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

TL;DR

This paper investigates the Hasse principle for bilinear symmetric forms over rings of integers in global function fields, linking the principle's validity to properties of the Picard group of the curve's affine part.

Contribution

It establishes a bijection between isomorphism classes of forms and certain étale cohomology groups, and characterizes when the Hasse principle holds based on the Picard group's order.

Findings

The set of isomorphism classes is bijective with the Picard group modulo 2.

The Hasse principle holds if and only if the Picard group has odd order.

For rank 2 forms, the principle depends on the integral closure being a UFD.

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.