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%%% %
%%% %David W. Lewis, Claus Scheiderer and Thomas Unger
%%% %
%%% %A Weak Hasse Principle for
%%% %Central Simple Algebras with an Involution
%%% %
%%% %Version of 11/05/2001
%%% %
%%% %
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\begin{document}
%%%%% ------------- fill in your data below this line  -------------------
%%%%%    The following lines \Title ... \EndAddress must ALL be present
%%%%%    and in the given order.
\Title A Weak Hasse Principle for Central Simple
Algebras with an Involution
%%%%%    Put here the title. Line breaks will be recognized.
\ShortTitle Weak Hasse Principle
%%%%%    Running title for odd numbered pages, ONE line, please.
%%%%%    If none is given, \Title will be used instead.
\SubTitle
%%%%%    A possible subtitle goes here.
\Author David W. Lewis, Claus Scheiderer, Thomas Unger
%%%%%    Put here name(s) of authors. Line breaks will be recognized.
\ShortAuthor
%%%%%%   Running title for even numbered pages, ONE line, please.
%%%%%%   If none is given, \Author will be used instead.
\EndTitle
\Abstract
%%%%%    Put here the abstract of your manuscript.
The notions of totally indefinite and weakly
isotropic algebras with involution are introduced and
a proof is given of the fact that  a field satisfies the
Effective Diagonalization  Property (ED) if and only if it satisfies
the following weak Hasse principle: every totally indefinite
central simple algebra with involution of the first kind over the given
field is weakly
isotropic. This generalizes a known result from quadratic form
theory.
\EndAbstract
\MSC
%%%%%    1991 Mathematics Subject Classification:
16K20, 11E39, 12J15
\EndMSC
\KEY
%%%%%    Keywords and Phrases:
Real fields, central simple algebras, involutions, weak
Hasse principles, hermitian squares
\EndKEY
%%%%%    All 4 \Address lines below must be present. To center the last
%%%%%    entry, no empty lines must be between the following \Address
%%%%%    and \EndAddress lines.
\Address
%%%%%    Address of first Author here
David W. Lewis
Department of Mathematics
University College Dublin
Belfield, Dublin 4
Ireland
david.lewis@ucd.ie
\Address Claus Scheiderer
Fachbereich Mathematik
Universit\"at Duisburg
47048 Duisburg
Germany
claus@uni-duisburg.de
%%%%%    Address of second Author here etc.
\Address Thomas Unger
Department of Mathematics
University College Dublin
Belfield, Dublin 4
Ireland
thomas.unger@ucd.ie
\Address
\EndAddress
%%
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%%       before the first \end{document} is the command  \Addresses
%%
%%---------------------Here the prologue ends---------------------------------
%%--------------------Here the manuscript starts------------------------------

%=========================================================
%BEGIN DOCUMENT
%=========================================================

%\title{A Weak Hasse Principle for Central Simple Algebras with an Involution}
%\author{David W. Lewis}
%\author{Claus Scheiderer}
%\author{Thomas Unger}
%\address{Department of Mathematics\\University College Dublin\\ Belfield\\ Dublin 4\\ Ireland}
%\email{david.lewis@ucd.ie}
%\address{Fachbereich Mathematik\\Universit\"at Duisburg\\47048
%Duisburg\\ Germany}
%\email{claus@uni-duisburg.de}
%\address{Department of Mathematics\\University College Dublin\\ Belfield\\ Dublin 4\\ Ireland}
%\curraddr{Fakult\"at f\"ur Mathematik\\Universit\"at
%Regensburg\\93040 Regensburg\\Germany}
%\email{thomas.unger@ucd.ie}
%\date{\today}
%\keywords{Real fields, central simple algebras, involutions, weak
%Hasse principles, hermitian squares} \subjclass{16K20, 11E39, 12J15}



%\begin{abstract} The notions of totally indefinite and weakly
%isotropic algebras with involution are introduced and
%a proof is given of the fact that  a field satisfies the
%Effective Diagonalization  Property (ED) if and only if it satisfies
%the following weak Hasse principle: every totally indefinite
%central simple algebra with involution of the first kind over the given
%field is weakly
%isotropic. This generalizes a known result from quadratic form
%theory.
%\end{abstract}

%\maketitle





%=========================================================
\section{Introduction}
%=========================================================

Let $F$ be a field of characteristic different from two and let
$(A,\s)$ be a central simple algebra over $F$ with involution of
the first kind (i.e. $\s|_F=\1_F$). Recall that $\s$ is
called \emph{orthogonal} (resp. \emph{symplectic}) if $\s$ is
adjoint to a symmetric (resp. skew-symmetric) bilinear form, after
scalar extension to a splitting field of~$A$.

The connection between orthogonal involutions and quadratic forms
has been a motivation for extending quadratic form theoretic
concepts and theorems to the realm of algebras with involution (of
any kind). For example, the classical invariants (discriminant,
Clifford algebra, signature) of quadratic forms have been defined
for algebras with involution (see \cite{BOI}) and classification
theorems \`a la Elman and Lam \cite{MR50:4485} have been obtained
by Lewis and Tignol \cite{MR2000m:16027}. Some more examples
include: a Cassels-Pfister theorem \cite{MR97e:16043}, an
orthogonal sum for Morita-equivalent algebras with involution
\cite{D} and analogues of the Witt ring \cite{Lewsg,MR1765403}.

In this paper we will examine the extension to central simple algebras
with an involution of the first kind of
the following weak Hasse
principle for weak isotropy:
\begin{description}
\item[(WH)] \textsl{Every totally indefinite quadratic form over $F$ is weakly
isotropic}
\end{description}
and prove an analogue of the following theorem due to
Prestel~\cite{Pres} and Elman et al.~\cite{ELP}:
\begin{theorem}\label{WHThm}
$F$ satisfies \textup{(WH)} $\iff$ $F$ satisfies the Strong Approximation
Property.
\end{theorem}
In particular, we will show that every totally indefinite central simple
$F$-algebra with involution of the first kind is weakly isotropic if and only if
$F$ satisfies the Effective Diagonalization Property.
Our result can also be re-interpreted to give a partial generalization
of a theorem of Lewis~\cite{Lsum}
on sums of squares representing zero in a central simple algebra.

We mention that there is a refined (and more difficult) version of
Theorem~\ref{WHThm}, which holds for arbitrary base fields,
due to Br\"ocker \cite[3.9]{Br} and Prestel \cite[p. 93]{P}. It
says that if $\phi$ is a totally indefinite quadratic form over a
field $F$,  and if
for every valuation with real residue class field, at least
one residue class form of $\phi$ is weakly isotropic, then $\phi$
is weakly isotropic.
This
statement can also be found in \cite[3.7.12]{S}. Its
converse is easily seen to be true.




All involutions on central simple algebras considered in this paper are of the
first kind and all forms (quadratic, hermitian, \dots) are
assumed to be nonsingular.
Standard references are \cite{L} and \cite{S} for the theory of
quadratic forms, \cite{BOI} for central simple algebras with an
involution and \cite{P} for real fields.


%=========================================================
\section{Weakly isotropic and totally indefinite algebras}
%=========================================================
In this section we will generalize the notions of totally
indefinite and weakly isotropic quadratic forms to the setting of
central simple algebras $(A,\s)$ with an involution of the first
kind over a field $F$ of characteristic $\not= 2$.
We denote the space of orderings of $F$ by $X_F$ and an
arbitrary ordering of $F$ by $\sP$.


\begin{defi} Let $(A,\s)$ be a central simple $F$-algebra with
involution of the first kind.
A right ideal $I$ in $A$ is called \emph{isotropic} (with respect
to the involution $\s$) if for all $x$ and $y$ in $I$ we have that
$\s(x)y=0$. The algebra with involution $(A,\s)$  is called
\emph{isotropic} if $A$ contains a nonzero isotropic right ideal, or
equivalently, if there exists an idempotent $e\ne0$ in $A$ such that
$\s(e)e=0$ (see {\cite[6.A]{BOI}}). We also say that $(A,\s)$ is
\emph{anisotropic} if for $x\in A$, $\s(x)x=0$ implies $x=0$.
\end{defi}

Recall that a quadratic form $q$ over $F$ is weakly isotropic if
there exists an $n\in\N$ such that $n\x q$ is isotropic.

\begin{defi}\label{rem1}
The algebra with involution $(A,\s)$ is called \emph{weakly
isotropic} if there exist
nonzero $x_1,\ldots, x_n \in A$ such that $\s(x_1)x_1+\cdots
+\s(x_n)x_n=0$ and \emph{strongly anisotropic} otherwise.
\end{defi}

\begin{remark}
In \cite{U} an $n$-fold orthogonal sum $\bplus^n(A,\s)$ is defined
and it is shown there that $\bplus^n(A,\s)\cong (M_n(F),t)\ox_F
(A,\s)$, where $t$ denotes the transpose involution. This is on
the one hand in accordance with Dejaiffe's \cite{D} construction
of an orthogonal sum of two Morita-equivalent algebras with
involution and on the other hand what one would expect since $n\x
q=\<\underbrace{1,\ldots,1}_{n\x}\>\ox q$ and $\bplus^n(A,\s)$
reduces to $n\x q$ in the split case when $\s$ is adjoint to a
quadratic form $q$. In analogy with the quadratic form case, one
could define $(A,\s)$ to be weakly isotropic by requiring that
$\bplus^n(A,\s)$ is isotropic for some positive integer~$n$ and it
is easy to see that this condition is equivalent with the one
given in Definition~\ref{rem1}.
\end{remark}

Let $(D, \vartheta)$ be a central division algebra over $F$ with
involution of the first kind and $(V,h)$ an
$\varepsilon$-hermitian form over $(D, \vartheta)$,
$\varepsilon=\pm 1$. Recall \cite[4.A]{BOI} that the adjoint
involution $\s_h$ of $h$ on $\End_D(V)$ is implicitly defined
by
$$h\bigl(x, f(y)\bigr)=h\bigl(\s_h(f)(x),y\bigr)\quad \text{ for } x,y\in V\text{ and }
f\in \End_D(V)$$
and that $\s_h$ is also of the first kind.

Just as for quadratic forms, we say that the
$\varepsilon$-hermitian form $h$ is weakly isotropic if there
exists a positive integer $n$ such that $n\x h$ is isotropic.


\begin{lemma} Let $(D, \vartheta)$, $(V,h)$ and $\s_h$ be as above.
Then $(\End_D(V), \s_h)$ is weakly isotropic if and only if $h$ is
weakly isotropic. More precisely, there exist $f_1,\ldots, f_n \in
\End_D(V)$ such that $\s_h(f_1)f_1+\cdots + \s_h(f_n)f_n=0$ if and
only if there exist $x_1,\ldots, x_n\in V$ such that
$h(x_1,x_1)+\cdots + h(x_n,x_n)=0$.
\end{lemma}

\begin{proof}
The lemma is folklore, and we only give the argument since
we couldn't find a suitable reference. It suffices to show that
$(\End_D(V), \s_h)$ is isotropic if and only if $h$ is
isotropic.

If $\sigma_h$ is isotropic,
there is $0\ne f\in\End_D(V)$ with $\sigma_h(f)f=0$. Choose $v\in
V$ with $f(v)\ne0$. Then
$$0=h\bigl(\sigma_h(f)(f(v)),\,v\bigr)=h\bigl(f(v),\,f(v)\bigr)$$
shows that $h$ is isotropic. Conversely, if $h(v,v)=0$ for some
$v\in V$, then $\sigma_h(f)f=0$ for any $f\in\End_D(V)$ with $f(V)
\subset vD$.
\end{proof}

\begin{cor} Let $(\End_F(V), \s_q)$ be a split algebra with involution,
adjoint to a quadratic form $q$ on $V$.
Then there exist $f_1,\ldots, f_n \in
\End_F(V)$ such that $\s_q(f_1)f_1+\cdots + \s_q(f_n)f_n=0$ if and
only if there exist $x_1,\ldots, x_n\in V$ such that
$q(x_1)+\cdots + q(x_n)=0$.
%Then
%$(\End_F(V), \s_q)$
%is weakly isotropic if and only if the quadratic form $q$ is
%weakly isotropic.
\label{corsplit}
\end{cor}


Now suppose that $F$ is a real field and that $\sP$ is an ordering of $F$. In
\cite{LT} Lewis and Tignol defined the signature of an algebra $(A,\s)$ with
involution of the first kind as
$$\sig_\sP\s=\sqrt{\sig_\sP T_\s},$$
where
$T_\s$ is the involution trace form, defined by $T_\s(x):=
\Trd_A(\s(x)x), \forall x\in A$. If $(A,\s)$ is split with
orthogonal involution,
$(A,\s)\cong (\End_F(V), \s_q)$, then Lewis and Tignol showed that
$\sig_\sP \s_q=|\sig_\sP q|$.

Recall that a quadratic form $q$ over $F$ is called totally indefinite if
it is indefinite for each ordering $\sP$ of $F$, i.e. $|\sig_\sP
q|<\dim q$ for each $\sP$.

\begin{defi}
The algebra with involution $(A,\s)$ is called \emph{indefinite}
for the ordering $\sP$ of $F$ if $\sig_\sP \s<\deg A$  and
\emph{totally indefinite} if it is indefinite for each ordering
$\sP$ of~$F$.
\end{defi}

%=========================================================
\section{The weak Hasse principle}
%=========================================================

We now have all the ingredients ready to generalize (WH) to:

\begin{description}
\item[(WHA)] \textsl{Every totally indefinite algebra with involution
of the first\\
\phantom{(WHA} kind over $F$ is weakly isotropic.}
\end{description}

In \cite[Ch.~5]{UPhD} Unger showed that (WHA) holds for
fields with a unique ordering, algebraic number fields and $\R(t)$. These
fields are some of the standard examples of SAP fields, as described
below.


\begin{defi}\label{def:SAP}
The field $F$ satisfies the \emph{Strong
Approximation Property} (or is SAP, for short)
if the following equivalent conditions hold:
\begin{itemize}
\item[(i)] Every clopen subset of $X_F$
has the form $\{\sP\in X_F\,|\,a>_\sP0\}$ for some
$a\in \Fd$.
\item[(ii)] For all $a,b\in\Fd$ the quadratic form $\<1,a,b,-ab\>$
is weakly isotropic.
\item[(iii)] Every quadratic form $q$ such that a power of $q$ is weakly
isotropic, is itself weakly isotropic.
\item[(iv)] For any two disjoint closed subsets $X,Y$ of $X_F$, there exists
  an $a\in \Fd$ such that $a>_\sP0, \forall \sP\in X$ and $a<_\sP 0, \forall
  \sP\in Y$.
\item[(v)] For every  (Krull) valuation $v: \Fd \twoheadrightarrow \Gamma$
with value group $\Gamma$ and real residue class field $\bar F_v$,
either (a) or (b) holds:
\begin{itemize}
\item[(a)] $\Gamma=2\Gamma$;
\item[(b)] $|\Gamma/2\Gamma|=2$ and $\bar F_v$ has a unique
ordering.
\end{itemize}
\end{itemize}
\end{defi}

Condition (iv) is the original definition of SAP fields, due to Knebusch
et al. \cite[Thm. 12]{KRW}. The equivalence (i) $\iff$ (iv) is given in
\cite[Thm. 12, Cor. 13]{KRW}. Prestel \cite[(2.2), (3.1)]{Pres} showed
(ii) $\iff$ (iii) $\iff$ (v) $\iff$ $F$ is a Pasch field, while the equivalence
$F$ is SAP $\iff$ $F$ is Pasch can be found in \cite[Thm. C]{ELP}. The notion
of a Pasch field was first introduced by Prestel; for a definition we refer
the reader to
\cite{Pres}. Additional references for SAP fields are the monographs by Lam~\cite{Lam2}
and Prestel~\cite{P}.

\begin{example}\label{SAPexamp}
Here are some examples of SAP fields:
\begin{itemize}
\item[(1)] Fields with only one ordering.
\item[(2)] Algebraic number fields.
\item[(3)] Fields of transcendence degree $\leq 1$ over a real-closed field,
  e.g. $\R(t)$.
\item[(4)] $F((t))$ if $F$ has at most one ordering.
\end{itemize}
The following fields are not SAP:
\begin{itemize}
\item[(5)] The rational function field $\Q(x)$.
\item[(6)] The rational function field $F(x,y)$, where $F$ is any
  real field.
\end{itemize}
\end{example}

Based on the results in \cite[Ch.~5]{UPhD} it was tempting to
think that (WHA) would hold for all SAP fields. The Strong
Approximation Property is definitely required, for if $F$ is not
SAP, we can construct a counterexample as follows:
There exist $a,b\in \Fd$ such that $q:=\<1,a,b,-ab\>$ is
strongly anisotropic. Hence the algebra $(A,\s)=(\End_F(F^4),
\s_q)$ is strongly anisotropic by Corollary~\ref{corsplit}.
However, the form $T_\s=q\ox q$ is equal to
\begin{eqnarray*}
q\ox q &=& q\perp aq\perp bq\perp -abq\\
&=& \<1,a,b,-ab,a,1,ab,-b,b,ab,1,-a,-ab,-b,-a,1\>\\
&=& 6\x \<1,-1\> \perp \<1,1,1,1\>,
\end{eqnarray*}
so $T_\s$ is in fact isotropic and hence totally indefinite.
Therefore the orthogonal involution $\s$ is totally indefinite.

For quadratic forms, this argument was of course already known in
the 1970's, as testified by Theorem~\ref{WHThm}. We merely
presented it from the point of view of algebras with
involution.

A counterexample in the symplectic case can be constructed by
tensoring the previous algebra with the quaternion division
algebra $(-1,-1)_F$ equipped with the canonical (symplectic)
involution, which is strongly anisotropic.

As it turns out, a property stronger than SAP is needed, the
Effective Diagonalization Property, first defined
by Ware \cite{W}, which we will describe now.

\begin{defi}
A quadratic form $\<a_1,\ldots, a_n\>$ is \emph{effectively diagonalizable}
if it is isometric to a form $\<b_1,\ldots,b_n\>$ satisfying
$b_i\in\sP \implies b_{i+1}\in\sP$ for all $1\leq i <n$ and all
$\sP\in X_F$. The field $F$ satisfies the \emph{Effective
Diagonalization Property} (or is ED, for short)
if every quadratic form over $F$ is effectively
diagonalizable.
\end{defi}
The class of ED fields is a proper subclass of the class of SAP
fields.

\begin{example} The field $\Q((t))$ is SAP, but not ED.
\end{example}

Prestel and Ware \cite{PW} proved the following
characterization theorem:
\begin{theorem}\label{EDcrit}
$F$ is \textup{ED} if and only if for every
(Krull) valuation $v:\Fd\twoheadrightarrow \Gamma$ with value
group $\Gamma$ and real residue class field $\bar F_v$,
we have $|\Gamma/2\Gamma|\leq 2$ and $\bar
F_v$ is euclidean in case $|\Gamma/2\Gamma|= 2$.
\end{theorem}
(Recall that a field is \emph{euclidean} if it is uniquely ordered and
every positive element is a square.)
They also showed:
\begin{theorem}
If $F$ is \textup{ED} then every $2$-extension of $F$ is also \textup{ED}. (In
particular, the pythagorean closure of $F$ is \textup{ED}.)
\end{theorem}
(Recall that an extension $K$ of $F$ is called a
$2$-\emph{extension} of $F$ if $K$ is contained in the quadratic
closure of $F$.)

\begin{remark}
The ED property also played an important role in the
classification theorems of Lewis and Tignol \cite{MR2000m:16027}.
\end{remark}



Our generalization of Theorem~\ref{WHThm} reads:
\begin{theorem}\label{LSUthm}
$F$ is \textup{ED} $\iff$ $F$ satisfies \textup{(WHA)}.
\end{theorem}
The proof will follow from the results below
(Theorems~\ref{EDimplWHA} and \ref{WHAimplED}).

\begin{lemma} Let $(A,\s)$ be a central simple algebra
with involution of the first kind over
$F$. Let $d\in F$ be a sum of squares, and
let $K=F(\sqrt{d})$. Suppose that $(A\ox_F K, \s_K)$ is weakly isotropic. Then
$(A,\s)$ is weakly isotropic.\label{totpos}
\end{lemma}

\begin{proof} We may assume that $(A\ox_F K, \s_K)$ is isotropic.
Hence there exist $x,y\in A$, not both zero such that
$$\bigl(\s(x)+\s(y)\sqrt{d}\bigr)\bigl(x+y\sqrt{d}\bigr)=0.$$
Separating, this implies $\s(x)x+d\s(y)y=0$. Suppose $d=d_1^2+\cdots +
d_r^2$ with $d_i\in F$, then
$$\s(x)x+\s(d_1y)(d_1y)+\cdots + \s(d_ry)(d_ry)=0,$$
i.e. $(A,\s)$ is weakly isotropic.
\end{proof}


\begin{lemma}\label{psap}
Let $F$ be a pythagorean \textup{SAP} field and $A$ a central simple
algebra of exponent $2$ over $F$. Then $A$ is Brauer-equivalent to
a quaternion division algebra $(-1,f)_F$ for some $f\in\Fd$.
\end{lemma}

\begin{proof}
By a well-known theorem of Merkurjev, $A$ is Brauer-equivalent to
a tensor product of finitely many quaternion division algebras
over $F$. Without loss of generality, we may assume that $A$ is
Brauer-equivalent to $(a,b)_F\ox_F (a',b')_F$ for certain
$a,a',b,b'\in \Fd$, and that $(a,b)_F$ and $(a',b')_F$ do not
split.

Since $(a,b)_F$ is a division algebra, its norm form
$\<1,-a,-b,ab\>$ is anisotropic. Hence
$\<a,b,-ab\>$ is anisotropic. Since $F$ is
SAP, the quadratic form $\<1,a,b,-ab\>$ is weakly isotropic, and
thus isotropic, since $F$ is pythagorean. Hence
$\<1,a,b,-ab\>\simeq \<1,-1,c,d\>$ for certain $c,d\in \Fd$.
Comparing determinants, we get $\<1,a,b,-ab\>\simeq \<1,-1,c,c\>$,
which implies $\<a,b,-ab\>\simeq \<-1,c,c\>$,
and thus $(a,b)_F \cong (-1,c)_F$.

Similarly, $(a',b')_F \cong (-1,c')_F$ for some $c\in \Fd$, and
so $A$  is Brauer-equivalent to $(-1,cc')_F$.
\end{proof}


\begin{theorem}\label{EDimplWHA}
Assume that $F$ is \textup{ED}, then $F$ satisfies \textup{(WHA)}.
\end{theorem}

\begin{proof}
Let $(A,\s)$ be totally indefinite. We will show that
$(A,\s)$ is weakly isotropic.

If $A$ is split,  the theorem is true by
Theorem~\ref{WHThm} (when $\s$ is orthogonal) or trivial (when
$\s$ is symplectic).

If the degree of $A$ is odd, then $A$ is split and $\s$ is
orthogonal.  So we are done in this case. Hence we
may assume that $A$ is not split and $\deg A=n=2m$ is even.

Since $F$ is ED, its pythagorean closure is ED.
By Lemma~\ref{totpos} we may replace $F$
by its pythagorean closure. (The pythagorean closure $F_{\mathrm{pyth}}$ is in general
an infinite extension of $F$ but, for any given algebra $A$, we
only need to pass to a finite extension of $F$, sitting inside
$F_{\mathrm{pyth}}$. Then we apply  Lemma~\ref{totpos} finitely many times.)
So we assume from now on that $F$ is a
pythagorean ED field.

By Lemma~\ref{psap}, $A$ is
Brauer-equivalent to a quaternion division algebra $D:=(-1,f)_F$ for some
$f\in \Fd$. So now we have
$$(A,\s)\cong (\End_D(D^m),\s_h)\cong (M_m(D),\s_h),$$
where $\s_h$ is the adjoint involution of a form $h:D^m\x D^m\too
D$, which is hermitian or skew-hermitian with respect to quaternion
conjugation $\bar{\phantom{x}}$ on $D$, according to whether $\s$ is
symplectic or orthogonal.

Suppose first that $\s$ is symplectic, so that $h$ is hermitian.
By \cite[7.6.3]{S} there exists a basis $\{e_1,\ldots,e_m\}$ of $D^m$
over $D$ which is orthogonal with respect to $h$. Let $\gl_i=h(e_i,e_i)$ for
$i=1,\ldots,m$. Lewis and Tignol \cite[Cor. 2]{LT} showed that
$\gl_i\in F$ for all $i=1,\ldots,m$ and that
$$T_\s=\<2\> \ox N\ox \gL\ox \gL$$
where $N$ is the norm form of $D$ and $\gL=\<\gl_1,\ldots, \gl_m\>$.
By assumption $T_\s$ is totally indefinite, and hence
weakly isotropic. Then $N\ox
\gL\ox \gL$ is weakly isotropic and so $N\ox N\ox
\gL\ox \gL$ is weakly isotropic. Since $F$ is SAP, this implies (by
Definition~\ref{def:SAP}(iii)) that $N\ox \gL$ is weakly isotropic. Since
$h(x,x)= \gl_1N(x_1)+\cdots +\gl_mN(x_m)$ for $x=(x_1,\ldots, x_m)\in D^m$,
this implies that the hermitian form $h$ is weakly isotropic over $D$ and
hence that $(A,\s)$ is weakly isotropic.

Suppose next that $\s$ is orthogonal, so that $h$ is
skew-hermitian. Put $K=F(\sqrt{f})$ (note that $f$ is not a square, since $D$ is
a division algebra).
Over $K$, the algebra $A$
splits. Fixing a splitting of $A_K$, the involution $\s_K$ on
$A_K$ becomes the adjoint involution of a quadratic form $q$ over
$K$ (and $q$ is unique up to scaling by elements of $K^\x$). By
our hypothesis, the quadratic form $T_\s$ is totally indefinite
over $F$. So $(T_\s)_K\simeq q\ox q$ is totally indefinite over
$K$, which implies that $q$ is totally indefinite over $K$.
Being a $2$-extension of the ED field $F$, the field $K$ is SAP,
and so $q$ is weakly isotropic over $K$ by Theorem~\ref{WHThm}. In
other words, the algebra with involution $(A,\s)\ox_F K$ is weakly
isotropic, which implies that the skew-hermitian form
$h$ becomes weakly isotropic over $K$ (i.e. as a form over
$D_K\cong M_2(K)$).
From this we will now deduce that that the form $h$ itself is
weakly isotropic, i.e. that the algebra $(A,\s)$
is weakly isotropic.

Replacing $h$ by $N\x h$ for $N\gg 0$ if necessary,
there are $x,y\in D^m$, not both zero, such that
$h_K(x+y\sqrt{f},x+y\sqrt{f})=0$. This implies $h(x,x)+fh(y,y)=0$.
If $h(y,y)=0$, it follows that $h$ is (weakly) isotropic and we
are done. Otherwise, $u:=h(y,y)\in D$ is a non-zero pure
quaternion (since $h$ is skew-hermitian) and $h$ has a
diagonalization
$$h\simeq \<-fu,\ldots\>.$$
Now let $d:=u^2=-\Nrd(u)\in \Fd$. Then $d<0$ on $\{\sP\in X_F |
f<_\sP 0\}$ and therefore $D_{\FP}\cong (d,f)_\FP$ for all
orderings $\sP \in X_F$ (here $\FP$ denotes the real closure of
$F$ with respect to $\sP$). Hence $D\cong (d,f)_F$
by Pfister's local-global principle (note that the Witt ring of $F$ is torsion
free, since  $F$ is pythagorean; see \cite[2.4.10--11]{S}),
and there
exists a pure quaternion $v\in D$ with
$$v^2=-\Nrd(v)=f\quad\text{and}\quad uv+vu=0.$$
Thus $\bar{v}=\Nrd(v)v^{-1}=-fv^{-1}$, and so
$$\bar{v}uv=-fv^{-1}uv=-f(-u)=fu.$$
Therefore $h(yv,yv)=\bar{v}uv=fu$ and
$h$ also has a diagonalization
$$h\simeq \<fu,\ldots\>.$$
This implies that $h\perp h\simeq \<-fu,fu,\ldots\>$, which is
isotropic. In other words, $h$ is weakly isotropic and hence $\s$
is weakly isotropic. We are done.
\end{proof}


\begin{theorem}\label{WHAimplED}
For any non-\textup{ED} field $F$, there is an algebra $(A,\s)$ with
involution of the first kind (and of either type) over $F$ which
is strongly anisotropic but totally indefinite.
\end{theorem}

\begin{proof}
The statement  is clear if the field is
not SAP (there is an involution which is totally indefinite and
strongly anisotropic, as explained just after Example~\ref{SAPexamp}),
 so we concentrate on the case of a SAP
field which is not ED.

Let $F$ be such a field. Then $F$ has a (Krull) valuation $v$
whose value group $\Gamma$ satisfies $\Gamma/2\Gamma=\Z/2\Z$, and whose
residue field $\bar F_v$ is real without being euclidean (this follows from
Definition~\ref{def:SAP}(v) and Theorem~\ref{EDcrit}).
Let $\pi\in F^\x$ with $v(\pi)\notin2\Gamma$, and let $a\in F^\x$ be
a $v$-unit whose residue class in $\bar F_v$ is a sum of squares but
not a square. We choose $a$ such that $a$ is a sum of squares in
$F$,
and consider the quaternion (division) algebra $A=(a,\pi)$ over
$F$. Let $1,i,j,k=ij$ be the standard $F$-basis of $A$, satisfying
$i^2=a$, $j^2=\pi$ and $k^2=-a\pi$. Let $h$ be the (diagonal)
skew-hermitian form $h=\<j,k\>$ over $(A, \bar{\phantom{x}})$ (where
$\bar{\phantom{x}}$ denotes the standard (symplectic) involution on $A$), and let
$\sigma$ be the adjoint involution of $h$ on $M_2(A)$. We claim
that $\sigma$ is totally indefinite,
but not weakly isotropic.

To show this, let $L=F(\sqrt a)$. We fix the splitting $\phi\colon
A_L\isomap M_2(L)$ over $L$ given by
$$i\mapsto\begin{pmatrix}\sqrt a&0\\0&-\sqrt a\end{pmatrix}\quad
\text{and}\quad j\mapsto\begin{pmatrix}0&1\\\pi&0\end{pmatrix}.$$
Under $\phi$, $h$ corresponds to a similarity class of quadratic
forms $q$ (of rank~$4$) over $L$. We are going to calculate $q$.

For $x\in A^\x$ with $x+\bar{x}=0$, let $\sigma_x$ be the (orthogonal)
involution on $A$ given by $\sigma_x(z)=x^{-1}\bar{z}x$. Under $\phi$,
$\sigma_x\otimes\1$ corresponds to a similarity class of quadratic
forms $q_x$ over $L$. Writing $J:=\begin{pmatrix}0&-1\\1&0\end
{pmatrix}$ we have $q_x=J\cdot \phi(x)$. In particular, taking
$x=j$ and $x=k$, we find
$$q_j=\<-\pi,1\>\quad\text{and}\quad
q_k=\<\pi\sqrt a,\sqrt a \>.$$
Thus
$$q\simeq\<1,\sqrt a,-\pi,\pi\sqrt a\>.$$
The form $q$ is totally indefinite. Since
the extension $L/F$ is totally real, the involution trace form
$T_\sigma$ (over $F$) is totally indefinite as well and hence $\s$
is totally indefinite.

On the other hand, the residue forms of $q$ with respect to $v$
are $\<1,\sqrt{\bar a}\>$ and $\<-1,\sqrt{\bar a}\>$ (note that
$\bar{\phantom{x}}$ denotes taking residue classes here). Neither
of them is totally indefinite. Hence $q$ is strongly anisotropic.
Therefore $\s$ cannot be weakly isotropic.

The symplectic case can be treated again by tensoring our algebra
with the quaternion division algebra $(-1,-1)_F$, equipped with
quaternion conjugation.
\end{proof}

Putting everything together now yields a proof of
Theorem~\ref{LSUthm}.

%=========================================================
\section{Sums of hermitian squares}
%=========================================================


In \cite{Lsum}, Lewis proved the following theorem, settling a conjecture
of Leep et~al.~\cite{LSW}.


\begin{theorem}
Let $A$ be a central simple algebra over a field $F$ of
characteristic $\not=2$. Then $0$ is a nontrivial sum of squares, i.e. there
exist nonzero $x_1,\ldots, x_\ell\in A$ such that $0=x_1^2+ \cdots +
x_\ell^2$, if and only if the trace form $T_A$ is weakly isotropic.
\end{theorem}

The natural adaptation of this theorem in the setting of algebras
with involution of the first kind is an easy consequence of the
work we have done hitherto:

\begin{defi}
Let $(A,\s)$ be a central simple algebra with involution of the first kind
over a field $F$ and
$x\in A$. Then $\s(x)x$ is called a \emph{hermitian square} in~$A$.
\end{defi}

\begin{theorem}\label{shsq}
Let $(A,\s)$ be a central simple algebra with involution of the first kind
over an \textup{ED}-field $F$. Then $0$ is a nontrivial sum of
hermitian squares, i.e. there
exist nonzero $x_1,\ldots, x_\ell\in A$ such that $0=\s(x_1)x_1+ \cdots +
\s(x_\ell)x_\ell$,
if and only if the involution trace form $T_\s$ is weakly isotropic.
\end{theorem}


\begin{proof}
The necessary condition follows trivially (and does not require ED) by
simply taking the reduced trace of both sides of
$0=\sum_{i=1}^\ell \s(x_i)x_i$.

For the sufficient condition, suppose that $T_\s$ is weakly
isotropic. Then $T_\s$, and hence $\s$, is totally indefinite.
Therefore $\s$ is weakly isotropic (since $F$ is ED), i.e.
there exist nonzero
$x_1,\ldots, x_\ell\in A$ such that $\s(x_1)x_1+ \cdots +
\s(x_\ell)x_\ell=0$.
\end{proof}

\begin{remark}
For several special classes of algebras with involution of the
first kind, the condition on $F$ can be relaxed and
the conclusion of Theorem~\ref{shsq} will still hold.
This happens for example
\begin{itemize}
\item[(1)] when $(A,\s)$ is an algebra of index $2$ with symplectic
involution over a SAP field $F$;
\item[(2)] when $(Q,\s)$ is a quaternion algebra with
involution of the first kind over a field $F$ of characteristic not
$2$;
\item[(3)] when $(A,\s)\cong (Q_1,\s_1)\ox_F\cdots \ox_F
(Q_\ell,\s_\ell)$ is a multi-quaternion algebra over a
field $F$ of characteristic not $2$ and each $\s_i$ is an
arbitrary involution of the first kind.
\end{itemize}
For proofs, see \cite[Ch.~5]{UPhD}.
\end{remark}


Finally, we obtain a version of Springer's theorem for strongly
anisotropic involutions:

\begin{cor}
Let $(A,\s)$ be a central simple algebra with involution of the first kind
over an \textup{ED}-field $F$ and let $K/F$ be any finite
extension of odd degree. If $(A,\s)$ is strongly anisotropic, then
$(A\ox_F K, \s_K)$ is (strongly) anisotropic.
\end{cor}

\begin{proof}
Since $\s$ is strongly anisotropic, $T_\s$ is strongly anisotropic
by Theorem~\ref{shsq}. By Springer's theorem (see e.g.
\cite[2.5.3]{S}), $(T_\s)_K=T_{\s_K}$ is strongly
anisotropic over $K$. Hence $\s_K$ is strongly anisotropic by
contraposition of the trivial direction of Theorem~\ref{shsq}.
\end{proof}



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

This research was supported by
the TMR research network (ERB~FMRX~CT-97-0107) on
``$K$-theory and algebraic groups''.





%=========================================================

\begin{thebibliography}{Knight}


\bibitem{Br}
L.~Br\"ocker, Zur Theorie der quadratischen Formen \"uber formal
reellen K\"orpern, \emph{Math. Ann.} \textbf{210} (1974), 233--256.

\bibitem{D} I. Dejaiffe, Somme orthogonale d'alg\`ebres \`a
involution et alg\`ebre de Clifford, \emph{Comm.  Algebra} {\bf 26} (1998), 1589--1612.

\bibitem{MR1765403}
I.~Dejaiffe, D.~W. Lewis, J.~P. Tignol,
\newblock Witt equivalence of central simple algebras with
involution,
\newblock {\em Rend. Circ. Mat. Palermo (2)} \textbf{49}(2) (2000), 325--342.


\bibitem{ELP} R. Elman, T.Y. Lam, A. Prestel, On some Hasse
principles over formally real fields, \emph{Math. Z.} {\bf 134}
(1973), 291--301.


\bibitem{MR50:4485}
R. Elman, T.~Y. Lam,
\newblock Classification theorems for quadratic forms over fields,
\newblock {\em Comment. Math. Helv.} {\bf 49} (1974), 373--381.

\bibitem{KRW} M. Knebusch, A. Rosenberg, R. Ware,
Structure of {W}itt rings, quotients of abelian group rings,
and orderings of fields, \emph{Bull. Amer. Math. Soc.} {\bf 77} (1971),
205--210.


\bibitem{BOI} M.-A. Knus, A.S. Merkurjev, M. Rost, J.-P. Tignol,
\emph{The Book of Involutions}, Coll. Pub.~{\bf 44}, Amer. Math. Soc.,
Providence, RI (1998).

\bibitem{L} T.Y. Lam, \emph{The Algebraic Theory of Quadratic Forms},
Reading, Mass. (1973).


\bibitem{Lam2} T.Y. Lam, \emph{Orderings, Valuations and Quadratic Forms},
CBMS Notes {\bf 52}, Amer. Math. Soc. (1983).


\bibitem{LSW} D.B. Leep, D.B. Shapiro, A.R. Wadsworth, Sums of squares in division
algebras, \emph{Math. Z.} {\bf 190} (1985), 151--162.


\bibitem{Lsum} D.W. Lewis, Sums of squares in central simple
algebras, \emph{Math. Z.} {\bf 190} (1985), 497--498.


\bibitem{Lewsg} D.W. Lewis, The Witt semigroup of central simple algebras with
  involution, \emph{Semigroup Forum} {\bf 60} (2000), 80--92.


\bibitem{LT} D.W. Lewis, J.-P. Tignol, On the signature of an
involution, \emph{Arch. Math.} {\bf 60} (1993), 128--135.

\bibitem{MR2000m:16027}
D.W. Lewis, J.-P. Tignol,
\newblock Classification theorems for central simple algebras with
involution,
\newblock {\em Manuscripta Math.} {\bf 100}(3) (1999), 259--276.
\newblock With an appendix by R. Parimala.

\bibitem{Pres} A. Prestel, Quadratische Semi-Ordnungen und
quadratische Formen, \emph{Math. Z.} {\bf 133} (1973), 319--342.

\bibitem{P} A. Prestel, \emph{Lectures on Formally Real Fields},
Lecture Notes in Mathematics {\bf 1093}, Springer-Verlag, Berlin
(1984).

\bibitem{PW} A. Prestel, R. Ware, Almost isotropic quadratic
forms, \emph{J. London Math. Soc.} (2) {\bf 19} (1979), 214--244.

\bibitem{S}  W. Scharlau, \emph{Quadratic  and  Hermitian  Forms},
{Grundlehren  Math.  Wiss.} {\bf 270}, Springer-Verlag,  Berlin
(1985).

\bibitem{MR97e:16043}
J.-P. Tignol,
\newblock A {C}assels-{P}fister theorem for involutions on central simple
  algebras,
\newblock {\em J. Algebra}  \textbf{181}(3) (1996), 857--875.


\bibitem{UPhD} T. Unger, \emph{Quadratic Forms and Central Simple Algebras with
Involution}, Ph.D. Thesis, National University of Ireland, Dublin
(2000), unpublished.

\bibitem{U} T. Unger, Clifford algebra periodicity for central
simple algebras with an involution, \emph{Comm. Algebra}  {\bf 29}(3)
(2001), 1141--1152.

\bibitem{W} R. Ware, Hasse principles and the $u$-invariant over
formally real fields, \emph{Nagoya Math.~J.} {\bf 61} (1976),
117--125.

\end{thebibliography}

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