Parametrization of ideal classes in rings associated to binary forms

By Melanie Matchett Wood. 2010

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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.

View Paper: https://www.semanticscholar.org/paper/a47fd7b5136fae5b96a85f6a2a21b05744d3e265