Higher composition laws II: On cubic analogues of Gauss composition

Abstract

In our first article [2] we developed a new view of Gauss composition of binary quadratic forms which led to several new laws of composition on various other spaces of forms. Moreover, we showed that the groups arising from these composition laws were closely related to the class groups of orders in quadratic number fields, while the spaces underlying those composition laws were closely related to certain exceptional Lie groups. In this paper, our aim is to develop analogous laws of composition on certain spaces of forms so that the resulting groups yield information on the class groups of orders in fields; that is, we wish to obtain genuine cubic analogues of Gauss composition. The fundamental object in our treatment of quadratic composition [2] was the space of 2 x 2 x 2 cubes of integers. In particular, Gauss composition arose from the three different ways of slicing a cube A into two 2 x 2 matrices Mi, Ni (i = 1, 2, 3). Each such pair (Mi, Ni) gives rise to a binary quadratic form QA(x, y) = Qi (x, y), defined by Qi(x, y) = -Det(Mix + Niy). The Cube Law of [2] declares that as A ranges over all cubes, the sum of [Qi], [Q2], [Q3] is zero. It was shown in [2] that the Cube Law gives a law of addition on binary quadratic forms that is equivalent to Gauss composition. Various other invariant-theoretic constructions using the space of 2 x 2 x 2 cubes led to several new composition laws on other spaces of forms. Furthermore, we showed that each of these composition laws gave rise to groups that are closely related to the class groups of orders in quadratic fields. Based on the quadratic case described above, our first inclination for the case might be to examine 3 x 3 x 3 cubes of integers. A 3 x 3 x 3 cube C can be sliced (in three different ways) into three 3 x 3 matrices Li, Mi, Ni (i = 1,2,3). We may therefore obtain from C three ternary forms fl(x, y, Z), f2(x, y,z), f3(x, y,z), defined by