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

Let C be a smooth projective curve defined over the finite field F_q (q is odd) and let K=F_q(C) be its function field. Any finite set S of closed points of C gives rise to an integral domain O_S:=F_q[C−S] in K. We show that given an O_S-regular quadratic space (V,q) of rank n≥3 the set of genera in the proper classification of quadratic O_S-spaces isomorphic to (V,q) in the flat or étale topology, is in 1:1 correspondence with Br2(O_S) thus there are 2^|S|−1 genera. If (V,q) is isotropic, then Pic (O_S)/2 classifies the forms in the genus of (V,q). For n≥5 this is true for all genera, hence the full classification is via the abelian group H_ét ²(O_S,μ_₂).