 
Summary: SIGNATURES OF HERMITIAN FORMS
VINCENT ASTIER AND THOMAS UNGER
ABSTRACT. Signatures of quadratic forms have been generalized to hermitian forms
over algebras with involution. In the literature this is done via Morita theory, which
causes sign ambiguities in certain cases. The main result of this paper consists of a
method for resolving this problem, using properties of the underlying algebra with
involution.
1. INTRODUCTION
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
