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