Equivalence relations of quadratic forms in characteristic 2 and quasilinear p-forms

Abstract

This thesis deals with quadratic forms and quasilinear p-forms in positive characteristic. In this setting, there are three well-known equivalence relations -- similarity, birational equivalence, and stable birational equivalence. Inspired by an algebraic characterization of motivic equivalence of quadratic forms over fields of characteristic other than two, this thesis defines a new equivalence relation -- the Vishik equivalence. The thesis is divided into chapters based on the kind of forms treated: quasilinear p-forms (over fields of characteristic p), totally singular quadratic forms, nonsingular quadratic forms, and singular quadratic forms (all of them over fields of characteristic 2). The main goal is to compare the four above-mentioned equivalences for each of those kinds of forms. We also derive some consequences of two forms being equivalent for each of the four equivalences separately. In particular, we give a new characterization of the stable birational equivalence for quadratic forms. Moreover, we provide some new results regarding the isotropy of quasilinear p-forms over field extensions.