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

Abstract

An orthogonal involution $σ$ 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 $q_σ$ 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 $σ$ if and only if they hold for $q_σ$. 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 $q_σ$ 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 $q_σ$ does not determine the isomorphism class of $σ$, 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.