Parametrization of ideal classes in rings associated to binary forms

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We give a parametrization of the ideal classes of rings associated to integral binary forms by classes of tensors in Z² Zⁿ Zⁿ. This generalizes Bhargava's work on Higher Composition Laws, which gives such parametrizations in the cases 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.