Pfister’s theorem for orthogonal involutions of degree 12

Abstract

Abstract. We use the fact that a projective half-spin representation of Spin 12 has an open orbit to generalize Pfister's result on quadratic forms of dimension 12 in I 3 to orthogonal involutions.In his seminal paper [Pf], Pfister proved strong theorems describing quadratic forms of even dimension ≤ 12 that have trivial discriminant and Clifford invariant, i.e., that are in I 3 . His results have been extended to quadratic forms of dimension 14 in I 3 by Rost, see [R] or [Ga]. One knows also extensions of these theorems where quadratic forms are replaced by central simple algebras with orthogonal involution, except in degree 12. The purpose of this paper is to complete this picture by giving the extension in the degree 12 case. The principle underlying the quadratic forms results and our extension is that a projective half-spin representation of Spin n for even n has an open orbit precisely for n ≤ 14, cf. [R], [I], and [SK].Let us first recall what is already known. We consider quadratic forms q of even dimension (resp. central simple algebras with orthogonal involution (A, σ) of even degree) with trivial discriminant and Clifford invariant. If q has dimension < 8, then q is hyperbolic by the Arason-Pfister Hauptsatz [L, X.5.1]. This also holds in the non split case: if A has degree < 8, then σ is hyperbolic, see e.g. [Ga 08, 1.4] or [Q, 4.4]. If q has dimension 8, then q is similar to a 3-Pfister form [L, X.5.6]; if A has degree 8, then (A, σ) is isomorphic to a tensor product ⊗ 3 i=1 (Q i , σ i ) of quaternion algebras with orthogonal involution [KMRT, 42.11]. If A has degree 10 or 14, then A is necessarily split [Ga 08, 1.5], so there is no interesting generalization of the theorem on quadratic forms. The remaining case is where q has dimension 12, where Pfister proved that q is isomorphic to φ ⊗ ψ for some 1-Pfister φ and 6-dimensional form ψ with trivial discriminant, see [Pf, pp. 123, 124] or [Ga, 17.13]. In Theorem 3.1 below, we prove an analogous statement for (A, σ) in case A has degree 12. We do not use Pfister's theorem on 12-dimensional quadratic forms in our proof, so we obtain his result as a corollary (Corollary 3.2).The paper comes in three sections. First, we study quadratic extensions of algebras with involution, and in particular orthogonal extensions of unitary involutions, as considered in [BP, App. 2], [ET, §3], and [QT, 2.14]. Second, we show how to construct an algebra with orthogonal involution that has trivial discriminant and Clifford invariant from any exponent 2 algebra with unitary involution. Third, we prove that in degree 12, this construction produces every central simple algebra with orthogonal involution that has trivial discriminant and Clifford invariant.In the language of linear algebraic groups, our results are as follows. Our construction takes a group of type 2

Decomposability of orthogonal involutions in degree 12

“…As opposed to this, Pfister's theorem does extend to algebras with orthogonal involutions. This was already known in dimension ≤ 10, and partial results in dimension 12 were discussed in [4] and [10]. The main result of this paper is Theorem 1.3, which is an improved version of these dimension 12 analogues, obtained by using the descent theorem for unitary involutions in degree 6 proven in [11,Th.…”

“…But A is split since A 0 and H 0 are Brauer-equivalent, hence (A, σ) ≃ Ad ψ for some 12-dimensional quadratic form ψ with e 1 (ψ) = e 2 (ψ) = 0. By Pfister's result (see (4)), there is a decomposition ψ ≃ ψ 0 ⊗ β for some 6-dimensional form ψ 0 with e 1 (ψ 0 ) = 0 and some 2-dimensional form β, hence another decomposition of (A, σ) as in Theorem 1.3:…”

“…To see this as a consequence of Corollary 2.1, consider a decomposition D = H ′ ⊗ H into quaternion subalgebras. Let Γ D , Γ H ′ , Γ H , Γ F be the value groups of D, H ′ , H, F for the valuation v, so Γ F = Z 4 and Γ D = ( 1 2 Z) 4 . By [14,Cor.…”

“…The first part of the argument for the converse is borrowed from [4]. More precisely, assume condition (a) holds.…”

Tits p‐indexes of semisimple algebraic groups

“…By Tits's Witt‐type Theorem, it is equivalent to describe the semisimple anisotropic kernel up to isogeny, meaning a group where is a central simple ‐algebra of degree 12 and exponent 2 (in particular for some of degree 4), is an orthogonal involution with trivial discriminant, and the even Clifford algebra is . All such are obtained by the construction in the paper , which takes as inputs a quadratic étale ‐algebra , a central simple ‐algebra of degree 6 and exponent 2, and a unitary involution on . That is, every such produces an and thereby a group of type with semisimple anisotropic kernel of type or with more circled vertices, and every such is obtained in this way.…”