Abstract
In 2001, M. Bhargava stunned the mathematical world by extending Gauss's 200-year-old group law on integral binary quadratic forms, now familiar as the ideal class group of a quadratic ring, to yield group laws on a vast assortment of analogous objects. His method yields parametrizations of rings of degree up to 5 over the integers, as well as aspects of their ideal structure, and can be employed to yield statistical information about such rings and the associated number fields. In this paper, we extend a selection of Bhargava's most striking parametrizations to cases where the base ring is not Z but an arbitrary Dedekind domain R. We find that, once the ideal classes of R are properly included, we readily get bijections parametrizing quadratic, cubic, and quartic rings, as well as an analogue of the 2x2x2 cube law reinterpreting Gauss composition for which Bhargava is famous. We expect that our results will shed light on the analytic distribution of extensions of degree up to 4 of a fixed number field and their ideal structure.
Highlights
The mathematics that we will discuss has its roots in the investigations of classical number theorists—notably Fermat, Lagrange, Legendre, and Gauss
Quadruples (a, (M1, M2, M3), θ, β) where a is an ideal class of R, Mi are lattices of rank 2 over R, θ : 2M1 ⊗ 2M2 ⊗ 2M3 → a3 is an isomorphism, and β : M1 ⊗ M2 ⊗ M3 → a is a trilinear map whose three partial duals βi : Mj ⊗ Mk → aMi∗ ({i, j, k} = {1, 2, 3}) have image a full-rank sublattice
When learning about Gauss composition over Z, one must sooner or later come to a problem that vexed Legendre: If one considers quadratic forms up to GL2Zchanges of variables, a group structure does not emerge because the conjugate forms ax2 ± bxy + cy2, which ought to be inverses, have been identified
Summary
The mathematics that we will discuss has its roots in the investigations of classical number theorists—notably Fermat, Lagrange, Legendre, and Gauss Quadruples (a, (M1, M2, M3), θ , β) where a is an ideal class of R, Mi are lattices of rank 2 over R (up to isomorphism), θ : 2M1 ⊗ 2M2 ⊗ 2M3 → a3 is an isomorphism, and β : M1 ⊗ M2 ⊗ M3 → a is a trilinear map whose three partial duals βi : Mj ⊗ Mk → aMi∗ ({i, j, k} = {1, 2, 3}) have image a full-rank sublattice Under this bijection, we get identifications 2S ∼= a and Ii ∼= Mi. In particular R may have characteristic 2, the frequent factors of 1/2 in Bhargava’s exposition notwithstanding, and by weakening the nondegeneracy condition, we are able to include balanced triples in degenerate rings. 5, we prove Bhargava’s parametrization of balanced ideal triples (itself a generalization of Gauss composition) over a Dedekind domain. We will be concerned with algebras of ranks 2, 3, and 4, which we call quadratic, cubic, and quartic algebras (or rings) respectively
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