Some minimisation algorithms in arithmetic invariant theory

Abstract

We extend the work of Cremona, Fisher and Stoll on minimising genus one curves of degrees 2,3,4,5, to some of the other representations associated to genus one curves, as studied by Bhargava and Ho. Specifically we describe algorithms for minimising bidegree (2,2)-forms, 3 x 3 x 3 cubes and 2 x 2 x 2 x 2 hypercubes. We also prove a theorem relating the minimal discriminant to that of the Jacobian elliptic curve.