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

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 $H^2_{\text{\'et}}(\mathcal{O}_S,\underline{\mu}_2)$.