Orthogonal involutions on central simple algebras and function fields of Severi-Brauer varieties

TL;DR

This paper explores the relationship between orthogonal involutions on central simple algebras and associated quadratic forms over function fields, revealing limitations in classifying involutions using these forms and invariants.

Contribution

It demonstrates that certain orthogonal involutions become totally decomposable over function fields, and shows that associated quadratic forms do not uniquely determine the involutions, challenging previous assumptions.

Findings

Existence of non-totally decomposable involutions becoming totally decomposable over function fields

Examples of nonisomorphic involutions with similar quadratic forms

The $e_3$ invariant is not sufficient for classification in specific degrees

Abstract

An orthogonal involution $\sigma$ on a central simple algebra $A$, after scalar extension to the function field $\mathcal{F}(A)$ of the Severi--Brauer variety of $A$, is adjoint to a quadratic form $q_\sigma$ over $\mathcal{F}(A)$, which is uniquely defined up to a scalar factor. Some properties of the involution, such as hyperbolicity, and isotropy up to an odd-degree extension of the base field, are encoded in this quadratic form, meaning that they hold for the involution $\sigma$ if and only if they hold for $q_\sigma$. As opposed to this, we prove that there exists non-totally decomposable orthogonal involutions that become totally decomposable over $\mathcal{F}(A)$, so that the associated form $q_\sigma$ is a Pfister form. We also provide examples of nonisomorphic involutions on an index $2$ algebra that yield similar quadratic forms, thus proving that the form $q_\sigma$ does not determine the isomorphism class of $\sigma$, even when the underlying algebra has index $2$. As a consequence, we show that the $e_3$ invariant for orthogonal involutions is not classifying in degree $12$, and does not detect totally decomposable involutions in degree $16$, as opposed to what happens for quadratic forms.