Discriminant of symplectic involutions (Q1011940)

Central simple \(K\)-algebras of degree \(2n\) with a symplectic involution are classified, when \(\text{char}(F) \neq 2\), by the set \(H^1(F,\text{PGSp}_{2n})\). For \(n\) divisible by \(4\), the authors construct the unique invariant \(\Delta \!:\! H^1(-,\text{PGSp}_{2n}) \rightarrow H^3(-,\mu_2)\), such that: (1) \(\Delta(A,\sigma) = 0\) if \(\sigma\) is hyperbolic; and (2) \(\Delta(A,\sigma) - \Delta(A,\sigma')\) equals the relative invariant \((\text{Nrp}(s))_2 \cup [A]\) where \(s\) is a \(\sigma\)-symmetric element such that \(\text{Int}(s) \circ \sigma' = \sigma\). Here \(\text{Nrp}\) is the Pfaffian reduced norm. No such invariant exists if \(n\) is not divisible by \(4\). This new invariant determines decomposability of the involution for algebras of degree \(8\): \(\Delta(A,\sigma) = 0\) iff \((A,\sigma)\) is a tensor product of three quaternion algebras. Two interesting interpretations are given for \(\Delta\) in the case of algebras of degree \(2n = 8\). First, \(\Delta\) is shown to be the composition \(H^1(*,\text{PGSp}_8) \rightarrow H^1(*,E_6) \rightarrow H^3(*,\mathbb{Q}/\mathbb{Z}(2))\), where the first map is induced by the inclusion \(\text{PGSp}_8 \subset E_6\), the second map is the Rost invariant for \(E_6\), and the image indeed lands in \(H^3(*,\mu_2)\). Secondly, for every \(2\)-central symmetric element \(u \in A\), \(\Delta(A,\sigma)\) is the Arason invariant \(e_3(q_u)\), where \(q_u \in I^3(F)\) is the quadratic form defined by \(q_u(x) = \frac{1}{2}\text{tr}_{F[u]/F}(x^2)\) on the \(10\)-dimensional space of zero trace symmetric elements which commute with \(u\).