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 F(A) of the Severi–Brauer variety of A, is adjoint to a quadratic form qσ over 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 qσ. As opposed to this, we prove that there exists non-totally decomposable orthogonal involutions that become totally decomposable over F(A), so that the associated form qσ 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 qσ does not determine the isomorphism class of σ, even when the underlying algebra has index 2. As a consequence, we show that the e3 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.