Minimisation of 2-coverings of genus 2 Jacobians

TL;DR

This paper presents an algorithm for minimising pairs of quadratic forms with applications to simplifying the Jacobian of genus 2 curves, aiding in computational arithmetic geometry tasks.

Contribution

It introduces a novel algorithm for minimising specific pairs of quadratic forms under constraints, advancing methods in genus 2 Jacobian analysis.

Findings

Algorithm effectively minimises quadratic form pairs

Improves computational techniques for genus 2 Jacobians

Facilitates 2-descent processes in arithmetic geometry

Abstract

An important problem in computational arithmetic geometry is to find changes of coordinates to simplify a system of polynomial equations with rational coefficients. This is tackled by a combination of two techniques, called minimisation and reduction. We give an algorithm for minimising certain pairs of quadratic forms, subject to the constraint that the first quadratic form is fixed. This has applications to 2-descent on the Jacobian of a genus 2 curve.