mumford's classic "Lectures on Theta", where he describes how to obtain an explicit model for hyperelliptic Jacobians as the union of several affine pieces described as the vanishing locus of explicit polynomial equations. In this article, we extend this work to give explicit equations for the group law on a dense open set. One can view these equations as generalizations of the usual chord-based group law on elliptic curves.