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

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. Removing one closed point $C^{\text{af}}=C-\{\mathrm{\infty }\}$ results in an integral domain ${\mathcal{O}}_{\{\mathrm{\infty }\}}={\mathbb{F}}_q[C^{\text{af}}]$ of $K$, over which we consider a non-degenerate bilinear and symmetric form $f$ with orthogonal group ${\underset{‾}{\mathbf{\text{O}}}}_V$. We show that the set ${\text{Cl}}_{\mathrm{\infty }}({\underset{‾}{\mathbf{\text{O}}}}_V)$ of ${\mathcal{O}}_{\{\mathrm{\infty }\}}$-isomorphism classes in the genus of $f$ of rank $n>2$, is bijective as a pointed set to the abelian groups $H^2_{$\acute{\text{e}}\text{t}$}($\mathcal{O}$_{{\infty}},\underline{\mu}_2) \cong $\text{Pic}$(C^$\text{af}$)/2$,i\mathrm{.}e\mathrm{.}isaninvariantof$C^$\text{af}$$. We then deduce that any such $f$ of rank $n>2$ admits the local-global Hasse principal if and only if $\mathrm{\mid }\text{Pic}(C^{\text{af}})\mathrm{\mid }$ is odd. For rank $2$ this principle holds if the integral closure of ${\mathcal{O}}_{\{\mathrm{\infty }\}}$ in the splitting field of ${\underset{‾}{\mathbf{\text{O}}}}_V{\otimes }_{{\mathcal{O}}_{\{\mathrm{\infty }\}}}K$ is a UFD.