A Springer theorem for Hermitian forms. (Q818571)

Let \(F\) be a discrete Henselian field, whose residue field has characteristic different than \(2\), and let \(\pi\) be a uniformizer. A classical theorem of Springer essentially reduces the theory of quadratic forms over \(F\) to that of its residue field, in the sense that a quadratic form \(q = q_1 \perp \pi q_2\) is anisotropic iff \(q_1\) and \(q_2\) are. This paper proves analogous results for \(\varepsilon\)-Hermitian forms over division algebras. Let \((D,\sigma)\) be a central finite dimensional division algebra with involution over a field \(F\), and assume \(F^{\sigma}\) is Henselian. An \(\varepsilon\)-Hermitian form on a vector space \(V\) over \(D\) is a sesquilinear form \(V\times V \rightarrow D\) satisfying \(h(y,x) = \varepsilon \sigma(h(x,y))\). Such forms are always diagonalizable, namely can be brought to the form \(h(x,y) = \sum_{i=1}^{n}\sigma(x_i)a_i y_i\). It is shown that a unit form (namely one in which \(a_1,\dots,a_n\) are units) is isotropic iff its residue form is; two unit forms are isomorphic iff their residues are. For general forms, the decomposition \(h = h_1 \perp \pi h_2\) into unit forms is shown to be unique, with \(h\) anisotropic iff \(h_1\) and \(h_2\) are. These results are phrased in greater generality, for an arbitrary discrete value group. The mechanism developed for Hermitian forms is also used to study involutions on \(\text{End}_D(V)\), and a Springer theorem for these is proved as well.