Minimisation of 2-coverings of genus 2 Jacobians

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.