Algebraic Geometry Seminar
The local-global dimension spectrum of quadratic forms
Speaker: Asher Auel, Yale University
Location: Warren Weaver Hall 317
Date: Tuesday, October 17, 2017, 3:30 p.m.
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, and its stable variant, 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.