Parametrization of ideal classes in rings associated to binary forms

Details Parametrization of ideal classes in rings associated to binary forms Parametrization of ideal classes in rings associated to binary forms

2010-08-27

arxiv.org/abs/1008.4781v1

Mathematics

Number Theory

Scientific paper

We give a parametrization of the ideal classes of rings associated to integral binary forms by classes of tensors in $\mathbb Z^2\tensor \mathbb Z^n\tensor \mathbb Z^n$. 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 $\mathbb Z$ is replaced by an arbitrary base scheme $S$, and geometric constructions of the modules from the tensors in the parametrization.

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