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 $n\ge 3$, the set of genera in the proper classification of quadratic ${\mathcal{O}}_S$-spaces isomorphic to $(V,q)$ in the flat or étale 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 $n\ge 5$ this is true for all genera, hence the full classification is via the abelian group $H_{\text{eˊt}}^2({\mathcal{O}}_S,{\underline{\mu }}_2)$.

19 pages, no figures

References in corpus (1)

• The Hasse principle for bilinear symmetric forms over a ring of integers of a global function field

Cited by in corpus (2)

• On the genera of semisimple groups defined over an integral domain of a global function field
• The twisted forms of a semisimple group over an ${\mathbb{F}}_q$-curve