Parametrization of ideal classes in rings associated to binary forms

Abstract

Abstract. We parametrize the ideal classes of rings associated to integral binary forms by classes of tensors in ℤ 2 ⊗ ℤ n ⊗ ℤ n ${\mathbb {Z}} ^2\otimes {\mathbb {Z}} ^n\otimes {\mathbb {Z}} ^n$ . This generalizes Bhargava's work on Higher Composition Laws, which gives such parametrizations in the cases n = 2 , 3 $n=2,3$ . We also obtain parametrizations of 2-torsion ideal classes by symmetric tensors. Further, we give versions of these theorems when ℤ is replaced by an arbitrary base scheme S, and geometric constructions of the modules from the tensors in the parametrization.