Local-global principles for quadratic forms and strong linkage
5 October 2018
Campus Middelheim, G.005 - Middelheimlaan 1 - 2020 Antwerpen (route: UAntwerpen, Campus Middelheim
Organization / co-organization:
Department of Mathematics and Computer Science
Karim Johannes Becher, Arno Fehm
PhD defence Parul Gupta, Faculty of Science - Department of Mathematics and Computer Science
Quadratic forms are homogeneous polynomials of degree two. A key problem in the study of quadratic forms over fields is to determine whether a given quadratic form is isotropic i.e. whether it has a non-trivial zero.
The Hasse-Minkowski Theorem states that a quadratic form over a global field is isotropic if and only if it is isotropic over all completions with respect to discrete valuations and real places. Theorems of this type are referred to as local-global principles or Hasse principles. In 2012, it was shown that quadratic forms of dimension at least three over algebraic function fields, i.e. finitely generated extensions of transcendence degree one, over a non-dyadic complete discretely valued field satisfy the local-global principle for isotropy with respect to the set of all divisorial discrete valuations.
One of the main goals in this thesis is to study local-global principles for isotropy of quadratic forms over algebraic function fields over certain complete discretely valued fields, and their possible refinements.
We study the local-global principles for isotropy of quadratic forms of dimension 3 and 4 over the rational function field over a non-dyadic henselian discretely valued field of characteristic zero, in the cases where residue field is algebraically closed or finite of odd characteristic.
Results on local-global principles often imply a bound on the u-invariant of a field, which is defined as the maximum of the dimensions of anisotropic quadratic forms. We obtain necessary and sufficient conditions on a field to have u-invariant 8, and relate this for certain fields of u-invariant 8 to the local-global principle for the class of quadratic forms containing a 6-dimensional Pfister neighbor.
For a field, we study the property, introduced by Elman and Lam, called strong linkage, that any finite number of quaternion algebras have a common slot. Global fields are motivating examples for this property. We show that strong linkage holds for algebraic function fields over a complete discretely valued field with algebraically closed residue field and also for fraction fields of two-dimensional complete local domains with algebraically closed residue field.