The local-global principle for quadratic forms over function fields

Abstract: The Hasse-Minkowski theorem says that a quadratic form over a global field admits a nontrivial zero if it admits a nontrivial zero everywhere locally. Over more general fields of arithmetic and geometric interest, the failure of the local-global principle is often controlled by auxiliary structures of interest, such as torsion points of the Jacobian and the Brauer group. I will explain work with V. Suresh on the failure of the local-global principle for quadratic forms over function fields varieties of dimension at least two. The counterexamples we construct are controlled by higher unramified cohomology groups and involve the study of Calabi-Yau varieties of generalized Kummer type that originally arose from number theory. Along the way, we need to develop an arithmetic version of a result of Gabber on the nontriviality of certain unramified cohomology classes on products of elliptic curves.

algebraic geometrynumber theory

Audience: researchers in the topic

SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

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