Multipliers of improper similitudes
Let \(A\) be a central simple algebra of degree \(2n\) over a field \(k\) (of char. not \(2\)), with an involution \(\sigma\) of orthogonal type. An element \(g \in A^{\times}\) is a similitude if \(\mu(g) = \sigma(g)g\) is central. The reduced norm of such an element is \(\pm \mu(g)^n\); the similitude is proper if its reduced norm is \(\mu(g)^n\), improper otherwise. Let \(G(A)\) denote the image of \(\mu\) in \(k^{\times}\) . Elements of \(G(A)\) are called multipliers. The relation of \(G(A)\) to the discriminant \(d\) of \((A,\sigma)\) was studied by \textit{A. S. Merkurjev} and \textit{J.-P. Tignol} in [J. Reine Angew. Math. 461, 13--47 (1995; Zbl 0819.16015)]. In this paper the similitudes are related to the next invariant, namely the Clifford algebra \(C = C(A,\sigma)\). Let \(Z\) denote the center of \(C\), a quadratic étale \(k\)-algebra. Let \(H^i(k)\) denote the \(i\)th Galois cohomology of \(k\) with coefficients in the \(2\)-primary group of roots of unity (on which the absolute Galois group acts by raising the natural action to the \(i-1\) power). In particular \([d] \in H^1(k)\) and \([C] \in H^2(k)\). Let \(H^2(k)/A\) denote the quotient of \(H^2(k)\) by the subgroup \(\left<[A]\right>\), and \(H^3(k)/A\) the quotient of \(H^3(k)\) by \(k^\times \cup [A]\). Merkurjev and Tignol showed that \(\lambda \cup d = 0\) in \(H^2(k)/A\) for any multiplier \(\lambda\). The main results of this paper are as follows: First, when \(A\) is split by \(Z\), there is an element \(\gamma \in H^2(k)\) whose restriction to \(Z\) equals \(C\); then \(\lambda \cup \gamma = 0\) in \(H^3 / A\) for any multiplier \(\lambda\). Next, let \(\lambda\) be a proper multiplier, then \(\lambda = \text{Norm}_{Z/k}(z)\) for some \(z \in Z^\times\), and \(\text{cor} (z \cup [C]) = 0\) in \(H^3(k)/A\) (here \(\text{cor}\) is the corestriction from \(Z\) to \(k\)). An inverse to these results is given when \(A\) has degree at most \(6\): If \(A\) is split by \(Z\), then \(\lambda \in k^{\times}\) is a multiplier iff \(\lambda \cup d = 0\) and \(\lambda \cup \gamma = 0\) as above. If \(A\) is not split by \(Z\), then \(\lambda \in k^\times\) is a multiplier iff \(\lambda = \text{Norm}_{Z/k}(z)\) for \(z\) for which \(\text{cor} (z \cup [C]) = 0\). A complete description of the multipliers is also given when \(k[\sqrt{-1}]\) has cohomological dimension at most \(2\) and \(A\) is split by \(Z\). The proofs are elegant and informative, making use of the theory of quadratic forms and their invariants.
central simple algebras with involutions
scientific article; zbMATH DE number 2095724