Failure of the local-global principle for isotropy of quadratic forms over function fields

TL;DR

This paper demonstrates that the local-global principle for isotropy of quadratic forms fails over certain function fields, using advanced geometric and cohomological techniques involving Kummer varieties and elliptic curves.

Contribution

It introduces a novel counterexample to the local-global principle for quadratic forms over function fields of higher transcendence degree, employing new results in unramified cohomology.

Findings

Counterexamples to local-global principle for quadratic forms

New nontriviality results for unramified cohomology

Application of generalized Kummer varieties

Abstract

We prove the failure of the local-global principle, with respect to discrete valuations, for isotropy of quadratic forms over function fields of transcendence degree at least 2 over algebraically closed fields. Our construction involves generalized Kummer varieties as well as a new nontriviality result for the unramified cohomology of products of elliptic curves over discretely valued fields, which can be viewed as an arithmetic version of a theorem of Gabber.