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\title{Signatures of hermitian forms}
\author{Vincent Astier}
\author{Thomas Unger}
\address{School of Mathematical Sciences\\ University College Dublin\\ Belfield\\
Dublin~4\\ Ireland} 
\email{vincent.astier@ucd.ie, thomas.unger@ucd.ie}


\date{4 March 2010}
\thanks{The authors were partially supported by the Science Foundation Ireland 
Research Frontiers Programme (project no. 07/RFP/MATF191).}
\keywords{Central simple algebras,
involutions, quadratic and hermitian forms, signatures, hermitian squares, torsion, Witt groups}
\subjclass[2000]{16K20, 11E39, 13J30}


\begin{document}


\begin{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. 




\end{abstract}


\maketitle

\section{Introduction}

Signatures of quadratic forms over formally real fields have been generalized in \cite{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 \cite{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~\ref{thm2.5})  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. 


\section{Preliminaries} \label{sec2}



\subsection{Central Simple Algebras with Involution} The general reference for this section is  \cite[2.A, 2.C]{BOI}.
Let $F$ be a field of characteristic not two
and let $A$ be a \emph{central simple $F$-algebra},
i.e., $Z(A)=F$ and $A$ has no nontrivial two-sided ideals. We always assume that $\dim_F(A)$ is finite. It can be shown that $\dim_F(A)$ is a square. We call $\deg(A):=\sqrt{\dim_F(A)}$ the \emph{degree} of $A$. Let $m=\deg(A)$.

An \emph{involution} $\s$ on $A$ is an anti-automorphism of $A$ of period two. Throughout the paper we assume that $\s$ is  $F$-linear, 
 i.e., the restriction of $\s$ to $F$ is the identity on $F$. Such involutions are also said to be \emph{of the first kind}.
Let 
\[\Sym(A,\s)=\{a\in A\mid \s(a)=a\} \text{ and } \Skew(A,\s)=\{a\in A\mid \s(a)=-a\}.\] Then $\s$ is either \emph{orthogonal} (or, \emph{of type $+1$}) if $\dim_F\Sym(A,\s)=m(m+1)/2$, or  \emph{symplectic} (or, \emph{of  type $-1$}) if $\dim_F\Sym(A,\s)=m(m-1)/2$.  By the Skolem-Noether theorem, two $F$-linear involutions $\s$ and $\tau$ on $A$ differ by an inner automorphism: 
\[\tau = \Int(u)\circ \s\]
for some $u\in A^\x$ such that $\s(u)=\pm u$. Here $\Int(u)(x):=uxu^{-1}$ for $x\in A$. The in\-vo\-lu\-tions $\tau$ and $\s$ are of the same type if and only if $\s(u)=u$. 

We denote by $\Sym(A,\s)^\x$ and $\Skew(A,\s)^\x$ the sets of invertible elements in  $\Sym(A,\s)$ and $\Skew(A,\s)$, respectively.


\begin{examples}\mbox{}\label{ex2.1}
\begin{enumerate}[(1)]
\item $(F,\id_F)$: the field $F$ is trivially a  central simple $F$-algebra. The identity map $\id_F$ is an orthogonal involution.

\item $(M_n(F),t)$: the algebra of $n\x n$-matrices with entries from $F$ is a  central simple $F$-algebra. The transposition map $t$  is an orthogonal involution.

\item $((a,b)_F, \bbar)$: the quaternion algebra determined by $a,b\in F^\x$ with $F$-basis $\{1,i,j,k\}$ satisfying $i^2=a$, $j^2=b$ and $ij=-ji=k$ is a central simple $F$-algebra. It is a division algebra if and only if the (quadratic) norm form $\<1,-a,-b,ab\>$ is anisotropic over $F$.
Quaternion conjugation $\bbar$, determined by $\bar{i}=-i$, $\bar{j}=-j$, and thus $\bar{k}=-k$ is the unique symplectic involution on $(a,b)_F$. Quaternion conjugation is often denoted by $\gamma$ instead of $\bbar$.

\item $((a,b)_F, \vt)$: the involution $\vt$ defined on the quaternion algebra $(a,b)_F$
by $\vt(i)=-i$, $\vt(j)=j$, $\vt(k)=k$ is orthogonal.
\end{enumerate}
\end{examples}


\subsection{$\ve$-Hermitian Spaces and Forms} 
The general reference for this section is \cite[Chap. I]{Knus}. Treatments of the general and division cases can also be found in \cite{GB} and \cite{Le}, respectively.

Let $A$ be a central simple $F$-algebra, equipped with an $F$-linear involution $\s$. Let $\ve\in \{-1,1\}$. An \emph{$\ve$-hermitian space} over $(A,\s)$ is a pair $(M,h)$, where $M$ is a finitely generated right $A$-module and $h:M\x M\too A$ is a sesquilinear form such that $h(y,x)=\ve \s(h(x,y))$ for all $x,y \in M$.
We call $(M,h)$ a \emph{hermitian space} when $\ve=1$ and a \emph{skew-hermitian space} when $\ve=-1$. Consider the left $A$-module $M^*=\Hom_A(M,A)$ as a right $A$-module via the involution $\s$. The form $h$ induces an $A$-linear map $h^*:M\too M^*, x\mapstoo h(x,\cdot)$. We call $(M,h)$ \emph{nonsingular} if $h^*$ is an isomorphism. 
All spaces occurring in this paper are assumed to be nonsingular.  If it is clear what $M$ is, we simply write $h$ instead of $(M,h)$ and speak of a \emph{form} instead of a space. 


Witt cancellation and Witt decomposition hold for  $\ve$-hermitian spaces $(M,h)$ over 
$(A,\s)$. Furthermore, if $A=D$ is a division algebra (so that $M\cong D^n$ for some integer $n$)  such that $(D,\s,\ve)\not=(F,\id_F, -1)$, then $h$ can be diagonalized: there exist  elements $a_1,\ldots, a_n\in \Sym(D,\s)^\x$ such that
\[h(x,y)=\sum_{i=1}^n \s(x_i) a_i y_i,\ \forall x,y \in D^n.\]
In this case we use the shorthand notation
\[h=\<a_1,\ldots, a_n\>_\s,\]
which resembles the notation used for diagonalized quadratic forms. If $A$ is central simple we can certainly consider diagonal hermitian forms defined on free $A$-modules of finite rank, but some hermitian forms over $(A,\s)$  may not be diagonalizable. 


Let $S_\ve (A,\s)$ denote the commutative monoid of isometry classes of $\ve$-hermitian spaces over $(A,\s)$ under orthogonal sum. In this paper we consider $\ve$-hermitian spaces $(M,h)$ up to isometry, and so identify them with their class in $S_\ve (A,\s)$.
Let $W_\ve(A,\s)$ denote the Witt group (or, more precisely,  the $W(F)$-module)
of Witt classes of $\ve$-hermitian spaces over $(A,\s)$.
When $\ve=1$ we drop the subscript and simply write $S(A,\s)$ and $W(A,\s)$.

\subsection{Adjoint Involutions}\label{adj.invo} The general reference for this section is  \cite[4.A]{BOI}.
Let $A$ be a central simple $F$-algebra, equipped with an $F$-linear involution $\s$. Let $(M,h)$ be an $\ve$-hermitian space over $(A,\s)$. The algebra $\End_A(M)$ is again central simple over $F$ since $M$ is finitely generated \cite[1.10]{BOI}.
The involution $\ad_h$ on $\End_A(M)$, defined by
\[h(x, f(y))=h(\ad_h(f)(x),y),\ \forall x,y \in M, \forall f\in \End_A(M)\]
is called the \emph{adjoint involution} of $h$. The involution $\ad_h$ is $F$-linear and
\[\type(\ad_h)=\ve\, \type(\s).\]
Furthermore, every $F$-linear involution on $\End_A(M)$ is of the form $\ad_h$ for some $\ve$-hermitian form $h$ over $(A,\s)$ and the correspondence between $\ad_h$ and $h$ is unique up to a multiplicative factor  in $F^\x$ in the sense that $\ad_h=\ad_{\lambda h}$ for every $\lambda \in F^\x$. 

By a theorem of Wedderburn there exists an $F$-division algebra $D$ (unique up to isomorphism) and a  finite-dimensional
right $D$-vector space $V$ such that $A\cong \End_D(V)$.
Thus $A\cong M_m(D)$   for some positive integer $m$. Furthermore, if there is an $F$-linear involution $\s$ on $A$, then there is an $F$-linear involution $\bbar$ on $D$ and an $\ve_0$-hermitian   form $\vf_0$ over $(D,\bbar)$  with $\ve_0\in \{-1,1\}$
such that $(A,\s)$ and $(\End_D(V), \ad_{\vf_0})$ are isomorphic as algebras with involution. In matrix form
$\ad_{\vf_0}$ is described as follows:
\[\ad_{\vf_0}(X)=\Phi_0\bar{X}^t\Phi_0^{-1},\ \forall X\in M_m(D),\]
where $\Phi_0\in \GL_m(D)$ is the Gram matrix of $\vf_0$. Thus $\bar{\Phi_0}^t=\ve_0 \Phi_0$. 



\subsection{Hermitian Morita Theory}
We refer to \cite[\S1]{BP1}, \cite{FM},  \cite[Chap. 2--3]{GB}, \cite[Chap.~I, \S9]{Knus},  or \cite{Le1} for more details. Let $(M,h)$ be an $\ve$-hermitian space over $(A,\s)$.
One can show that the algebras with involution $(\End_A(M),\ad_h)$ and $(A, \s)$ are Morita equivalent:  for every $\mu\in \{-1,1\}$
there is an equivalence between the categories
$\Herm_\mu(\End_A(M),\ad_h)$ and $\Herm_{\ve \mu}(A,\s)$ of non-singular $\mu$-hermitian forms over $(\End_A(M),\ad_h)$   and non-singular $\ve \mu$-hermitian forms over $(A,\s)$, respectively (where the morphisms are given by isometry), cf. \cite[Thm. 9.3.5]{Knus}. This equivalence  respects isometries, orthogonal sums and hyperbolic forms. It
 induces isomorphisms
\[S_\mu (\End_A(M), \ad_h) \cong S_{\ve\mu}(A,\s) \text{ and } W_\mu (\End_A(M), \ad_h) \cong  W_{\ve\mu}(A,\s) \]
of commutative monoids and $W(F)$-modules, respectively. 
The
Morita equivalence and the isomorphisms are 
not canonical.  One of the reasons is that
$\ad_{h}=\ad_{\lambda h}$ for any $\lambda\in F^\x$, as observed above.

The algebras with involution $(A,\s)$ and $(D, \bbar)$ are also Morita equivalent. 
For future use, it will be convenient to decompose this 
Morita equivalence  into three non-canonical
equivalences of categories, the last two of which we will call \emph{scaling} and \emph{collapsing}. For computational purposes later on, we describe them in matrix form. We follow the approach of \cite{LU2}:
\[\Herm_{\ve} (A,\s)\xrightarrow{\phantom{xxx}}
\Herm_{\ve} (M_m(D), \ad_{\vf_0}) \xrightarrow{\text{ scaling  }} 
\Herm_{\ve_0 \ve} (M_m(D), \bbar^t) \xrightarrow{\text{ collapsing  }}
 \Herm_{\ve_0 \ve} (D, \bbar).
\]

\textbf{Scaling:} Let $(M,h)$ be an $\ve$-hermitian space over $(M_m(D), \ad_{\vf_0})$. Scaling is given by 
\begin{equation}\label{eq.scal}
(M,h)\mapstoo (M, \Phi_0^{-1}h).
\end{equation} 
Note that $\Phi_0^{-1}$ is only determined up to a scalar factor in $F^\x$ since 
$\ad_{\vf_0}=\ad_{\lambda \vf_0}$ for any $\lambda\in F^\x$.

\textbf{Collapsing:} Recall that $M_m(D)\cong \End_D(D^m)$ and
that we always have $M \cong (D^m)^k\cong M_{k,\, m}(D)$ for some integer $k$. Let $h: M\x M\too M_m(D)$ be an $\ve_0 \ve$-hermitian form with respect to $\bbar^t$. Then
\[h(x,y)=\bar{x}^tBy,\ \forall x,y\in M_{k,\, m}(D),\]
where $B\in M_k(D)$ satisfies $\bar{B}^t=\ve_0\ve B$, so that $B$ determines an $\ve_0\ve$-hermitian form $b$ over $(D,\bbar)$. Collapsing is then given by 
\[(M,h)\mapstoo (D^k, b).\]



\section{Signatures of Hermitian Forms} 


In order to introduce signatures of hermitian forms, we deal with a special case first. Let $F$ be a real closed field and let $\HH=(-1,-1)_F$ denote Hamilton's quaternion division algebra over $F$. Let $\bbar$ be quaternion conjugation on $\HH$ and let $h\simeq\<a_1,\ldots, a_n\>_{\bbar}$ be a hermitian form over $(\HH,\bbar)$. Now, $a_1,\ldots, a_n\in \Sym(\HH,\bbar)=F$ and so we can consider the quadratic form $q=\<a_1,\ldots, a_n\>$. We define the \emph{signature of $h$}, denoted $\sign h$, to be the signature of  the quadratic form $q$. Note that this definition is independent of the choice of elements $a_1,\ldots, a_n$ by a theorem of Jacobson \cite{J}.

For the general case we follow the approach of \cite[\S3.3,\S3.4]{BP2}:
let $F$ be a formally real field and let $(A,\s)$ be a central simple $F$-algebra with $F$-linear involution. Let $h$ be a hermitian form over $(A,\s)$. 
Consider an ordering $P\in X_F$, the space of orderings of $F$. In order to define the signature of $h$ at $P$ we do the following:
Extend scalars to the real closure $F_P$ of $F$ at $P$. The extended algebra with involution $(A\ox_F F_P, \s\ox \id_{F_P})$ is then Morita equivalent to an $F_P$-division algebra with $F_P$-linear involution $(D_P, \vt_P)$, where $\s\ox \id_{F_P}$ is adjoint to
an $\ve_P$-hermitian form $\vf_P$ over $(D_P, \vt_P)$ and $\ve_P \in \{-1,1\}$.
By a famous theorem of Frobenius the only division algebras with center the real closed field $F_P$ are $F_P$ itself and $\HH_P$, Hamilton's quaternions over $F_P$. Furthermore, we may choose $\vt_P=\id_{F_P}$ (often simply denoted $\id$) in the first case and $\vt_P=\bbar$ in the second case by Morita theory (scaling).
Thus we may take
\begin{equation}\label{frob}
(D_P,\vt_P)= (F_P, \id_{F_P}) \text{ or } (D_P,\vt_P)= (\HH_P,\bbar).
\end{equation}
The Morita equivalence induces an isomorphism 
\begin{equation}\label{mor.iso}
\M_{P}: S(A\ox_F F_P, \s\ox \id_{F_P}) \simtoo S_{\ve_P}(D_P, \vt_P)
\end{equation}
which is not canonical (for instance, the form $\vf_P$ is only determined up to a nonzero scalar factor).
Since Morita equivalence preserves isometries of hermitian forms, we may define
the signature of $h$ at $P$ to be equal to the signature of  $\M_{P}(h\ox F_P)$. 


When $\ve_P=1$, the form $\M_{P}(h\ox F_P)$ is either quadratic over $F_P$, in which case its signature is obtained in the usual way, or hermitian over $(\HH_P, \bbar)$, in which case the signature is computed as in the special case above.

When $\ve_P=-1$, the forms $\vf_P$ and $\M_{P}(h\ox F_P)$ are both skew-hermitian over $(\HH_P, \bbar)$ or alternating over $F_P$. Since skew-hermitian forms over $(\HH_P, \bbar)$ are always torsion  \cite[Thm.~10.3.7]{Sch} and alternating forms over $F_P$ are always hyperbolic, it makes sense to define $\sign_P \M_{P}(h\ox F_P) =\sign_P \vf_P=0$ in those cases. We  call the orderings $P\in X_F$ for which $\ve_P=-1$ the \emph{$(A,\s)$-nil orderings} of $F$, or simply the \emph{nil orderings} of $F$ if the context is clear. We denote the set of $(A,\s)$-nil orderings of $F$ by $\Nil(A,\s)$.




 A different choice of Morita equivalence between $(A\ox_F F_P,\s\ox \id_{F_P})$ and $(D'_P,\vt'_P)$, say, 
may at most result in a sign change for the signature.  This follows from the computations in \cite[pp. 54--55]{GB}.  (Note that such a sign change may occur, cf. Remark~\ref{rem5.2} below.)
We fix a Morita equivalence for each ordering $P\in X_F$.  



In light of these remarks we now make the following

\begin{defi}  We define the \emph{signature of $h$ at $P$}, denoted $\sign^\nc_P h$, as follows:
\[\sign^\nc_P h:= \begin{cases}
\sign_P \M_{P}(h\ox F_P) & \text{if } \ve_P=1\\
0 & \text{if } \ve_P=-1\\
\end{cases},\]
where the superscript $\nc$ indicates the dependence on the choice of Morita equivalence discussed earlier. 
\end{defi}

\begin{remark} An attempt to make this definition canonical is 
more  problematic than suggested in \cite[\S3.3, \S3.4]{BP2}, see Section~\ref{algo.part}.
\end{remark}

The following table summarizes all the possibilities:
\medskip

\begin{small}
\begin{center}
\begin{tabular}{|c||c|c|}\hline
\backslashbox{$(D_P,\vt_P)$}{$\s$} & orthogonal & symplectic\\ \hline\hline
$(\HH_P,\bbar)$ &
	\begin{tabular}{l}
		$\ve_P=-1$ \\
		$\vf_P$, $\M_{P}(h\ox F_P)$: skew-hermitian\\
		\\
		$\boxed{\sign^\nc_Ph:=0}$\\
		\\
	\end{tabular}
 & 
 	\begin{tabular}{l}
		$\ve_P=1$ \\
		$\vf_P$, $\M_{P}(h\ox F_P)$: hermitian\\
		\\
		$\boxed{\sign_P^\nc h:=\sign_P \M_{P}(h\ox F_P)}$\\
		\\
	\end{tabular}
 \\ \hline
$(F_P,\id_{F_P})$ &
	\begin{tabular}{l}
		$\ve_P=1$ \\
		$\vf_P$, $\M_{P}(h\ox F_P)$: quadratic \phantom{xxxxx}\\
		\\
		$\boxed{\sign_P^\nc h:=\sign_P \M_{P}(h\ox F_P)}$\\
		\\
	\end{tabular}
 & 
 	\begin{tabular}{l}
		$\ve_P=-1$ \\
		$\vf_P$, $\M_{P}(h\ox F_P)$: alternating \phantom{xx}\\
		\\
		$\boxed{\sign_P^\nc h:=0}$\\
		\\
	\end{tabular}
 \\ \hline
\end{tabular}
\end{center}
\end{small}
\medskip





By the properties of Morita equivalence, the signature of a hyperbolic form will be zero and
\[\sign^\nc_P (h_1\perp h_2)=\sign^\nc_P h_1 + \sign^\nc_P h_2\]
for all hermitian forms $h_1,
h_2$ over $(A,\sigma)$.
Thus $\sign^\nc_P$ induces a homomorphism of additive groups $W(A,\s)\too \Z$ for each $P\in X_F$.


\begin{remark}\label{rem5.2} Let $h$ be a hermitian form over $(A,\s)$, let $P \in X_F$ and let $\lambda_P \in F_P^\x$. If we replace $\vf_P$ by $\lambda_P \vf_P$ in the computation of $\sign^\nc_P h$ above, the final result is multiplied by the sign of $\lambda_P$.
 This follows  from considering the scaling part of Morita equivalence, cf. \eqref{eq.scal}.\end{remark}


Let $h$ be a hermitian form over $(A,\s)$ 
and let $P\in X_F\sm \Nil(A,\s)$ (so that $\ve_P=1$).  Let $\mathscr{B}$ be an $F$-basis of $A$.
The isomorphism $\M_{P}$ in \eqref{mor.iso} can be decomposed into three isomorphisms as follows:
\begin{equation}\label{seq5}
\begin{split}
\xymatrix@R=5pt{
S(A\ox_F F_P, \s\ox \id_{F_P})\ar[r]^--{\xi_P^*}& S(M_m(D_P), \ad_{\vf_P}) \ar[r]^--{\text{scaling}} &S(M_m(D_P), {\vt_P}^t)
\ar[r]^--{\text{collapsing}} & S(D_P, \vt_P)\\
h\ox F_P\ar@{|->}[r] & \xi_P^*(h\ox F_P)\ar@{|->}[r] & \Phi_P^{-1} \xi_P^*(h\ox F_P) \ar@{|->}[r] & \M_{P}(h\ox F_P)
}
\end{split}
\end{equation}
Here $\xi_P^*$ is the commutative monoid isomorphism  induced by the isomorphism
\[\xi_P: (A\ox_F F_P, \s\ox\id_{F_P})\simtoo (M_m(D_P), \ad_{\vf_P})\] 
discussed in the context of Wedderburn's theorem in \S\ref{adj.invo}. The scaling matrix $\Phi_P$ is the matrix of the form $\vf_P$ with respect to the basis $\xi_P(\mathscr{B})$ of $M_m(D_P)$.




\begin{example}\label{ex5.4}
We describe how to compute the signature of a diagonal hermitian form $h$ over $(A,\s)$ at an ordering $P\in X_F$. Assume that $h=\<a_1,\ldots, a_n\>_\s$ with respect to some $F$-basis $\mathscr{B}$ of $A$. Note that
$a_1,\ldots, a_n \in \Sym(A,\s)^\x$. By the properties of the signature we have
\[\sign^\nc_P \<a_1,\ldots, a_n\>_\s = \sum_{i=1}^n \sign^\nc_P \<a_i\>_\s,\]
so it suffices to do the computation for a form  $\<a\>_\s$ of rank $1$ (with $a\in \Sym(A,\s)^\x$). 

 If $\ve_P=-1$, the ordering $P$ is $(A,\s)$-nil and $\sign^\nc_P\<a\>_\s=0$ for that ordering. Thus we assume that $\ve_P=1$. 


We push $\<a\>_\s$  through the sequence $\eqref{seq5}$. The first  step gives us:
\[\<a\ox 1\>_{\s\ox\id} \mapstoo  \xi_P^*(\<a\ox 1\>_{\s\ox\id})=    \<\xi_P (a\ox 1)\>_{\ad_{\vf_P}},\]
 where 
 \[\xi_P (a\ox 1) \in \Sym(M_m(D_P), \ad_{\vf_P}).\] 
 
 For the second step, 
 let $\Phi_p$ be the matrix of the form $\vf_P$ with respect to $\xi_P(\mathscr{B})$. It is easy to see that
 \[\Phi_P^{-1}  \<\xi_P (a\ox 1)\>_{\ad_{\vf_P}} = \< \Phi_P^{-1}\xi_P (a\ox 1)\>_{{\vt_P}^t}.
 \]
 
 For the third step, note that
 $\Phi_P^{-1}   \xi_P (a\ox 1) \in \Sym(M_m(D_P), {\vt_P}^t)$.
 Since $(D_P,\vt_P)$ is either $(F_P,\id_{F_P})$ or $(\HH_P,\bbar)$, the matrix $\Phi_P^{-1}   \xi_P (a\ox 1)$ is either symmetric or hermitian and thus corresponds to a quadratic or a hermitian form  $\psi_P$ of dimension $m$ over $(D_P,\vt_P)$.
  We then have
 \[\sign^\nc_P \<a\>_\s:= \sign_P \psi_P.\]
\end{example}

We give a simple illustration of this method (more elaborate examples will be given later in Propositions~\ref{prop6.3} and \ref{prop6.4}):

\begin{example} 
Let $F$ be the Laurent series field $\R(\!(x)\!)$. Then
$X_F=\{P_1, P_2\}$, where $x>_{P_1} 0$ and $x<_{P_2} 0$. Consider the
quaternion algebra $D=(-1,-x)_F$. This is a division algebra over $F$ since
its norm form $\<1,1,x,x\>$ is anisotropic over $F$. 
 Let $A:=M_2(D)$ be equipped with the conjugate transpose involution $\s=\bbar^t$, where $\bbar$ denotes quaternion conjugation. Then $\s$ is a symplectic involution.
Consider the hermitian
form $h=\<\bigl(\begin{smallmatrix} 1 &0\\0&1
\end{smallmatrix}\bigr)\>_{\s}$ over $(A,\s)$.  
Now $A\ox_F F_{P_1} \cong M_2(\HH_{P_1})$ since $x$ is a
square in $F_{P_1}$ and $A\ox_F F_{P_2} \cong M_4(F_{P_2})$ since $-x$ is a
square in $F_{P_2}$. We see that the ordering $P_2$ is $(D,\bbar)$-nil, so
that $\sign_{P_2}^\nc h=0$. Following the steps in Example~\ref{ex5.4} we get
$\sign_{P_1}^\nc h = \pm 2$ since $\s$ becomes adjoint to the hermitian form $\vf_{P_1}=\<1\>_{\bbar^t}$ over $(M_2(\HH_{P_1}), \bbar^t)$ after scalar extension to the real closure of $F$ at $P_1$.
(As observed in Remark~\ref{rem5.2}, only knowing $\vf_{P_1}$ up to sign only gives us the signature up to sign. In Section~\ref{algo.part}
we will explain how a choice of sign can be made.)
\end{example}



\begin{lemma}\label{lem5.5} Let $P \in X_F$ and let $\vf_P$ be as above. Then $\sign^\nc_P \<1\>_\s =\sign_P \vf_P$.
\end{lemma}

\begin{proof} This is trivially true for the $(A,\s)$-nil orderings of $F$. Thus assume that $P\in X_F$ is not nil. We extend scalars to the real closure of $F$ at $P$, $\<1\>_\s \mapstoo \<1\>_\s\ox F_P=
 \<1\ox 1\>_{\s\ox\id}$ and push $ \<1\ox 1\>_{\s\ox\id}$  through the sequence $\eqref{seq5}$, as illustrated in Example~\ref{ex5.4}:
\[\<1\ox 1\>_{\s\ox\id} \mapstoo \xi^*_P (\<1\ox 1\>_{\s\ox\id}) = \<\xi_P(1\ox 1)\>_{\ad_{\vf_P}}
=\<I_m\>_{\ad_{\vf_P}}\mapstoo \Phi_P^{-1}\<I_m\>_{\ad_{\vf_P}}=
\<\Phi_P^{-1}\>_{{\vt_P}^t}.\]
(Note that  $\xi_P (1\ox 1)=I_m$, the $m\x m$-identity matrix in $M_m(D_P)$ since  $\xi_P$ is an algebra homomorphism.) The matrix $\Phi_P^{-1}$ now corresponds to a quadratic form over $F_P$ or a hermitian form over $(\HH_P, \bbar)$. In either case $\Phi_P^{-1}$ is congruent to $\Phi_P$.  Thus $\sign^\nc_P \<1\>_\s =\sign_P \vf_P$. 
\end{proof}



In \cite{LT}, Lewis and Tignol defined the \emph{signature of the involution $\s$ at $P\in X_F$} as follows:
\[\sign_P \s:= \sqrt{\sign_P T_\s},\]
where $T_\s$ is the \emph{involution trace form} of $(A,\s)$ which is a quadratic form over $F$ defined by $T_\s(x):=\Trd_A(\s(x)x)$, for all $x\in A$. Here $\Trd_A$ denotes the reduced trace of $A$.

\begin{examples} \mbox{}
\begin{enumerate}[(1)]
\item Let $(A,\s)=(M_n(F), t)$. Then $T_\s\simeq n^2\x \<1\>$. Hence  $\sign_P\s=n$ for all $P \in X_F$.

\item Let $(A,\s)=((a,b)_F, \bbar)$. Then $T_\s \simeq \<2\>\ox \<1,-a,-b,ab\>$.  Hence
$\sign_P \s=2$ for all $P\in X_F$ such that $a<_P 0, b<_P 0$ and $\sign_P=0$ for all other $P\in X_F$. Note that $N=\<1,-a,-b,ab\>$ is the norm form of $A$.

\end{enumerate}
\end{examples}

\begin{remark}\label{rem3.8}
 Let $(A,\s)$ and $(B,\tau)$ be two central simple $F$-algebras with $F$-linear involution. 
\begin{enumerate}[(1)]
\item Consider the tensor product $(A\ox_F B, \s\ox\tau)$.
Then $T_{\s\ox\tau}=T_\s \ox T_\tau$ and so $\sign_P(\s\ox\tau)=(\sign_P \s)(\sign_P \tau)$ for all $P \in X_F$.

\item If $(A,\s)\cong (B,\tau)$, then $T_\s \simeq T_\tau$ so that $\sign_P\s= \sign_P \tau$ for all $P\in X_F$.
\end{enumerate}

\end{remark}


\begin{remark} \label{rem3.9}
Pfister's local-global principle holds for algebras with involution $(A,\s)$ and also for hermitian forms $h$ over such algebras,  \cite{LU1}:
\[\sign_P \s=0,\ \forall P\in X_F \iff \text{$(A,\s)$ is weakly hyperbolic}\]
(i.e., $\s$ is the adjoint involution of a torsion form)
and
\[\sign^\nc_P h=0,\ \forall P\in X_F \iff \text{the class of $h$ in $W(A,\s)$ is torsion}.\]
\end{remark}



\begin{remark}\label{rem4.6} The map $\sign \s$ is continuous from $X_F$
(equipped with the Harrison topology, see \cite[Chapter VIII 6]{Lam} for a
definition) to $\Z$ (equipped with the discrete topology). Indeed: define the map
$\sqrt{\phantom{x}}$ on $\Z$ by setting $\sqrt{k}=-1$ if $k$ is not a
square in $\Z$. Since $\Z$ is equipped with the discrete topology, this map
is continuous. Since $T_\s$ is a quadratic form, the map $\sign T_\s$ is
continuous from $X_F$ to $\Z$ (by \cite[Proposition 6.6]{Lam}). Thus, by
composition, $\sign \s = \sqrt{\sign T_\s}$ is continuous from $X_F$ to
$\Z$.
\end{remark}



\begin{lemma}\label{lem5.6} 
Let $P\in X_F$. Then 
\[\sign_P \s = \lambda_P\,  |\!\sign_P \vf_P|,\]
where $\lambda_P=1$ if $(D_P,\vt_P)=(F_P,\id_{F_P})$ and $\lambda_P=2$ if $(D_P,\vt_P)=(\HH_P,\bbar)$.
$($If we want to indicate the dependence of $\lambda_P$ on $(A,\s)$ we will write $\lambda_{P, A, \s}$.$)$ In particular, if $P\in \Nil (A,\s)$, then $\sign_P\s = \sign_P \vf_P=0$.
\end{lemma}

\begin{proof}   This is a reformulation of  \cite[11.11]{BOI} or of \cite[Theorem~1]{LT} and part of its proof.
\end{proof}


\begin{lemma}\label{h-adh} Let $(M,h)$ be a hermitian space over $(A,\s)$.  Let $P\in X_F$. 
Then
\[\sign_P \ad_h = \lambda_P\, |\! \sign_P^\nc h|,\]
with $\lambda_P$ as defined in Lemma~\ref{lem5.6}. In particular,
\[\sign_P^\nc h= 0 \Leftrightarrow \sign_P \ad_h=0.\]
\end{lemma}

\begin{proof} Assume first that $P\in \Nil(A,\s)$. Then $\sign_P^\nc h=0$. Consider the adjoint involution $\ad_h$ on $\End_A(M)$.  After extension of scalars to $F_P$ we have Morita equivalences
\[\Herm (\End_A(M)\ox_F F_P, \ad_h \ox \id_{F_P}) \too \Herm(A\ox_F F_P, \s\ox \id_{F_P})\too \Herm_{-1} (D_P,\vt_P)\] 
and $\ad_h \ox \id_{F_P}$ is adjoint to a skew-hermitian form over $(D_P, \vt_P)$.
Note that $\ad_h$ and $\s$ are of the same type since $h$ is hermitian. Thus $\ad_h$ and $\vt_P$ are of opposite type since $P\in \Nil(A,\s)$. Therefore $P\in \Nil(\End_A(M), \ad_h)$. By Lemma~\ref{lem5.6} we conclude that $\sign_P \ad_h=0$.

Next, assume that $P\in X_F\sm \Nil(A,\s)$. Without loss of generality we may replace $F$ by its real closure at $P$.  Consider the Morita equivalence
\[\Herm(A,\s) \too \Herm (D,\vt)\]
with $(D,\vt) = (\HH, \bbar)$ or $(D,\vt)=(F,\id)$. Let $(N,b)$ be the hermitian space over $(D,\vt)$ corresponding to $(M,h)$ under this Morita equivalence.
Then $\sign^\nc h=\sign b$. By \cite[Remark 1.4.2]{BP1} we have $(\End_A(M), \ad_h)\cong (\End_D(N), \ad_b)$ so that $\sign \ad_h=\sign \ad_b$.
By  \cite[Theorem~1]{LT} or \cite[11.11]{BOI} we have $\sign \ad_b=\lambda\, |\!\sign b|$ with $\lambda=1$ if $(D,\vt)=(F,\id)$ and $\lambda=2$ if $(D,\vt) = (\HH, \bbar)$. We conclude that $\sign \ad_h=\lambda\, |\!\sign^\nc h|$.
\end{proof}

\begin{remark}\label{sign0-clopen}
  Since by Remark \ref{rem4.6} the total signature of an involution on $A$
  is continuous, it follows from Lemma~\ref{h-adh} that if $h$ is any hermitian form over
  $(A,\s)$, then the set $\{P \in X_F \mid \sign_P h = 0\}$ is clopen.
\end{remark}


\begin{cor}\label{cor6.13} Let $(M,h)$ be a hermitian space over $(A,\s)$ and let $a\in \Sym(A,\s)^\x$. Consider the hermitian space $(M, ah)$ over $(A, \Int(a)\circ\s)$.  Let $P\in X_F$. Then
\[\sign_P^\nc (ah)=\pm \sign_P^\nc h.\]
\end{cor}

\begin{proof} An easy computation shows that the involutions $\ad_h$ and $\ad_{ah}$ coincide on $\End_A(M)$. Hence they have the same signature at $P\in X_F$. The conclusion now follows from Lemma~\ref{h-adh}.
\end{proof}

 In other words, scaling by an invertible element at most changes the sign of the signature. 
 Scaling by $-1$ gives an instance where a sign change of the signature occurs.
This is contrary to what is claimed in \cite[p. 662]{BP2}.


\begin{lemma}\label{lem2.1} 
 Let $a\in \Sym(A,\s)^\x$. For any $P\in X_F$ we have
\[\sign^\nc_P \<a\>_{\s} = \pm \frac{1}{\lambda_P} \sign_P(\Int(a^{-1})\circ \s ),\]
with $\lambda_P$ as defined in Lemma~\ref{lem5.6}. 
\end{lemma}

\begin{proof}
The involution $(\Int(a^{-1})\circ\s)\ox \id_{F_P}$ on $A\ox_F F_P$ is adjoint to some form  $\vf_P$. We have
\[\sign^\nc_P  \<1\>_{\Int(a^{-1})\circ \s} = \sign_P \vf_P =\pm \frac{1}{\lambda_P} \sign_P \bigl(\Int(a^{-1})\circ \s\bigr),\]
by Lemmas~\ref{lem5.5} and \ref{lem5.6}.
By Corollary~\ref{cor6.13}, we have
\[\sign^\nc_P \<a\>_{\s} = \pm\sign_P \<1\>_{\Int(a^{-1})\circ \s}.\]
The result now follows.
\end{proof}




\begin{lemma} Let $(A,\s)$ and $(B,\tau)$ be central simple $F$-algebras, equipped with $F$-linear involutions.  Let $a\in \Sym(A,\s)^\x$ and $b\in \Sym(B, \tau)^\x$. For any $P\in X_F$ we have
\[\sign_P \<a\ox b\>_{\s\ox\tau} = \pm \mu_P \sign_P\<a\>_\s \sign_P \<b\>_\tau,\]
where $\mu_P=4$  if $A\ox_F F_P$ and $B\ox_F F_P$ are both non-split, and $\mu_P=1$ otherwise.
\end{lemma}


\begin{proof} By Lemma~\ref{lem2.1}  and the fact that the signature of involutions is multiplicative we have
\begin{align*}
\sign_P \<a\ox b\>_{\s\ox\tau} &= \pm \frac{1}{\lambda_{P, A\ox B, \s\ox\tau}} \sign_P\Bigl( \Int\bigl( (a\ox b)^{-1}\bigr)\circ (\s\ox\tau)\Bigr)\\
&= \pm \frac{1}{\lambda_{P, A\ox B, \s\ox\tau}} \sign_P \Bigl( (\Int(a^{-1})\circ \s) \ox (\Int(b^{-1}) \circ\tau)\Bigr)\\
&= \pm \frac{1}{\lambda_{P, A\ox B, \s\ox\tau}} \sign_P  (\Int(a^{-1})\circ \s) \sign_P  (\Int(b^{-1}) \circ\tau)\\
&=\pm \frac{\lambda_{P,A,\s} \lambda_{P,B,\tau}}{\lambda_{P, A\ox B, \s\ox\tau}} \sign_P \<a\>_\s \sign_P \<b\>_\tau.
\end{align*}
Letting $\mu_P = \lambda_{P,A,\s} \lambda_{P,B,\tau} / \lambda_{P, A\ox B, \s\ox\tau}$,
its value can be determined by a case analysis. 
\end{proof}


\section{An Algorithm for choosing the Sign of the Signature}\label{algo.part}

In quadratic form theory the signature of the form $\<1\>$ is always $1$ at any ordering of the ground field. In contrast, the signature of the hermitian form $\<1\>_\s$ over $(A,\s)$ may not even always be positive and could  very well be zero, cf. Lemma~\ref{lem5.5}. 

In order to pursue the analogy with the quadratic forms case, it seems natural to require of the signature map at $P$ from $W(A,\s)$ to $\Z$ that the signature of $\<1\>_\s$ be positive. This is precisely the approach taken in \cite[\S3.3, \S3.4]{BP2}, where the form in $\{\vf_P, -\vf_P\}$ is chosen, whose signature at $P$ is nonnegative, cf. Remark~\eqref{rem5.2}. The effect of this choice is to make the signature of    $\<1\>_\s$  positive by Lemma~\ref{lem5.5}.

However, it is possible that the signature of  $\<1\>_\s$ at $P$ (i.e. $\sign_P \vf_P$) is zero, in which case the approach taken in \cite{BP2}  no longer works and the signature map at $P$ from $W(A,\s)$ to $\Z$ remains defined only up to sign.

In order to fill this gap, our approach consists of replacing the form $\<1\>_\s$ by
a finite number of rank one hermitian forms $\<b_1\>_\s, \ldots, \<b_\ell\>_\s$ over $(A,\s)$, having the property that at any ordering $P\in X_F$ at least one of them has nonzero signature.
We start by proving the existence of  the elements $b_1,\ldots, b_\ell$. As our proof makes use of Merkurjev's theorem \cite{Me} we will deal with the case of multi-quaternion algebras with decomposable involution first.


\begin{remark}\mbox{}\label{rem4.1}
\begin{enumerate}[(1)]
\item Let $(A,\s)$, $(B,\tau)$ and $(C, \upsilon)$ be central simple $F$-algebras with $F$-linear involution such that $(A,\s)\cong (B, \tau)\ox_F (C,\upsilon)$, then 
\begin{equation}\label{type.eq}
\type(\s) = \type (\tau)\cdot \type (\upsilon), 
\end{equation}
cf. \cite[2.23]{BOI}.

\item Assume  that $A$ is a biquaternion algebra with decomposable involution $\s$. If $\s$ is orthogonal, then it is not difficult to see that  there exist quaternion algebras with orthogonal involution $(Q_1, \s_1)$, $(Q_2,\s_2)$ and quaternion algebras with symplectic involution $(Q'_1,\gamma_1)$, $(Q'_2, \gamma_2)$ such that
\[(A,\s) \cong (Q_1, \s_1)\ox_F (Q_2,\s_2) \cong (Q'_1,\gamma_1)\ox_F (Q'_2, \gamma_2),\]
cf. \cite[\S2]{ST}. On the other hand, if $\s$ is decomposable symplectic, then it follows from \eqref{type.eq} that one of the quaternion components  has to be endowed with the canonical (symplectic) involution, and the other with an orthogonal involution. 
\end{enumerate}
\end{remark}

\begin{lemma}  Let $(A,\s)=(Q_1,\s_1)\ox_F \cdots \ox_F (Q_n,\s_n)$  be a multi-quaternion algebra with decomposable $F$-linear involution.  Let $P\in X_F\setminus \Nil(A,\s)$. 
Then the number of indices  $i\in \{1,\ldots, n\}$ such that  $P\in \Nil (Q_i,\s_i)$ is even.
\end{lemma}

\begin{proof} Recall that $P\in \Nil (Q_i,\s_i)$ if and only if $Q_i\ox_F F_P \cong \HH_P$ (resp. $M_2(F_P)$)
in case $\s_i$ is orthogonal (resp. symplectic). The statement now follows from an easy, but tedious,  
case analysis depending on the type of $\s$ and the parity of $n$.
\end{proof}



\begin{prop}\label{thm2.4}
Let $(A,\s)=(Q_1,\s_1)\ox_F \cdots \ox_F (Q_n,\s_n)$  be a multi-quaternion algebra with decomposable $F$-linear involution. There exists a finite subset $S=\{a_1,\ldots, a_\ell\}$ of $\Sym(A,\s)^\x$ such that for every $P\in X_F\setminus \Nil(A,\s)$  there is an index $r\in \{1,\ldots,\ell\}$ such that
\[\sign_P \bigl( \Int(a_r)\circ \s \bigr) \not=0.\] 
\end{prop}

\begin{proof}  We will carry out the proof in three steps.

(1) Assume that $n=1$, so that $A=(a,b)_F$ for certain elements $a,b\in F^\x$. 
For a positive integer $t$ and $a_1,\ldots, a_t \in F^\x$, recall the Harrison set notation 
\[H(a_1,\ldots, a_t):=\{P\in X_F \mid a_1 >_P 0,\ldots, a_t>_P 0\}.\]
Observe that
\[X_F= H(a,b)\dotcup H(a,-b)\dotcup H(-a,-b)\dotcup H(-a,b).\] 

If $\s$ is symplectic, then $X_F\setminus \Nil(A,\s)=H(-a,-b)$ and thus $\sign_P \s =2$ for all $P\in H(-a,-b)$ since $T_\s$ is the norm form of $A$ (up to a factor $\<2\>$). Thus we may take  $S=\{1\}$.

Next assume that $\s$ is orthogonal.  In this case 
\[X_F\setminus \Nil(A,\s)=H(a,b)\dotcup H(a,-b)\dotcup H(-a,b).\]
Consider the orthogonal involution $\vt$, defined by
\[\vt(1)=1,\ \vt(i)=-i,\ \vt(j)=j,\ \vt(k)=k,\]
where $\{1,i,j,k\}$ denotes the usual $F$-basis of $A$. Since any two involutions differ by an inner automorphism, there exists a $q\in\Sym(A,\s)^\x$ such that $\vt = \Int(q)\circ \s$. 
Consider also the involutions $\tau=\Int(j)\circ \vt$ and $\omega=\Int(k)\circ\vt$. After 
computing the involution trace forms of $\vt$, $\tau$ and $\omega$
we see that 
\begin{align*}
\sign_P \vt =2 &\text{ for all } P\in H(-a,b),\\ 
\sign_P \tau =2 &\text{ for all } P\in H(a,b),\\ 
\sign_P \omega =2 &\text{ for all } P\in H(a,-b).
\end{align*}
Thus we may take $S=\{q, jq, kq\}$.
This settles the case $n=1$.

(2) Next assume that $n=2$, so that $A=(a,b)_F \ox_F (c,d)_F$ for certain elements $a,b,c,d \in F^\x$ and $\s=\s_1\ox \s_2$.

\textbf{$\boldsymbol{\s}$  orthogonal $\boldsymbol{(n=2)}$.} We may assume that $\s_1$ is orthogonal on $Q_1=(a,b)_F$ and that $\s_2$ is orthogonal on $Q_2=(c,d)_F$, cf. Remark~\ref{rem4.1}(2). We have $P\in X_F\sm \Nil(A,\s)$ if and only if $A\ox_F F_P \cong M_4(F_P)$. Hence, $P\in X_F\sm \Nil(A,\s)$ if and only if 
\[Q_1\ox_F F_P\cong M_2(F_P)\quad\text{and}\quad Q_2\ox_F F_P\cong M_2(F_P)\]
or
\[Q_1\ox_F F_P\cong \HH_P\quad\text{and}\quad Q_2\ox_F F_P\cong \HH_P.\]
Thus 
\begin{align*}
X_F\sm \Nil(A,\s) &= \Bigl[\Bigl(H(a,b) \dotcup H(a,-b)\dotcup H(-a,b)\Bigr) \cap \Bigl(H(c,d) \dotcup H(c,-d) \dotcup H(-c,d)\Bigr)\Bigr]\\
&\qquad \dotcup\ \Bigl[ H(-a,-b)\cap H(-c,-d)\Bigr]\\
&=\Bigl[\Bigl(X_F\sm \Nil(Q_1, \s_1)\Bigr) \cap \Bigl(X_F\sm \Nil(Q_2, \s_2)\Bigr)\Bigr] \\
&\qquad \dotcup\ \Bigl( \Nil(Q_1,\s_1) \cap \Nil (Q_2,\s_2)\Bigl).
\end{align*}
We first consider $\Bigl(X_F\sm \Nil(Q_1, \s_1)\Bigr) \cap \Bigl(X_F\sm \Nil(Q_2, \s_2)\Bigr)$. By the $n=1$ case there exist involutions $\pi_{i,1},\pi_{i,2},\pi_{i,3}$ on $Q_i$ for $i=1,2$  such that 
\[\pi_{i,k}=\Int(a_{i,k})\circ \s_i\text{ for some } a_{i,k}\in \Sym (Q_i,\s_i)^\x\]
for $k=1,2,3$ and for every  $P\in X_F\sm \Nil(Q_i, \s_i)$,
one of $\sign_P \pi_{i,1}, \sign_P \pi_{i,2}, \sign_P \pi_{i,3}$ is equal to $2$. 
We consider all possible products
\[\varrho_{k,\ell}=\pi_{1,k} \ox \pi_{2,\ell} = \Int(a_{1,k}\ox a_{2,\ell}) \circ (\s_1\ox \s_2)\]
for $k,\ell\in\{1,2,3\}$.
Note that $a_{1,k}\ox a_{2,\ell} \in \Sym(A,\s)^\x$. 
Then for each $P\in \Bigl(X_F\sm \Nil(Q_1, \s_1)\Bigr) \cap \Bigl(X_F\sm \Nil(Q_2, \s_2)\Bigr)$, one of
\[\sign_P \varrho_{k,\ell} = (\sign_P \pi_{1,k})(\sign_P \pi_{2,\ell})\]
is equal to $4$.


Secondly, we consider $\Nil(Q_1,\s_1) \cap \Nil (Q_2,\s_2)$. For $i=1,2$, 
let $\gamma_i$ denote the unique symplectic involution on $Q_i$. Then there exist $a_i \in \Skew(Q_i,\s_i)^\x$ such that $\gamma_i=\Int(a_i)\circ \s_i$. Let 
\[\gamma=\gamma_1\ox \gamma_2=\Int(a_1\ox a_2) \circ (\s_1\ox \s_2).\]
Then $a_1\ox a_2 \in \Sym(A,\s)^\x$. Furthermore,
\[\sign_P \gamma = (\sign_P \gamma_1)(\sign_P \gamma_2) =4\]
for all $P\in \Nil(Q_1,\s_1) \cap \Nil (Q_2,\s_2)$. Thus we may take $S=\{a_{1,k}\ox a_{2,\ell} \mid k,\ell =1,2,3\} \cup \{a_1\ox a_2\}$.

\textbf{$\boldsymbol{\s}$  symplectic $\boldsymbol{(n=2)}$.} We may assume that
 $\s_1$ is orthogonal on $Q_1=(a,b)_F$ and that $\s_2=\gamma_2$ is the unique symplectic involution on $Q_2=(c,d)_F$, cf. Remark~\ref{rem4.1}(2). We have $P\in X_F\sm \Nil(A,\s)$ if and only if $A\ox_F F_P \cong M_2(\HH_P)$. Hence, $P\in X_F\sm \Nil(A,\s)$ if and only if 
\[Q_1\ox_F F_P\cong M_2(F_P)\quad\text{and}\quad Q_2\ox_F F_P\cong \HH_P\]
or
\[Q_1\ox_F F_P\cong \HH_P\quad\text{and}\quad Q_2\ox_F F_P\cong M_2(F_P).\]
Thus 
\begin{align*}
X_F\sm \Nil(A,\s) &= \Bigl[\Bigl(H(a,b) \dotcup H(a,-b)\dotcup H(-a,b)\Bigr) \cap 
H(-c,-d)\Bigr]\\
&\qquad \dotcup\ \Bigl[ H(-a,-b)\cap \Bigl(H(c,d) \dotcup H(c,-d) \dotcup H(-c,d)\Bigr)\Bigr]\\
&=\Bigl[\Bigl(X_F\sm \Nil(Q_1, \s_1)\Bigr) \cap \Bigl(X_F\sm \Nil(Q_2, \gamma_2)\Bigr)\Bigr] \\
&\qquad \dotcup\ \Bigl( \Nil(Q_1,\s_1) \cap \Nil (Q_2,\gamma_2)\Bigl).
\end{align*}
We first consider $\Bigl(X_F\sm \Nil(Q_1, \s_1)\Bigr) \cap \Bigl(X_F\sm \Nil(Q_2, \gamma_2)\Bigr)$. By the $n=1$ case there exist involutions $\pi_{1},\pi_{2},\pi_{3}$ on $Q_1$   such that 
\[\pi_{k}=\Int(a_{k})\circ \s_i\text{ for some } a_{k}\in \Sym (Q_1,\s_1)^\x\]
for $k=1,2,3$ and for every  $P\in X_F\sm \Nil(Q_1, \s_1)$,
one of $\sign_P \pi_{1}, \sign_P \pi_{2}, \sign_P \pi_{3}$ is equal to $2$.  Also, $\sign_P \gamma_2=2$ 
for every  $P\in X_F\sm \Nil(Q_2, \gamma_2)$. For $k=1,2,3$
we consider, as before, all possible products
\[\pi_k \ox \gamma_2 = \Int(a_k \ox 1)\circ (\s_1\ox \gamma_2).\]
Note that $a_k \ox 1 \in \Sym(A,\s)^\x$. 
We have that  for each $P\in \Bigl(X_F\sm \Nil(Q_1, \s_1)\Bigr) \cap \Bigl(X_F\sm \Nil(Q_2, \gamma_2)\Bigr)$, one of
\[\sign_P(\pi_k\ox \gamma_2)=(\sign_P \pi_{k})(\sign_P \gamma_{2})\]
is equal to $4$.

Secondly, we consider $\Nil(Q_1,\s_1) \cap \Nil (Q_2,\gamma_2)$. Let $\gamma_1$ denote the unique symplectic involution on $Q_1$. Then there exists $a_1\in\Skew(Q_1,\s_1)^\x$ such that $\gamma_1=\Int(a_1)\circ \s_1$. Hence $\sign_P\gamma_1=2$ for all $P\in \Nil(Q_1,\s_1)=X_F\sm\Nil(Q_1,\gamma_1)$. 

Let $\tau_2$ be an orthogonal involution on $Q_2$. Note that $\Nil(Q_2,\gamma_2)=X_F\sm \Nil(Q_2, \tau_2)$. There exists an element $a_2\in \Skew(Q_2,\gamma_2)^\x$ such that $\tau_2=\Int(a_2)\circ \gamma_2$. By the case $n=1$, there are $b_1,b_2,b_3 \in \Sym (Q_2,\tau_2)^\x$ such that for every $P\in \Nil(Q_2,\gamma_2)$ one of  $\sign_P \Int(b_i)\circ \tau_2$ is equal to $2$. For $k=1,2,3$, let 
\[\pi_k:=\Int(b_k)\circ \tau_2= \Int(b_k)\circ\Int(a_2)\circ \gamma_2=\Int(b_k a_2)\circ\gamma_2
\]
and note that $b_ka_2 \in \Skew (Q_2,\gamma_2)^\x$. We consider all possible products
\[\gamma_1 \ox \pi_k = \Int( a_1\ox b_ka_2)\circ (\s_1\ox \gamma_2).\]
Observe that $a_1\ox b_ka_2 \in \Sym (A,\s)^\x$. We have that for each $P \in \Nil(Q_1,\s_1) \cap \Nil (Q_2,\gamma_2)$, one of
\[\sign_P( \gamma_1\ox \pi_k)=(\sign_P \gamma_1)(\sign_P \pi_k)
\]
is equal to $4$. Thus we may take $S=\{a_k\ox 1 \mid k=1,2,3\} \cup \{ a_1\ox b_ka_2 \mid k=1,2,3\}$.

(3) Assume finally that $n\geq 3$. 

\textbf{$\boldsymbol{\s}$  orthogonal ($\boldsymbol{n\geq 3}$).}  We may assume that $\s_i$ is orthogonal on $Q_i$ for $i=1,\ldots, n$, cf. Remark~\ref{rem4.1}(2). 
We have $X_F\sm\Nil (A,\s) =\{P\in X_F \mid A\ox_F F_P \cong M_{2^n}(F_P)\}$. For $P\in X_F$, let 
\[\delta_P = \bigl\vert \bigl\{i \in \{1,\ldots, n\} \mid Q_i\ox_F F_P \cong \HH_P\bigr\}\bigr\vert.\]
Then $X_F\sm\Nil(A,\s)=\{P\in X_F \mid \delta_P \text{ is even}\}$.
Since $X_F\sm \Nil(A,\s)$ is a finite union of sets of the form $\{P \in X_F \mid \delta_P=2m\}$ for certain $m\in \N$, it suffices to prove the theorem for a fixed $m\in\N$ and  for the set of  orderings $\{P\in X_F \mid \delta_P=2m\}$. The general statement  will then follow by taking the union of the different sets $S$ obtained in this way. Therefore we only consider orderings in $Y=\{P \in X_F \mid \delta_P=2m\}$ for a fixed $m\in \N$.
 After relabeling indices we may assume that $Q_i\ox_F F_P \cong \HH_P$ if and only if $1\leq i \leq 2m$. After regrouping we can thus write
\begin{align*}
(A,\s)&= (Q_1\ox_F Q_2, \s_1\ox \s_2) \ox_F\cdots \ox_F  (Q_{2m-1}\ox_F Q_{2m},\s_{2m-1}\ox \s_{2m}) \\
&\qquad \ox_F (Q_{2m+1}, \s_{2m+1}) \ox_F \cdots \ox_F (Q_n, \s_n).
\end{align*}
Observe now that $P\in Y$ implies that 
$P\in X_F \sm \Nil (Q_{2i+1}\ox_F Q_{2i+2}, \s_{2i+1}\ox \s_{2i+2})$ for $i=0,\ldots, m-1$ and $P\in X_F\sm \Nil(Q_\ell, \s_\ell)$ for $\ell= 2m+1,\ldots, n$.
We now use the cases $n=1$ and $n=2$ and products of involutions to settle this case.

\textbf{$\boldsymbol{\s}$  symplectic $\boldsymbol{(n\geq 3)}$.}  We may assume that $\s_i$ is orthogonal on $Q_i$ for $i=1,\ldots, n-1$ and that $\s_n=\gamma_n$ is symplectic on $Q_n$, cf. Remark~\ref{rem4.1}(2). We have $X_F\sm\Nil (A,\s) =\{P\in X_F \mid A\ox_F F_P \cong M_{2^{n-1}}(\HH_P)\}$. For $P\in X_F$, let 
\[\delta_P = \bigl\vert \bigl\{i \in \{1,\ldots, n\} \mid Q_i\ox_F F_P \cong \HH_P\bigr\}\bigr\vert.\]
Then $X_F\sm\Nil(A,\s)=\{P\in X_F \mid \delta_P \text{ is odd}\}$.
By an argument similar to the one in the previous case, it suffices to successively consider the following two sets of orderings: 

Case a: $\{P\in X_F\sm\Nil(A,\s) \mid Q_n\ox_F F_P\cong \HH_P\}$.

Case b: $\{P\in X_F\sm\Nil(A,\s) \mid Q_n\ox_F F_P\cong M_2(F_P)\}$.


In Case a, after relabeling, we may assume that $Q_i\ox_F F_P \cong \HH_P$ if and only if $i\in \{1,\ldots, 2m\}\cup \{n\}$. After regrouping we may write
\[(A,\s)= (A_1,\tau_1) \ox_F (Q_{2m+1}, \s_{2m+1}) \ox_F \cdots \ox_F (Q_n,\s_n),\]
where  $(A_1,\tau_1)=(Q_1\ox_F Q_2, \s_1\ox \s_2) \ox_F\cdots \ox_F  (Q_{2m-1}\ox_F Q_{2m},\s_{2m-1}\ox \s_{2m})$.
We conclude by using the orthogonal $n=2$ case for $(A_1,\s_1)$ and the case $n=1$ for the other components together with products of involutions.

In Case b,  after relabeling, we may assume that $Q_i\ox_F F_P \cong \HH_P$ if and only if $i\in \{1,\ldots, 2m+1\}$. After regrouping we may write
\[(A,\s)= (A_1,\tau_1) \ox_F (Q_{2m+1}, \s_{2m+1}) \ox_F \cdots \ox_F (Q_n,\s_n),\]
where  $(A_1,\tau_1)=(Q_1\ox_F Q_2, \s_1\ox \s_2) \ox_F\cdots \ox_F  (Q_{2m-1}\ox_F Q_{2m},\s_{2m-1}\ox \s_{2m})$.
We conclude by using the orthogonal $n=2$ case for $(A_1,\s_1)$, the symplectic $n=2$ case for $(Q_{2m+1}\ox_F Q_n, \s_{2m+1}\ox \s_n)$ 
and the orthogonal $n=1$ case for the other components together with products of involutions.
\end{proof}

\begin{prop}\label{prop4.4} 
Let $(A,\s)$ be a central simple $F$-algebra equipped with an involution of the first kind. There exists an integer $k$ and a finite subset $\{b_1,\ldots, b_\ell\}$ of $\Sym(M_k(A),\s\ox t)^\x$ such that
 for every $P\in X_F\setminus \Nil(M_k(A),\s\ox t)$  there is an index $r\in \{1,\ldots,\ell\}$ such that
\[\sign_P \bigl( \Int(b_r)\circ (\s\ox t) \bigr) \not=0.\] 
\end{prop}

\begin{proof}  
Since $\s$ is of the first kind, the exponent of $A$ in the Brauer group of $F$ is at most $2$. Thus, by Merkurjev's theorem \cite{Me}, there exist $k,m\in \N$ such that
$M_k(A)\cong M_m(Q)\cong Q\ox_F M_m(F)$, where $Q=Q_1\ox_F \cdots \ox_F Q_n$ is a multi-quaternion algebra. Extend $\s$ to the involution $\s\ox t$ on $M_k(A)$, where $t$ denotes transposition. 
Then $\s\ox t \cong\Int(u)\circ (\tau\ox t)$ for an involution $\tau$ of the same type as $\s$ on $Q$ and an invertible element $u\in\Sym(Q\ox_F M_m(F), \tau\ox t)$. Without loss of generality we may assume that $\tau=\s_1\ox\cdots \ox\s_n$, where  $\s_i$ is an involution on $Q_i$ for $i=1,\ldots, n$. 



Consider the elements $a_1,\ldots, a_\ell \in Q$ whose existence is asserted by
Proposition~\ref{thm2.4}. For $i=1,\ldots, \ell$ let $b_i$ be the element in $M_k(A)$ which is mapped to $(a_i \ox I_m) u^{-1}$ under the isomorphism $M_k(A)\cong Q\ox_F M_m(F)$, where $I_m$ denotes the identity matrix in $M_m(F)$. Then  each $b_i \in \Sym(M_k(A),\s\ox t)^\x$. 

Let $P\in X_F\sm\Nil(M_k(A),\s\ox t)$. Observe that $\Nil(M_k(A),\s\ox t)=\Nil(Q,\tau)$ since $M_k(A)\cong M_m(Q)$ and $\s\ox t$ and $\tau$ are of the same type. By Proposition~\ref{thm2.4} there exists an index $r\in \{1,\ldots,\ell\}$ such that
\[\sign_P \bigl( \Int(a_r)\circ \tau \bigr) \not=0.\] 
Thus by Remark~\ref{rem3.8}(2) we have
\begin{align*}
\sign_P (\Int(b_r)\circ(\s\ox t)) &= \sign_P \Bigl(\Int\bigl( (a_r\ox I_m)u^{-1}\bigr)\circ \bigl(\Int(u)\circ (\tau\ox t)\bigr) \Bigr) \\
&= \sign_P \bigl(\Int (a_r\ox I_m) \circ \Int(u^{-1}) \circ \Int(u)\circ (\tau\ox t) \bigr) \\
&=\sign_P \bigl(\Int (a_r\ox I_m) \circ (\tau\ox t) \bigr) \\
&=\sign_P \bigl((\Int (a_r) \circ\tau)  \ox t \bigr) \\
&=\sign_P \bigl(\Int (a_r) \circ\tau\bigr)\cdot \sign_P (t)\\
&=m\sign_P \bigl(\Int (a_r) \circ\tau\bigr)\\
&\not= 0,
\end{align*}
which concludes the proof.
\end{proof}

\begin{cor}\label{nil-clopen} The set of $(A,\sigma)$-nil orderings of $F$ is clopen.
\end{cor}

\begin{proof}  We have $\Nil(A,\s)=\Nil(M_k(A), \s\ox t)$ for any $k\in \N$ since $\s$ and $\s\ox t$ are of the same type. By Proposition~\ref{prop4.4}
\[\Nil(M_k(A),\sigma\ox t) = \bigcap_{r=1}^\ell \{P \in X_F \mid \sign_P \bigl(\Int(b_r)\circ (\s\ox t)\bigr) = 0\}, \]
which is clopen by Remark~\ref{rem4.6}. 
\end{proof}




\begin{theorem}\label{thm2.5} 
Let $(A,\s)$ be a central simple $F$-algebra equipped with an involution of the first kind. There exists a finite subset $\{b_1,\ldots, b_\ell\}$ of $\Sym(A,\s)^\x$ such that for every $P\in X_F\setminus \Nil(A,\s)$  there is an index $r\in \{1,\ldots, \ell\}$ such that
\[\sign_P \bigl( \Int(b_r)\circ \s \bigr) \not=0.\] 
\end{theorem}

\begin{proof} Assume first that $A$ is split, i.e. $A\cong M_n(F)$. If $\s$ is symplectic then $\Nil(A,\s)=X_F$ and there is nothing to prove. If $\s$ is orthogonal, then there exists $a\in \Sym(A,\s)^\x$ such that $\s=\Int(a)\circ t$, where $t$ is the transpose involution. It follows that 
\[\sign_P (\Int(a^{-1})\circ \s) = \sign_P t = n\not=0\] 
for all $P\in X_F$.

Secondly assume that $A$ is not split, so that $A\cong M_n(D)$ for some $n\in \N$ and some division algebra $D$. Since $\s$ is an $F$-linear involution on $A$, there exists an $F$-linear involution $\vt$ on $D$. We  first show that for every $P\in X_F\sm \Nil(A,\s)$ there exists a $b_P \in \Sym(A,\s)^\x$ such that $\sign_P (\Int(b_P)\circ \s) \not=0$.

Let $P\in X_F\sm\Nil(A,\s)$. Assume for the sake of contradiction that $\sign_P \omega=0$ for every $F$-linear involution $\omega$ on $A$. Since such involutions are adjoint to
$n$-dimensional hermitian forms over $(D, \vt)$ and this correspondence is
one-to-one (up to a nonzero scalar factor), all hermitian forms of dimension $n$ over $(D,\vt)$ have signature zero at $P$ by Lemma~\ref{h-adh}. Let $d\in \Sym(D,\vt)^\x$ be arbitrary, then the $n$-dimensional hermitian form $n\x \<d\>_\vt$ has signature zero at $P$. This implies $\sign^\nc_P \<d\>_\vt=0$ for all $d\in \Sym(D,\vt)^\x$. Hence all hermitian forms over $(D,\vt)$ have signature zero at $P$. However, by Proposition~\ref{prop4.4} (with $D$ in the role of $A$) there exists $k\in \N$ and an involution $\tau$ on $M_k(D)$ such that $\sign_P \tau \not=0$. But $\tau$ is adjoint to some hermitian form over $(D,\vt)$ which should have zero signature at $P$, a contradiction with  Lemma~\ref{h-adh}. We conclude that there exists a $b_P \in \Sym(A,\s)^\x$ such that $\sign_P (\Int(b_P)\circ \s) \not=0$.

For $a\in \Sym(A,\s)^\x$ define 
\[U(a):= \{P\in X_F \mid \sign_P (\Int(a)\circ \s) \not=0\}.\]
By Remark~\ref{rem4.6} the set $U(a)$ is clopen in $X_F$. By the previous part of the proof we have
\[X_F\sm\Nil(A,\s) = \bigcup_{P\in X_F} U(b_P).\]
Since $X_F\sm\Nil(A,\s)$ is compact by Corollary~\ref{nil-clopen}, there exists $\ell\in \N$ and $b_1,\ldots, b_\ell \in \Sym(A,\s)^\x$ such that 
\[X_F\sm\Nil(A,\s) = \bigcup_{i=1}^\ell U(b_i).\]
\end{proof}


\begin{cor}\label{nil-clopen2} An ordering $P \in X_F$ is $(A,\sigma)$-nil if and only if
      $\sign^\nc_P h = 0$ for every hermitian form $h$ over $(A,\sigma)$ if and only if $\sign^\nc_P \<a\>_\s = 0$ for every $a\in\Sym(A,\sigma)^\x$ .
\end{cor}

\begin{proof}
By Theorem~\ref{thm2.5},
\[\Nil(A,\sigma) = \bigcap_{i=1}^\ell \{P \in X_F \mid \sign_P (\Int(b_i)\circ \s) = 0\}.\] 
The result then follows from  the definition of nil-ordering
and Lemma~\ref{lem2.1}
 (since $\sign_P \bigl( \Int(b_i)\circ \s \bigr)=| \sign^\nc_P \<b_i^{-1}\>_\s|$).
\end{proof}




\subsection*{The Algorithm}
Fix some tuple of elements $(b_1,\ldots, b_\ell)$ with properties as described in Theorem~\ref{thm2.5}. Observe that for each $P\in X_F$ we have 
\begin{equation}\label{alg4}
|\!\sign^\nc_P \<b_i\>_\s| =\frac{1}{\lambda_P}\sign_P(\Int(b_i)\circ\s)
\end{equation}
by Lemma~\ref{lem2.1} since $\<b_i^{-1}\>_\s\simeq \<b_i\>_\s$.
By Theorem~\ref{thm2.5} this implies that for each $P\in X_F\sm\Nil(A,\s)$ at least one of $\sign^\nc_P \<b_1\>_\s, \ldots, \sign^\nc_P \<b_\ell\>_\s$ is nonzero.

Therefore, for each  $P\in X_F\sm\Nil(A,\s)$
 we decide if the signature computation is performed with $\vf_P$ or $-\vf_P$ as follows:
\begin{enumerate}[$(i)$]
\item Let $i$ be the least element in $\{1,\ldots, \ell\}$ such that $\sign^\nc_P \<b_i\>\not=0$.


\item If $\sign^\nc_P \<b_i\>_\s >_P0$, we keep using $\vf_P$ for the signature computation at this ordering. 
If $\sign^\nc_P \<b_i\>_\s <_P0$, we replace $\vf_P$ by $-\vf_P$ in the computation of signatures at $P$ (which then makes  $\sign^\nc_P \<b_i\>_\s >_P0$). 
\end{enumerate}

Note that we may assume $b_1=1$, in which case our algorithm extends the  algorithm in \cite[\S3.3, \S3.4]{BP2}.

This algorithm depends on the choice of the tuple  $(b_1,\ldots, b_\ell)$ in Theorem~\ref{thm2.5}.  
Once such a choice is made for the algebra with involution $(A,\s)$ we can consider properties such as  the continuity of the total signature function $\sign(h): X_F \too \Z$ associated to a hermitian form $h$ over $(A,\s)$.

\begin{notation}
In the remainder of the paper when  writing $\sign_P$ instead of $\sign_P^\nc$ we mean that we use the above algorithm  for some fixed choice of a tuple $(b_1,\ldots, b_\ell)$.
\end{notation}





\section{Continuity of the Total Signature Map of Hermitian Forms}




\begin{lemma}\label{lembluefrog} 
There is a finite partition of $X_F$ into clopens
\[X_F= \Nil(A,\s) \dotcup \bigcupdot_{i=1}^s Z_i,\]
and there are $\alpha_1,\ldots, \alpha_s \in \Sym (A,\s)^\x$ such that $\sign\<\alpha_i\>_\s$ is constant non-zero on $Z_i$.
\end{lemma}

\begin{proof}
Let $b_1,\ldots, b_\ell$ be as in Theorem~\ref{thm2.5} and, for $r=1,\ldots, \ell$, let
\[Y_r:=\{ P\in X_F \mid \sign_P \<b_i\>_\s=0,\ i=1,\ldots, r\}.\]
Observe that each $Y_r$ is clopen since 
\[Y_r= \bigcap_{i=1}^r \{P\in X_F \mid \sign_P \bigl(\Int(b_i)\circ\s\bigr)=0\}.\]
We have
$Y_0:=X_F \supseteq Y_1 \supseteq \cdots \supseteq Y_{\ell-1} \supseteq Y_\ell= \Nil(A,\s)$
and therefore,
\[X_F=(Y_0\sm Y_1) \dotcup (Y_1\sm Y_2) \dotcup \cdots \dotcup (Y_{\ell-1}\sm Y_\ell) \dotcup \Nil(A,\s).\]
Let $r\in \{0,\ldots,\ell-1\}$ and consider $Y_r\sm Y_{r+1}$. By \eqref{alg4} 
the map $\sign \<b_{r+1}\>_\s$ is never $0$ on $Y_r\sm Y_{r+1}$ and only takes a finite number of values $k_1,\ldots, k_m$. 

\textbf{Claim:} There exists a $\lambda \in \{1,2\}$ such that 
\[\sign \<b_{r+1}\>_\s=\frac{1}{\lambda} \sign \bigl( \Int(b_{r+1})\circ \s\bigr)\]
on $Y_r\sm Y_{r+1}$.


\textbf{Proof of claim:} If $\s$ is orthogonal and $P\not\in \Nil(A,\s)$, then $(D_P,\vt_P)\cong (F_P,\id_{F_P})$. By Lemma~\ref{lem5.6} together with the definition of signature of a hermitian form (since $P\in Y_r\sm Y_{r+1}$) we have 
\[\sign_P \<b_{r+1}\>_\s= \sign_P \bigl( \Int(b_{r+1})\circ \s\bigr).\]

If $\s$ is symplectic and $P\not\in \Nil(A,\s)$, then $(D_P,\vt_P)\cong (\HH_P,\bbar)$. By Lemma~\ref{lem5.6} together with the definition of signature of a hermitian form (since $P\in Y_r\sm Y_{r+1}$) we have 
\[\sign \<b_{r+1}\>_\s= \frac{1}{2} \sign \bigl( \Int(b_{r+1})\circ \s\bigr).\]
So we simply take $\lambda=2$ if $\s$ is symplectic and 
$\lambda=1$  if $\s$ is orthogonal.

The claim gives us:
\[ \bigl( \sign \<b_{r+1}\>_\s\bigr)^{-1} (k_i) \cap (Y_r \sm Y_{r+1})=  \Bigl(\sign \bigl( \Int(b_{r+1})\circ \s\bigr)\Bigr)^{-1} (\lambda k_i)\cap (Y_r \sm Y_{r+1}),\]
which is clopen by Remark~\ref{rem4.6}. It follows that $Y_r \sm Y_{r+1}$ is covered by finitely many disjoint clopen sets on which the map $\sign \<b_{r+1}\>_\s$ has constant non-zero value. The result follows since the sets $Y_r \sm Y_{r+1}$ for $r=0,\ldots, \ell-1$
form a partition of $X_F\sm \Nil (A,\s)$.
\end{proof}


\begin{prop} Let $h$ be a hermitian form over $(A,\s)$. The total signature of $h$, 
\[\sign h : X_F \too \Z,\ P\mapstoo \sign_P h\]
is continuous.
\end{prop}

\begin{proof} We use the notation and the conclusion of Lemma~\ref{lembluefrog}. Since $\Nil(A,\s)$ and the sets $Z_i$ are clopen, it suffices to show that $(\sign h)|_{Z_i}$ is continuous for every $i=1,\ldots, s$. 

Let $i\in \{1,\ldots, s\}$ and let $k_i\in \Z\sm\{0\}$ be such that $\sign \<\alpha_i\>_\s=k_i$ on $Z_i$. Let $k\in \Z$. Then
\begin{align*}
\bigl((\sign h)|_{Z_i}\bigr)^{-1} (k) &= \{P \in Z_i \mid \sign_P h = k\}\\
&= \{P \in Z_i \mid k_i \sign_P h = k_i k\}\\
&= \{P \in Z_i \mid k_i \sign_P h = k \sign_P \<\alpha_i\>_\s \}\\
&= \{P \in Z_i \mid  \sign_P  ( k_i \x h \perp k\x \<-\alpha_i\>_\s)=0 \}.
\end{align*}
It follows from Lemma~\ref{h-adh} that
\[\bigl((\sign h)|_{Z_i}\bigr)^{-1} (k) =  \{P \in Z_i \mid  \sign_P \ad_{ k_i \x h \perp k\x \<-\alpha_i\>_\s}=0 \},\]
which is clopen by Remark~\ref{rem4.6}.
\end{proof}


\section{Torsion in Witt Groups and Sums of Hermitian Squares}

Let $F$ be a formally real field. It is well-known that the Witt ring of $F$ is torsion-free if and only if $F$ is pythagorean (i.e., every sum of squares in $F$ is a square in $F$),  see \cite[VIII, Theorem 4.1]{Lam}.  

Now let $(A,\s)$ be a central simple $F$-algebra equipped with an $F$-linear involution. 
A \emph{hermitian square} in $(A,\s)$ is an element of $A$ of the form $\s(x)x$ for some $x\in A$.
We denote the set of hermitian squares in $(A,\s)$ by $(A,\s)^2$ and the set of sums of hermitian squares in $(A,\s)$ by $\Sigma (A,\s)^2$. It is clear that
\[ (A,\s)^2 \subseteq \Sigma (A,\s)^2 \subseteq \Sym(A,\s).\]
We say that $(A,\s)$ is \emph{pythagorean} if $\Sigma (A,\s)^{2}=(A,\s)^{2}$, i.e.,
if every sum of hermitian squares in $(A,\s)$ is a hermitian square in $(A,\s)$.

We denote the torsion subgroup of $W(A,\s)$ by $W_t(A,\s)$. A fundamental result of Pfister is that $W_t(F)$ is $2$-primary. The torsion subgroup $W_t(A,\s)$ is $2$-primary as well, see \cite[Cor. 6.1]{Sch1} or \cite[Thm. 4.1]{Mah}.

In this section we will show that there is in general no obvious relation  between the property `torsion-free Witt group' ($W_t(A,\s)=0$) and the property `pythagorean'. 

An unsurprising exception is the following: 


\begin{prop} \label{prop6.1}
Let $D$ be a quaternion division algebra over a formally real  field $F$, equipped with quaternion conjugation $\bbar$.
Let $N$ be the norm form of $D$. 
\begin{enumerate}[$(1)$]
\item If $F$ is pythagorean, then $(D,\bbar)$ is pythagorean.
\item  $(D,\bbar)$ is pythagorean and $N$ is strongly anisotropic if and only if every weakly isotropic hermitian form over $(D,\bbar)$ is isotropic and is of dimension at least two. 
\item  $(D,\bbar)$ is pythagorean and $N$ is strongly anisotropic if and only if $W_t(D,\bbar)=0$.
\end{enumerate}
$($Note that $N$ is strongly anisotropic if and only if $\ell\x N$ is anisotropic for all $\ell \in\N$ if and only if every sum of nonzero hermitian squares is nonzero.$)$
\end{prop}

\begin{proof} (1) Follows from a computation with the norm form $N$ of $D$.

(2)  Let $h= \<a_1,\ldots, a_n\>_{\bbar}$ be a hermitian form over $(D,\bbar)$. Note that $a_1,\ldots, a_n \in\Sym(D,\bbar)=F$.
Assume that $\ell\x h$ is isotropic for some positive integer $\ell$. Then there exist $\ell$ vectors $(x_{11},\ldots, x_{1n}), \ldots, (x_{\ell 1}, \ldots, x_{\ell n})$ in $D^n$, not all zero, such that
\[a_1 (\bar{x_{11}}x_{11}+\cdots + \bar{x_{\ell 1}}x_{\ell 1}) + \cdots + a_n (\bar{x_{1n}}x_{1n}+\cdots + \bar{x_{\ell n}}x_{\ell n})=0.\]
Thus, by the hypotheses on $(D,\bbar)$ there exist $y_1,\ldots, y_n \in D$, not all zero, such that
\[a_1 \bar{y_1}y_1 + \cdots + a_n \bar{y_n}y_n=0,\]
i.e., $h$ is isotropic.  Note that $n\geq 2$ since $D$ is a division algebra.

Conversely, let $\alpha = \bar{x}x+\bar{y}y$ with $x, y\in D^\x$ and note that $\alpha \in F$. Then the hermitian form  $\<1,1,-\alpha, -\alpha\>_{\bbar}=2\x \<1,-\alpha\>_{\bbar}$ is isotropic. By the assumption the form $ \<1,-\alpha\>_{\bbar}$ is isotropic, so that there exists a $z\in D^\x$ such that $\alpha=\bar{z}z$.  Note that $\alpha\not=0$ since $D$ is a division algebra. Furthermore, the strong anisotropy of $N$ follows at once.



(3) Assume that $(D,\bbar)$ is pythagorean and that $N$ is strongly anisotropic. Let $h$ be a torsion hermitian form over $(D,\bbar)$. Since the torsion in $W(D,\bbar)$ is $2$-primary there exists a minimal positive integer $\ell$ such that $2^\ell \x h$ is hyperbolic.  Let $f$ be an anisotropic hermitian form which is in the Witt class of $h$ in $W(D,\bbar)$.
Then $2^\ell\x f$ is hyperbolic, and thus in particular isotropic, which implies that $f$ is isotropic by (2), a contradiction. 

Conversely, assume that $W_t(D,\bbar)=0$. Let $x_1,\ldots, x_n \in D^\x$ and assume for the sake of contradiction that $N(x_1)+\cdots +N(x_n) = 0$. Let $k$ be an integer such that $2^k\geq n$. Then $2^k\x \<1\>_{\bbar}$ is isotropic, and so the quadratic form $N\ox (2^k\x \<1\>)$ is isotropic and thus hyperbolic since it is a Pfister form. By a theorem of Jacobson \cite{J} the hermitian form $2^k\x \<1\>_{\bbar}$ is hyperbolic. Since $W_t(D,\bbar)=0$ we obtain that  $\<1\>_{\bbar}$ is hyperbolic, which is impossible.

Now let $\alpha = \bar{x}x+\bar{y}y$  with $x, y\in D^\x$ and note that $\alpha \in F^\x$. 
Then the hermitian form $\<1,1,-\alpha, -\alpha\>_{\bbar}$ is isotropic. Hence the quadratic form $N\ox \<1,1,-\alpha, -\alpha\>$ is isotropic, and thus hyperbolic since it is a Pfister form. But this implies that $2\x \<1,-\alpha\>_{\bbar}=\<1,1,-\alpha, -\alpha\>_{\bbar}$ is hyperbolic by Jacobson's theorem. Thus $ \<1,-\alpha\>_{\bbar}$ is hyperbolic by our assumption. Therefore $\alpha$ is a norm.
\end{proof}

\begin{remark} The converse of Proposition~\ref{prop6.1}(1) is not true. For example, $F=\Q$ is not pythagorean, but $((-1,-1)_\Q, \bbar)$ is pythagorean since every sum of four squares in $\Q$ is again a square in $\Q$.
\end{remark}




\begin{prop}
 Let $F$ be a formally real field. Consider Hamilton's quaternion algebra $\HH=(-1,-1)_F$ equipped with the orthogonal involution $\vt$ from Example $\ref{ex2.1}(4)$. 
Then:
\begin{enumerate}[$(1)$]
\item $W(\HH,\vt)=W_t(\HH,\vt)\ne 0$.
\item If $F$ is real closed, then $\Sym(\HH,\vt)=(\HH,\vt)^2$ and
$(\HH,\vt)$ is pythagorean.
\end{enumerate}
\end{prop}

\begin{proof} 
$(1)$ Let $h\in W(\HH,\vt)$. Since $\Nil (\HH,\vt)=X_F$ we have that $\sign_P h=0$ for all $P\in X_F$. Thus $h$ is a torsion form by Pfister's local-global principle (cf. Remark~\ref{rem3.9}). Hence $W_t(\HH,\vt)= W(\HH,\vt)\not = 0$. 

$(2)$ Let $u_0=\alpha_0+\gamma_0 j+\delta_0 k \in \Sym(\HH,\vt)$. For $u=\alpha+\beta i +\gamma j+ \delta k \in \HH$ we have
\[\vt(u)u=(\alpha^2+\beta^2-\gamma^2-\delta^2) +2(\alpha\gamma+\beta\delta)j +2(-\beta\gamma +\alpha\delta)k.\]
We will show that the equation $\vt(u) u=u_0$ has a solution $(\alpha,\beta,\gamma,\delta)\not=(0,0,0,0)$. Let $\gamma=0$. An easy computation shows that 
$\vt(u) u=u_0$ if and only if $2\alpha\delta=\delta_0$, $2\beta\delta=\gamma_0$ and
\[\delta^4+\alpha_0 \delta^2 -\frac{1}{4} (\delta_0^2+\gamma_0^2)=0.\]
The last equation is quadratic in $\delta^2$ with discriminant $\Delta=\alpha_0^2+\delta_0^2+\gamma_0^2$. Since $F$ is real closed (and thus pythagorean), $\Delta$ is a square. Let $\ve$ be the positive square root of $\Delta$. Then
\[\delta^2=\frac{-\alpha_0 \pm \ve}{2}.\]
Since $\Delta \geq \alpha_0^2$, we have $\ve\geq \alpha_0$ and so $(-\alpha_0+\ve)/2\geq 0$. Thus $\sqrt{(-\alpha_0+\ve)/2}$ exists, since $F$ is real closed. 

Finally, $(\HH,\vt)$ is pythagorean since $\Sigma (\HH,\vt)^2 \subseteq \Sym(\HH,\vt)$.
\end{proof}

This proposition shows that already for algebras with involution over a real closed base field,  `pythagorean' does not imply `torsion-free Witt group'. The following two propositions describe  examples which show that `torsion-free Witt group' does not imply  `pythagorean' either.




\begin{prop} \label{prop6.3}
Let $F=\R\br{x}\br{y}\br{z}\br{w}$ be the iterated Laurent series field in the unknowns $x,y,z,w$ over the field of real numbers $\R$. Consider the quaternion algebras $D_1=(x,y)_F$ and $D_2=(z,w)_F$ and the biquaternion algebra $D=D_1\ox_F D_2$. For $\ell=1,2$, let $\{1,i_\ell, j_\ell, k_\ell\}$ be the usual $F$-basis for $D_\ell$ and let $\s_\ell$ be the orthogonal involution on $D_\ell$ that sends $i_\ell$ to $-i_\ell$ and that fixes the other basis elements. Let $\s=\s_1\ox \s_2$ be the resulting orthogonal involution on $D$. Then:
\begin{enumerate}[$(1)$]
\item $D$ is a division algebra.
\item $W_t(D,\s)=0$.
\item $(D,\s)$ is not pythagorean.
\end{enumerate}
\end{prop}

\begin{proof} $(1)$ Let $v$ be the standard $(x,y,z,w)$-adic valuation on
$F$ (see for instance \cite[\S3]{Wad}).  Note that $F$ is Henselian
with respect to $v$. An application of Springer's theorem shows that the Albert form  $\<x,y,-xy, -z,-w,zw\>$ of $D$ is
anisotropic (we obtain six residue forms of dimension $1$ over $\R$, that
are necessarily isotropic). Hence $D$ is a division algebra, cf. \cite[Chap. III, Thm. 4.8]{Lam}.

$(2)$ Since $F$ is Henselian, the valuation $v$ extends uniquely to a
valuation on $D$ (see \cite[Thm. 2]{Mo}), which we also denote by $v$.
We now claim that the residue division algebra $\bar{D}$ is isomorphic to $\R$. The proof of this claim goes as follows:
Since $\ch( \R) = 0$, the division algebras $D_1$ and
$D_2$ are tame (in the sense of \cite[\S6]{j-w}). By \cite[Corollary
6.7]{j-w} we have $\Gamma_D \subseteq \Gamma_{D_1} + \Gamma_{D_2}$ (this
sum takes place in the divisible closure of $\Gamma_F$). We first compute
$\Gamma_{D_1}$. Since $i_1^2 = x$ and $v(x) = (1,0,0,0)$ we have $v(i_1) =
(1/2,0,0,0)$. Similarly $v(j_1) = (0,1/2,0,0)$ and $v(k_1) =
(1/2,1/2,0,0)$. Let $\gamma_1$ 
be the quaternion conjugation on $D_1$.
Since $v$ extends uniquely from $F$ to $D_1$ and $v \circ \gamma_1$ is a
valuation on $D_1$ we have $v(a) = v \circ \gamma_1 (a)$ for every $a \in
D_1$. In particular $v(\gamma_1(a)a) = 2v(a)$. If we write $a=a_0 + a_1i_1
+ a_2j_1 + a_3k_1$ we obtain $\gamma_1(a)a = a_0^2 -x a_1^2 - ya_2^2 +
xya_3^2$. Since the four terms in this sum have different valuation we get
\begin{equation*}
\begin{split}
  v(a) &= \frac{1}{2} \min\{\epsilon_0 0, \epsilon_1 v(x), \epsilon_2 v(y),
      \epsilon_3 v(xy)\} \\
       &= \frac{1}{2} \min\{\epsilon_0 0, \epsilon_1 (1,0,0,0), \epsilon_2
       (0,1,0,0), \epsilon_3 (1,1,0,0)\},
\end{split}       
 \end{equation*}
where $\epsilon_i = 0$ if $a_i=0$, and $1$ otherwise (for $i=0, \ldots, 3$;
this is to account for the presense or absence of $a_i$).

This yields $\Gamma_{D_1} = \frac{1}{2} \Z \times \frac{1}{2} \Z \times \Z
\times \Z$. A similar argument shows that $\Gamma_{D_2} = \Z \times \Z \times
\frac{1}{2} \Z \times \frac{1}{2} \Z$.
Since $\Gamma_D \subseteq \Gamma_{D_1} + \Gamma_{D_2}$ we get $\Gamma_D =
\frac{1}{2} (\Z \times \Z \times \Z \times \Z)$. In particular $[\Gamma_D :
\Gamma_F] = 16$ and by Draxl's
``Ostrowski Theorem'' (see \cite[Equation 1.2]{j-w}), we obtain $[\bar
{D} : \bar F] = 1$, i.e. $\bar{D} = \bar F = \R$. This proves the claim.


Using now that $W_t(\R)=0$ and also $W_{-1}(\R)=0$,
\cite[Theorem 3.7]{Larmour} implies that $W_t(D,\s)=0$.

$(3)$ Consider the sum of two hermitian squares
\begin{align*}
a&=\s(j_1\ox j_2 + 1\ox 1)(j_1\ox j_2 + 1\ox 1) + \s(i_1\ox j_2) (i_1\ox j_2)\\
&=(j_1\ox j_2 + 1\ox 1)^2 -(i_1\ox j_2)^2
\end{align*}
in $(D,\s)$. We will show that $a$ is not a hermitian square in $(D,\s)$ by
means of a signature computation. Let $P\in X_F$ be the ordering for which
$x,y,z,w>_P 0$ and let $F_P$ be the real closure of $F$ at $P$. Then
\[D\ox_F F_P \cong M_4(F_P).\]
If $a$ were a hermitian square, then the hermitian forms $\<a\>_\s$ and $\<1\>_\s$ over $(D,\s)$ would be isometric. We will shortly see, however, that $\sign_P^\nc \<a\>_\s=\pm 4$, while $\sign_P^\nc \<1\>_\s=0$. Thus the forms are not isometric and so $a$ is not a hermitian square.

In order to compute $\sign_P^\nc\<a\>_\s$ we follow the method of Example~\ref{ex5.4}. 
The algebra $D$ is generated by the basic tensors $i_1\ox 1$, $j_1\ox 1$, $1\ox i_2$ and $1\ox j_2$. 
We extend scalars to the real closure of $F$ at $P$, $D\too D\ox_F F_P$, and then apply the splitting isomorphism
\[\xi_P: (D\ox_F F_P, \s\ox\id_{F_P})\simtoo (M_4(F_P), \ad_{\vf_P})\]
induced by the algebra isomorphisms
\begin{align*}
\eta_\ell: D_\ell \ox_F F_P &\simtoo M_2(F_P)\\
i_\ell \ox 1 &\mapstoo \left[\begin{matrix}
1& 0\\
0&-1
\end{matrix}\right]\\
j_\ell \ox 1 &\mapstoo \left[\begin{matrix}
0& 1\\
1&0
\end{matrix}\right],\\
\end{align*}
for $\ell=1,2$. A straightforward computation shows that 
\[\xi_P(a\ox 1)=\left[
\begin{matrix}
1 & 0 & 0 & 2\\
0 & 1 & 2 & 0\\
0 & 2 & 1 & 0\\
2 & 0 & 0 &1
\end{matrix}\right].\]
Since $\s\ox\id_{F_P}$ is an orthogonal involution, $\vf_P$ is a quadratic form over $F_P$.  
Let $\Phi_P$ denote the Gram matrix of $\vf_P$. Since $\xi_P$ is an isomorphism of algebras with involution we have that
\[\xi_P\circ (\s\ox \id_{F_P}) = \ad_{\vf_P}\circ \xi_P,\]
from which it follows (by easy, but tedious computations) that we may take
\[\Phi_P=\pm\left[
\begin{matrix}
0 & 0 & 0 & 1\\
0 & 0 & 1 & 0\\
0 & 1 & 0 & 0\\
1 & 0 & 0 &0
\end{matrix}\right].\]
Since  $\Phi_P =\Phi_P^{-1}$ we have
\[\Phi_P^{-1}\xi_P(a\ox 1)=\pm \left[
\begin{matrix}
2 & 0 & 0 & 1\\
0 & 2 & 1 & 0\\
0 & 1 & 2 & 0\\
1 & 0 & 0 &2
\end{matrix}\right],\]
from which it follows that $\psi_P\simeq \pm \<1,1,1,1\>$. We conclude that $\sign_P^\nc \<a\>_\s=\pm 4$.

On the other hand,  since $\vf_P$ is clearly hyperbolic, it follows from 
Lemma~\ref{lem5.5} that $\sign_P^\nc\<1\>_\s=\sign_P \vf_P=0$.
\end{proof}



\begin{prop}\label{prop6.4} 
We use the same notation as in the previous proposition, except that we let $\s=\s_1\ox \gamma_2$, where $\gamma_2$ denotes quaternion conjugation on $D_2$, so that $\s$ is a symplectic involution on $D$.
 Then:
\begin{enumerate}[$(1)$]
\item $D$ is a division algebra.
\item $W_t(D,\s)=0$.
\item $(D,\s)$ is not pythagorean.
\end{enumerate}
\end{prop}

\begin{proof} (1) \& (2): identical to the proof of Proposition~\ref{prop6.3}(1) \& (2).

(3) The proof is similar to the proof of Proposition~\ref{prop6.3}(3). We explain the main differences.
Consider the sum of three hermitian squares
\begin{align*}
a&=\s(j_1\ox 1 + 1\ox 1)(j_1\ox 1 + 1\ox 1) + 2 \s(i_1\ox i_2) (i_1\ox i_2)\\
&=(j_1\ox 1 + 1\ox 1)^2 +2(i_1\ox i_2)^2
\end{align*}
in $(D,\s)$. Let $P\in X_F$ be the ordering for which $x,y>_P0$ and $z,w<_P0$. Then  
\[D\ox_F F_P=(D_1\ox_F D_2) \ox_F F_P \cong M_2(F_P) \ox_{F_P} \HH_P \cong M_2(\HH_P).\]
Let $\eta_1$ be as before and let $\eta_2$ be the isomorphism $D_2\ox_F F_P\simtoo  \HH_P$ defined by letting $\eta_2(i_2\ox 1)=i$ and $\eta_2(j_2\ox 1)=j$. Let $\xi_P$ be the induced isomorphism 
\[(D\ox_F F_P, \s\ox \id_{F_P} )\simtoo   (M_2(\HH_P), \ad_{\vf_P}).\]
This time the form $\vf_P$ is hermitian over $(\HH_P, \bbar)$ and a computation shows that we may take $\Phi_P=\pm \left[
\begin{matrix}
0 & 1\\
1 & 0 
\end{matrix}\right].$
Another computation shows that $\xi_P(a\ox 1) = \left[
\begin{matrix}
0 & 2\\
2 & 0 
\end{matrix}\right].$ Hence
\[\Phi_P^{-1}\xi_P(a\ox 1)=\pm\left[
\begin{matrix}
2 & 0\\
0 & 2 
\end{matrix}\right]\]
from which it follows that $\sign_P^\nc \<a\>_\s = \pm 2$. Since again $\sign_P^\nc \<1\>_\s=0$ it follows that $a$ cannot be hermitian square.
\end{proof}





%=========================================================
\section*{Acknowledgements}
%=========================================================

The authors thank Karim Becher and David Lewis for several discussions on this and related topics.





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\end{document}
