The Discriminant of a Decomposable Symplectic Involution

A formula is given for the discriminant of the tensor product of the canonical involution on a quaternion algebra and an orthogonal involution on a central simple algebra of degree divisible by 4. As an application, an alternative proof of Shapiro’s “Pfister Factor Conjecture” is given for tensor products of at most five quaternion algebras. Throughout this paper, the characteristic of the base field F is supposed to be different from 2. Recall from [4, (2.5)] that a symplectic (resp. orthogonal) involution on a central simple F -algebra A is a map A → A which after scalar extension to a splitting field K can be identified with the adjoint involution of a nonsingular alternating (resp. symmetric) bilinear form over a K-vector space. The discriminant of an orthogonal involution ρ on a central simple F -algebra of even degree was defined by Jacobson, Tits, and Knus–Parimala–Sridharan, see [4, §7]. It is an element of the Galois cohomology group H(F, μ2) ≃ F /F which we denote by disc ρ. For symplectic involutions σ, σ0 on a central simple F -algebra A of degree divisible by 4, a relative discriminant ∆σ0(σ) ∈ H (F, μ2) related to the Rost invariant of symplectic groups is defined in [3]. (The definition is recalled at the beginning of Section 1.) If the Schur index indA divides 1 2 deg A, then A carries a hyperbolic involution σ0, and we write simply ∆(σ) for ∆σ0(σ). Our main result relates the discriminants of symplectic and orthogonal involutions as follows: Main Theorem. Suppose B is a central simple F -algebra of degree divisible by 4 with orthogonal involution ρ and Q is a quaternion F -algebra with canonical (symplectic) involution γ. Let A = B⊗F Q and σ = ρ⊗γ, a symplectic involution on A. If indA divides 1 2 deg A, then, denoting by [Q] ∈ H(F, μ2) the cohomology class associated with Q, ∆(σ) = (disc ρ) ∪ [Q]. In the particular case where A is a tensor product of three quaternion algebras and σ is the tensor product of the canonical involutions, it follows that ∆(σ) = 0. We then use a theorem of Berhuy–Monsurrò–Tignol to derive a decomposition of σ where one of the factors is an orthogonal involution on a split algebra of degree 2, see Section 1. Research leading to this paper was initiated while the first author was visiting the Université catholique de Louvain with a grant from the “Secrétariat à la coopération internationale,” whose support is gratefully acknowledged. The second author is partially supported by the National Fund for Scientific Research (Belgium).