Outline and References for Project: Hasse Principle for Rational Function Fields,

Hasse-Minkowski’s theorem asserts that a quadratic form over a number field k admits a nontrivial zero if it does over completions at all places of k. One could look for analogues of Hasse principle for function fields. Let k be a field of characteristic not 2 and Ω a set of discrete valuations of k. Let k̂v denote the completion of k at v. We say that k satisfies Hasse principle with respect to Ω if every quadratic form over k which is isotropic over k̂v for all v ∈ Ω is isotropic. We say that k satisfies weak Hasse principle with respect to Ω if every quadratic form over k which is hyperbolic over k̂v for all v ∈ Ω is hyperbolic.