Parametrization of ideal classes in rings associated to binary forms

TL;DR

This paper generalizes Bhargava's parametrization of ideal classes in rings from binary forms to higher dimensions using tensor classes, including symmetric tensors for 2-torsion classes, with geometric interpretations.

Contribution

It introduces a new tensor-based parametrization of ideal classes in rings associated to binary forms for arbitrary dimensions, extending prior work to broader contexts.

Findings

Parametrization of ideal classes via tensor classes for arbitrary n

Extension of parametrization to 2-torsion ideal classes using symmetric tensors

Geometric constructions of modules from tensor parametrizations

Abstract

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.