Department of MathematicsGiven a quadratic form (homogeneous degree two polynomial) q over a field k, some basic questions one can ask are
- Does q have a non-trivial zero (is q isotropic)?
- Which non-zero elements of k are represented by q?
- Does q represent all non-zero elements of k (is q universal)?
Over a global field F, the Hasse-Minkowski Theorem, which is one of the first examples of a local-global principle, allows us to use answers to these questions over the completions of F to form answers to these questions over F itself. In this talk, we will focus primarily on quadratic forms over semi-global fields (function fields of curves over complete discretely valued fields), and see how a local-global principle of Harbater, Hartmann, and Krashen can be used to study universal quadratic forms over semi-global fields.