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, it is an interesting problem to quantify the extent to which the analogous local-global principle holds.

For example, over the function field of a complex curve, quadratic forms in 2 variables that locally admit zeros are in bijection with the 2-torsion points of the Jacobian, while every quadratic form in at least 3 variables admits a zero by Tsen's theorem.  I will introduce a new invariant, the local-global dimension spectrum of a field, which quantifies the failure of the local-global principle.  Finally, I will explain recent work with V. Suresh on bounding the local-global spectrum over function fields of complex varieties, which involves the study of certain Calabi-Yau varieties of generalized Kummer type.