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

Abstract

We prove the failure of the local-global principle, with respect to discrete valuations, for isotropy of quadratic forms in $2^n$ variables over function fields of transcendence degree $n\ge 2$ over an algebraically closed field of characteristic $\ne 2$. Our construction involves the generalized Kummer varieties considered by Borcea and by Cynk and Hulek as well as new results on the nontriviality of unramified cohomology of products of elliptic curves over discretely valued fields.