Algebraic Geometry Seminar

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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.

Synopsis:

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.