Content area
Abstract
The primary aim of this thesis is to derive higher analogues of Gauss's celebrated law of composition on the space of binary quadratic forms. Specifically, we show that Gauss's law is only one of at least ten such laws of composition that yield information on the class groups of algebraic number fields. We begin our investigation of higher composition laws by giving a new perspective on Gauss composition in a manner reminiscent of the group law on elliptic curves. We then proceed to derive new laws of composition on 2 x 2 x 2 cubes, binary cubic forms, pairs of binary quadratic forms, 2 x 3 x 3 boxes, and pairs of ternary quadratic forms. We show that the resulting groups in these spaces all have natural interpretations in terms of ideal classes of orders in algebraic number fields.
We also develop a theory of resolvent rings in order to explain how orders in number fields of low degree should be parametrized. The theory allows us, in particular, to obtain a new derivation of the Delone-Faddeev-Gross parametrization of cubic rings by means of binary cubic forms. More importantly, our perspective enables us to generalize the Delone-Faddeev-Gross result to the quartic case, yielding a parametrization of quartic rings by means of two ternary quadratic forms.
We use this new parametrization result for quartic rings, in the spirit of Davenport-Heilbronn, to compute the density of discriminants of S4-quartic fields, thus resolving this long-standing problem. In addition, our methods allow us also to compute the mean value of the size of the 2-class group of cubic fields. This result confirms, for the first time, a case of the Cohen-Martinet heuristics, and implies that at least 75% of totally real cubic fields have odd class number.
Finally, we expect that the composition laws presented here will have many additional applications, e.g., to the theory of automorphic forms on exceptional groups. We outline a few of these potential applications to indicate directions for future work.