A local-global principle for algebras with involution and Hermitian forms. (Q1407782)

Let \(A\) be a (finite-dimensional) central simple algebra over a field \(K\) of characteristic different from~\(2\), and let \(D\) be a central division \(K\)-algebra which is Brauer-equivalent to \(A\). Every involution (i.e., involutive anti-automorphism) on \(A\) is adjoint to a Hermitian or skew-Hermitian form on a \(D\)-vector space. An involution \(\sigma\) is called hyperbolic if it is adjoint to a hyperbolic form. It is called weakly hyperbolic if for some \(n\) the involution on the matrix algebra \(M_n(A)\) defined by \((a_{ij})\mapsto(\sigma(a_{ij}))^t\) is hyperbolic. The main result of the paper characterizes weakly hyperbolic involutions in terms of the associated trace form \(T_\sigma\) on \(A\), defined by \(T_\sigma(x)=T(\sigma(x)x)\), where \(T\) is the reduced trace. The authors show that \(\sigma\) is weakly hyperbolic if and only if \(T_\sigma\) has signature \(0\) at each ordering of the subfield \(k\subset K\) of \(\sigma\)-invariant elements. As a consequence, they obtain a similar result for Hermitian forms on \(A\) with respect to any involution \(\sigma\): a Hermitian form defines a torsion element in the Witt group \(W(A,\sigma)\) if and only if its signature is \(0\) at each ordering of \(k\). In the case where \(A=K\) and \(\sigma\) is the identity, this is Pfister's ``local-global principle'' for quadratic forms.