Hasse principles for quadratic forms over function fields

TL;DR

This paper explores the validity of Hasse principles for quadratic forms over various function fields, revealing numerous counterexamples to isotropy principles under specific valuation conditions.

Contribution

It provides new counterexamples to the Hasse principle for quadratic forms over certain function fields, extending previous results to lower dimensions and different valuation sets.

Findings

Counterexamples to Hasse principle for isotropy over purely transcendental extensions.

Counterexamples of lower dimension derived from higher-dimensional cases.

Analysis of Hasse principles with respect to various sets of discrete valuations.

Abstract

We investigate the Hasse principles for isotropy and isometry of quadratic forms over finitely generated field extensions with respect to various sets of discrete valuations. Over purely transcendental field extensions of fields that satisfy property $\mathscr{A}_i(2)$ for some $i$, we find numerous counterexamples to the Hasse principle for isotropy with respect to a relatively small set of discrete valuations. For finitely generated field extensions $K$ of transcendence degree $r$ over an algebraically closed field of characteristic $\ne 2$, we use the $2^r$-dimensional counterexample to the Hasse principle for isotropy due to Auel and Suresh to obtain counterexamples of lower dimensions with respect to the divisorial discrete valuations induced by a variety with function field $K$.