The Hasse-Minkowski theorem states that a quadratic form over a global field is isotropic if it is isotropic over all completions of the field. This is one of the first “local-global principles” for quadratic forms and implies that two quadratic forms over a global field are isometric if they are isometric over each completion. Over other fields, these local-global principles can be phrased in terms of discrete valuations on the field. In this talk, we explore these local-global principles for isotropy and isometry of quadratic forms over function fields with respect to various sets of discrete valuations. We will see that over rational function fields, it is “easy” to satisfy the local-global principle for isometry, but "hard" to satisfy the local-global principle for isotropy. More generally, over finitely generated field extensions of transcendence degree *r* over an algebraically closed field, we use the *2^r*-dimensional counterexample to the local-global principle for isotropy of Auel and Suresh to show that there exist counterexamples of dimension <*2^r* with respect to sets of discrete valuations arising naturally from geometric considerations.

### Algebra Seminar

Friday, April 8, 2022 - 3:30pm

#### Connor Cassady

University of Pennsylvania