Classification of Quadratic Forms over Skew Fields of Characteristic 2

Quadratic forms over division algebras over local or global fields of characteristic 2 are classified by an invariant derived from the Clifford algebra construction. Quadratic forms over skew fields were defined by Tits in [14] to investigate twisted forms of orthogonal groups in characteristic 2, and by C.T.C. Wall [16] in a topological context. The purpose of this paper is to obtain a classification of these generalized quadratic forms—which we call simply quadratic forms—over finitedimensional division algebras over local or global fields of characteristic 2 by means of “classical” invariants. When the characteristic of the base field is different from 2, the corresponding classification theorem is due to Bartels [3, Satz 5] over global fields and Tsukamoto [13, p. 363] over local fields. We define in section 2 a relative invariant of quadratic forms which plays the same rôle as the invariant introduced by Bartels in characteristic different from 2. The methods are different, however: our definition is based on Tits’ construction of Clifford algebras, whereas Bartels uses Galois cohomology with coefficients μ2 = {±1}, which is not available in characteristic 2. Another feature of the paper is that we systematically consider quadratic spaces from the viewpoint of their endomorphism algebra. We show in section 1 that every nonsingular quadratic form on a vector space V induces on its endomorphism algebra a quadratic pair, as defined in [10, (5.4)], so that quadratic pairs on EndV correspond bijectively to quadratic forms up to a scalar factor. In section 2, we discuss Arf invariants and Clifford algebras of quadratic forms, and use them to define the analogue of the Bartels invariant mentioned above. Section 3 collects results in the literature about the Witt group of (ordinary) quadratic forms over a field, and its behaviour under scalar extension to a separable quadratic extension. The same theme is discussed for generalized quadratic forms in section 4, where a criterion for a (generalized) quadratic form to become hyperbolic over a separable quadratic extension is given in terms of the adjoint quadratic pair. This result is crucial for the classification theorems of section 5, since the main idea of the proof (as in the work [4] of Bayer–Fluckiger and Parimala, which was the main inspiration for this part) is to reduce the orthogonal case to the unitary case by a quadratic extension. Our main classification theorems are the following: Date: September 26, 2000. The second author was supported in part by the National Fund for Scientific Research (Belgium) and by the TMR research network (ERB FMRX CT-97-0107) on “K-theory and algebraic groups.” 1 2 MOHAMED ABDOU ELOMARY AND JEAN-PIERRE TIGNOL Theorem A. Let F be a local or global field of characteristic 2, and let q, q be nonsingular quadratic forms of the same dimension over a central division F algebra. If q, q have the same Arf invariant and if the relative invariant c(q, q) vanishes, then q and q are isometric. Theorem B. Let F be a local or global field of characteristic 2, and let (σ, f), (σ, f ) be quadratic pairs on a central simple F -algebra A. If the Clifford algebras C(A, σ, f) and C(A, σ, f ) are F -isomorphic, then (σ, f) and (σ, f ) are conjugate. Our techniques can be applied also in characteristic different from 2, to yield similar results (except in section 3: the transfer map tr∗ is not onto if charF 6= 2). Indeed, it would be possible to give an exposition of our results valid in all characteristics; we refrained from this option for the sake of clarity, and because all these results are already known in characteristic different from 2: we refer to [10, (4.2)] for the relation between hermitian forms and their adjoint involution, to [5, Theorem 3.3] for the hyperbolicity criterion, to [4, Theorem 4.4.1] and [3, Satz 5] for the classification theorem for hermitian forms over division algebras and to [11, Proposition 6] for the classification of quadratic pairs. (In characteristic different from 2, a quadratic pair is uniquely determined by its orthogonal involution.) Thus, we assume throughout the paper that the characteristic of the base field F is 2. We let ℘(F ) = {x−x | x ∈ F} and for α ∈ F , β ∈ F we denote by [α, β) the quaternion F -algebra generated by two elements i, j subject to i − i = α, j = β and ji = ij + j. Abusing notations, we also denote by [α, β) the image of this algebra in the Brauer group Br(F ). 1. Quadratic forms and quadratic pairs Throughout this section, we let D be a finite-dimensional central division algebra over a field F of characteristic 2, and let V be a finite-dimensional right vector space over D. We assume D carries an involution θ which is the identity on F and let Sym(D, θ) = {x ∈ D | θ(x) = x}(= {x ∈ D | θ(x) = −x}), Alt(D, θ) = {x− θ(x) | x ∈ D}(= {x+ θ(x) | x ∈ D}). Following [15], [16] (see also [9, Chapter 14]), we call quadratic form on V any pair (ψ, q) where ψ : V × V → D is a hermitian form with respect to θ, and q : V → D/Alt(D, θ) is a map from V to the quotient of the additive group of D by Alt(D, θ) subject to the following conditions: (a) q(x + y)− q(x) − q(y) = ψ(x, y) + Alt(D, θ) for x, y ∈ V ; (b) q(xλ) = θ(λ)q(x)λ for x ∈ V and λ ∈ D. 1.1. Proposition. Let (ψ, q) be a quadratic form on V . 1. The hermitian form ψ is uniquely determined by q through condition (a). 2. For all x ∈ V , ψ(x, x) = q(x) + θ (