Explicit composition identities for higher composition laws

TL;DR

This paper provides explicit composition identities for higher composition laws introduced by Bhargava, generalizing Gauss composition to new algebraic structures involving forms and ideal classes.

Contribution

It derives explicit formulas for composition laws in five new algebraic settings, extending Bhargava's work on higher composition laws.

Findings

Explicit composition identities for five higher composition laws.

Connection between form classes and narrow class groups of quadratic rings.

Generalization of Gauss composition to complex algebraic structures.

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.