TAILIEUCHUNG - Đề tài " Higher composition laws II: On cubic analogues of Gauss composition "

Annals of Mathematics 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. | Annals of Mathematics Higher composition laws II On cubic analogues of Gauss composition By Manjul Bhargava Annals of Mathematics 159 2004 865 886 Higher composition laws II On cubic analogues of Gauss composition By Manjul Bhargava 1. Introduction 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 cubic 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 Mị Nị 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 cubic case might be to examine 3 X 3 X 3 cubes of integers. A3 X 3 X 3 cube C can be sliced in three different ways into three 3

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