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

Auel, Asher; Suresh, V.

doi:10.2140/ant.2024.18.1497

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

Asher Auel and V. Suresh

Vol. 18 (2024), No. 8, 1497–1513

DOI: 10.2140/ant.2024.18.1497

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 $n \geq 2$ over an algebraically closed field of characteristic $\neq 2$ . 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.

Keywords

quadratic forms, local-global principle, Hasse principle, function fields, unramified cohomology, elliptic curves

Mathematical Subject Classification

Primary: 11E04, 11R58, 12G05, 14G12

Secondary: 14H52, 14J32, 14J40, 19D45

Milestones

Received: 9 August 2022

Revised: 16 August 2023

Accepted: 18 September 2023

Published: 18 September 2024

Authors

Asher Auel
Department of Mathematics Dartmouth College Hanover, NH United States

V. Suresh
Department of Mathematics Emory University Atlanta, GA United States

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Vol. 18, No. 8, 2024