Orthogonal involutions on central simple algebras and function fields of Severi-Brauer varieties. (English) Zbl 1499.20118
Summary: 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.
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