ORTHOGONAL INVOLUTIONS ON CENTRAL SIMPLE ALGEBRAS AND FUNCTION FIELDS OF SEVERI-BRAUER VARIETIES - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Advances in Mathematics Année : 2018

ORTHOGONAL INVOLUTIONS ON CENTRAL SIMPLE ALGEBRAS AND FUNCTION FIELDS OF SEVERI-BRAUER VARIETIES

Résumé

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 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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hal-03815205 , version 1 (14-10-2022)

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  • HAL Id : hal-03815205 , version 1

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Anne Quéguiner-Mathieu, Jean-Pierre Tignol. ORTHOGONAL INVOLUTIONS ON CENTRAL SIMPLE ALGEBRAS AND FUNCTION FIELDS OF SEVERI-BRAUER VARIETIES. Advances in Mathematics, 2018. ⟨hal-03815205⟩
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