Discriminant of Symplectic Involutions

Abstract

We define an invariant of torsors under adjoint linear algebraic groups of type C n -equivalently, central simple algebras of degree 2n with symplectic involution-for n divisible by 4 that takes values in H 3 (F, 2 ). The invariant is distinct from the few known examples of cohomological invariants of torsors under adjoint groups. We also prove that the invariant detects whether a central simple algebra of degree 8 with symplectic involution can be decomposed as a tensor product of quaternion algebras with involution.