On the classification of quadratic forms over an integral domain of a global function field

TL;DR

This paper classifies quadratic forms over certain function field rings, relating their genera to Brauer groups and Picard groups, providing explicit counts and classifications for forms over these integral domains.

Contribution

It establishes a correspondence between the genus of quadratic forms over $\

Findings

Number of genera equals $2^{|S|-1}$ for regular quadratic spaces.

Genus classification is linked to the 2-torsion of the Brauer group.

Full classification for $n \\geq 5$ uses étale cohomology group $H^2_{\\text{ét}}$.

Abstract

Let $C$ be a smooth projective curve defined over the finite field $\mathbb{F}_q$ ($q$ is odd) and let $K=\mathbb{F}_q(C)$ be its function field. Any finite set $S$ of closed points of $C$ gives rise to an integral domain $\mathcal{O}_S:=\mathbb{F}_q[C-S]$ in $K$. We show that given an $\mathcal{O}_S$-regular quadratic space $(V,q)$ of rank $n \geq 3$, the set of genera in the proper classification of quadratic $\mathcal{O}_S$-spaces isomorphic to $(V,q)$ in the flat or \'etale topology, is in $1:1$ correspondence with ${_2\text{Br}}(\mathcal{O}_S)$, thus there are $2^{|S|-1}$ such. If $(V,q)$ is isotropic, then $\text{Pic}(\mathcal{O}_S)/2$ classifies the forms in the genus of $(V,q)$. For $n \geq 5$ this is true for all genera, hence the full classification is via the abelian group $H^2_{\text{\'et}}(\mathcal{O}_S,\underline{\mu}_2)$.