Involutions of degree at most 4 and function fields of a quadric in characteristic 2. (Involutions en degré au plus 4 et corps des fonctions d’une quadrique en caractéristique 2.) (French) Zbl 1120.16018

Laghribi, Ahmed

Bull. Belg. Math. Soc. - Simon Stevin 12, No. 2, 161-174 (2005).

Let \(A\) be a central simple algebra over a field \(F\) of characteristic \(2\) and let \(\sigma\) be an involution of the first kind on \(A\), i.e., \(\sigma\) is the identity on \(F\). Let \(\text{Alt}(A,\sigma)=\{x-\sigma(x)\mid x\in A\}\) and \(\text{Sym}(A,\sigma)=\{x\in A\mid x=\sigma(x)\}\). \(\sigma\) is said to be symplectic if it is adjoint to an alternating bilinear form, otherwise it is called orthogonal. A quadratic pair \((\sigma,f)\) consists of a symplectic involution \(\sigma\) and an \(F\)-linear map \(f\colon\text{Sym}(A,\sigma)\to F\) such that restricted to \(\text{Alt}(A,\sigma)\), \(f\) acts like the reduced trace. \(\sigma\) (resp. \((\sigma,f)\)) is said to be isotropic if there exists a nonzero right ideal \(I\subset A\) such that \(\sigma(I)I=0\) (resp. \(\sigma(I)I=0\) and \(f(I\cap\text{Sym}(A,\sigma))=0\)), anisotropic otherwise. \(\sigma\) (resp. \((\sigma,f)\)) is said to be hyperbolic (where we assume \(A\) to be of even degree in the situation of a quadratic pair) if there exists an idempotent \(e\in A\) with \(\sigma(e)=1-e\) (resp. \(f(s)=\text{Trd}_A(es)\) for all \(s\in\text{Sym}(A,\sigma)\)).
A natural problem is to find criteria for an anisotropic involution (or quadratic pair) to become isotropic or hyperbolic over a given field extension \(K\) of \(F\). Of particular interest is the case where \(K\) is the function field of a quadric. This problem has been studied in characteristic \(\neq 2\) by the author [Indag. Math., New Ser. 12, No. 3, 337-351 (2001; Zbl 1005.16018)] for algebras of degree at most \(8\), and by Dejaiffe for function fields of conics.
In the present paper, the author treats the case of characteristic \(2\) and with the above notations, he gives a complete characterization of those quadratic forms \(\varphi\) for which an anisotropic symplectic involution \(\sigma\) or an anisotropic quadratic pair \((\sigma,f)\) over \(F\) becomes hyperbolic over \(F(\varphi)\) in the cases where the degree of \(A\) is at most \(4\). He also gives a full answer to the isotropy question for anisotropic orthogonal involutions of quaternion algebras under such function field extensions, and to the hyperbolicity question for anisotropic symplectic involutions and for quadratic pairs of algebras of arbitrary degree over inseparable quadratic extensions, thus completing results by M. A. Elomary and J.-P. Tignol who considered the case of separable quadratic extensions [J. Algebra 240, No. 1, 366-392 (2001; Zbl 0986.11026)].

Reviewer: Detlev Hoffmann (Nottingham)

MSC:

16K20	Finite-dimensional division rings
11E04	Quadratic forms over general fields
11E81	Algebraic theory of quadratic forms; Witt groups and rings
16W10	Rings with involution; Lie, Jordan and other nonassociative structures

Keywords:

quadratic forms; function fields of quadrics; central simple algebras; algebras with involution; symplectic involutions; alternating bilinear forms; quadratic pairs; reduced traces; anisotropic involutions; anisotropic orthogonal involutions; quaternion algebras; function field extensions

Citations:

Zbl 1005.16018; Zbl 0986.11026