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 ${\underline{\text{O}}}_V$. We show that the set ${\text{Cl}}_{\infty }({\underline{\text{O}}}_V)$ 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 $H_{\text{eˊt}}^2({\mathcal{O}}_{\{\infty \}},{\underline{\mu }}_2)\cong \text{Pic}(C^{\text{af}})/2$, 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 ${\underline{\text{O}}}_V{\otimes }_{{\mathcal{O}}_{\{\infty \}}}K$ is a UFD.

10 pages, no figures

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