Signatures of Hermitian Forms

Signatures of quadratic forms over formally real fields have been generalized in [BP2] to hermitian forms over central simple algebras with involution over such fields. This was achieved by means of an application of Morita theory and a reduction to the quadratic form case. A priori, signatures of hermitian forms can only be defined up to sign, i.e., a canonical definition of signature is not possible in this way. In [BP2] a choice of sign is made in such a way as to make the signature of the form which mediates the Morita equivalence positive. A problem arises when that form actually has signature zero or, equivalently, when the rank one hermitian form represented by the unit element over the algebra with involution has signature zero, for it is not then possible to make a sign choice. In this paper, after introducing the necessary preliminaries (Section 2), we review the definition of signature of hermitian forms and study some of its properties, before proposing a method to address the problem mentioned above (Sections 3 and 4). Our main result (Theorem 4.6) shows that there exists a finite number of rank one hermitian forms over the algebra with involution, having the property that at any ordering of the base field at least one of them has nonzero signature. These rank one forms are used in an algorithm for making a sign choice, resolving the problem formulated above. In Section 5 we show that the resulting total signature map associated to any hermitian form is continuous. Finally, in Section 6 we show, using signatures, that in general there is no obvious connection between torsion in the Witt group of an algebra with involution and sums of hermitian squares in this algebra.