Explicit composition identities for higher composition laws

Abstract

In 2001, Bhargava proved a composition law for $2 \times 2 \times 2$ integer cubes, which generalized Gauss composition of integral binary quadratic forms. Furthermore, he derived four new composition laws defined on the following spaces: 1) binary cubic forms with triplicate middle coefficients, 2) pairs of binary quadratic forms with duplicate middle coefficients, 3) pairs of quaternary alternating 2-forms and 4) senary alternating 3-forms. In each of the five cases, there is a natural group action on the underlying space with a unique polynomial invariant called the discriminant, and a notion of projectivity for the elements of the space. The strategy behind Bhargava's approach is to construct a discriminant-preserving bijection between the set of orbits under the group action and the set of (tuples of) suitable ideal classes of quadratic rings. The projective ideal classes are equipped with a natural group structure and hence we get a group structure on the spaces of equivalence classes of projective forms of fixed discriminant $D$. In each case the class group of projective forms of discriminant $D$ has a natural interpretation in terms of the narrow class group of the quadratic ring of discriminant $D$. The aim of this paper is to give explicit composition identities (similar to Gauss' formulation of composition of binary quadratic forms) for these higher composition laws.