Some criteria for stably birational equivalence of quadratic forms

TL;DR

This paper establishes criteria for stably birational equivalence of quadratic forms over fields of characteristic not 2, using isotropy, representations, and Clifford group isomorphisms, with applications to Pfister forms.

Contribution

It introduces new criteria linking isotropy over function fields and Clifford group isomorphisms to stably birational equivalence of quadratic forms.

Findings

Criterion for isotropy over function fields in terms of representations

Application to stably birational equivalence of multiples of Pfister forms

Criterion based on isomorphisms of Clifford group quotients

Abstract

Let $\varphi$ and $\psi$ be quadratic forms over a field $K$ of characteristic different from 2. In this paper, we give a criterion for isotropy of $\varphi$ over the function field of $\psi$ in terms of representations and we apply it to stably birational equivalence of $\varphi$ and $\psi$. Then we use this criterion to investigate the case of stably birational equivalence of multiples of Pfister forms. Finally, we give a a criterion of stably birational equivalence of quadratic forms in terms of isomorphisms of quotients of special Clifford groups by the kernel of the spinor norm.