On the flat cohomology of binary norm forms

TL;DR

This paper investigates the structure of flat cohomology sets associated with binary quadratic forms, extending classical results on form classification to non-fundamental discriminants using cohomological methods.

Contribution

It describes the structure of flat cohomology sets for binary quadratic forms of non-fundamental discriminants, generalizing Gauss's classical classification results.

Findings

Describes the structure of $H^1_{fl}(Z, O_{d,m})$ for quadratic forms.

Extends Gauss's classification to non-fundamental discriminants.

Provides a cohomological framework for understanding quadratic form classification.

Abstract

Let $\mathcal{O}$ be an order of index $m$ in the maximal order of a quadratic number field $k=\mathbb{Q}(\sqrt{d})$. Let $\underline{\mathbf{O}}_{d,m}$ be the orthogonal $\mathbb{Z}$-group of the associated norm form $q_{d,m}$. We describe the structure of the pointed set $H^1_{\mathrm{fl}}(\mathbb{Z},\underline{\mathbf{O}}_{d,m})$, which classifies quadratic forms isomorphic (properly or improperly) to $q_{d,m}$ in the flat topology. Gauss classified quadratic forms of fundamental discriminant and showed that the composition of any binary $\mathbb{Z}$-form of discriminant $\Delta_k$ with itself belongs to the principal genus. Using cohomological language, we extend these results to forms of certain non-fundamental discriminants.