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\author{\sc{M.G. Mahmoudi}}
\date{December 2004}
\title{Hermitian forms and the $\u$-invariant}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}
\title{Hermitian forms and the $\u$-invariant}

\author{M. G. Mahmoudi}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}
We study the notion of hermitian $\u$-invariant. We give 
some estimates of the $\u$-invariant of a division algebra with 
involution in terms of
the $\u$-invariant of some subalgebras stable under
the involution. 
We also find some finiteness results for comparing the $\u$-invariant
of a division algebra with involution and that of its centre.
Some results about the values of this invariant are also given.
A description of the Tits index of some algebraic groups of classical
type over $\qp(t)$, $p\neq 2$ is given as an application.
\\\\
%\keywords{$\u$-invariant, hermitian form, anisotropic form, Tits index}
%\subclass{11E39, 11E81, 11E57}
Mathematics Subject Classification (2000): {11E39, 11E81, 11E57}
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
The $\u$-invariant is one of the most interesting invariants in the algebraic
theory of quadratic forms.
This invariant was introduced by Kaplansky over non formally real fields
and by Elman and Lam \cite{elman-lam} over formally real fields.
(In this work we always assume that characteristic $\neq2$).
 
Determination of this invariant for a given field,
has been of great importance in the literature. For a wide class of fields, 
the values of this invariant are not known and sometimes it is not clear
if the $\u$-invariant of such fields is finite. 
%One method to approach the 
Obtaining lower and upper bounds for this invariant and comparing 
the $\u$-invariant of a given field $K$ with the $\u$-invariant of its 
subfields or the fields containing $K$ are good approaches to the problem.  

Systems of quadratic forms and exact sequences of Witt groups are two powerful
tools used in the literature to obtain good lower and upper bounds for
the $\u$-invariant. For example, systems of quadratic forms have been used in 
\cite{leep} by Leep to obtain the bound $\u(L)\leqslant\frac{n+1}{2}\u(K)$
for a field extension $L/K$ of degree $n$. 
 
Using exact sequences of Witt groups is however less
common than using systems of quadratic forms, but one can mention 
an exact triangle of Witt groups 
used by Elman and Lam 
(cf.  \cite{elman-lam}, \cite{elman}) to compare the $\u$-invariant
of a field with the $\u$-invariant of a quadratic extension of this field.

In this work, we try to adapt these tools in the context of hermitian forms
over a division algebra with involution. To our best knowledge, 
the notion of the $\u$-invariant of a division algebra with involution
appears, for the first time, 
%in the works of Pfister 
in Pfister's paper 
\cite{pfister89} in
connection with systems of quadratic forms over a formally real field and 
with exception of \cite{pfister89}, this notion has not been further studied 
in the literature. 

Our initial motivation for studying this notion was to describe the Tits index 
of some classical groups over $\qp(t)$, $p\neq 2$ for which we would have need 
to obtain some information about the maximal dimension of anisotropic 
hermitian forms over some division algebras with involution over $\qp(t)$,
$p\neq 2$. 
The results concerning Tits indices are given in the last section of
this paper as applications of our main results.
We later come back to this subject.
\bigskip

Let $D$ be a division algebra with an involution $\sigma$ and let $\ve$ be an
element of the centre of $D$ with $\sigma(\ve)\ve=1$. The $\u$-invariant
with respect to $\ve$, is by 
definition the supremum over the dimension of anisotropic $\ve$-hermitian forms over
$(D,\sigma)$. This number is denoted by $\u(D,\sigma,\ve)$ (value
$\infty$ is admitted also). Let $E$ be a subdivision algebra of $D$ stable
under $\sigma$. We are interested in the following questions:

\begin{question}\label{question1}(Going up)

 Under which conditions does the finiteness of $\u(E,\sigma|_E,\ve)$
imply that of $\u(D,\sigma,\ve)$ and can one obtain an upper bound 
for $\u(D,\sigma,\ve)$  in terms of $\u(E,\sigma|_E,\ve)$?
\end{question}


\begin{question}\label{question2}(Going down)
 
Under which conditions does the  finiteness of $\u(D,\sigma,\ve)$
imply that of $\u(E,\sigma|_E,\ve)$ and can one obtain a lower bound 
for  $\u(D,\sigma,\ve)$  in terms of $\u(E,\sigma|_E,\ve)$?
\end{question}

%\begin{question}
%Let $D$ be a particular type of a division algebra with an involution $\sigma$.
%Another natural question is to ask about possible values for the $\u$-invariant 
%for some particular type of division algebra with involution. We 
%consider the case of a quadratic extension 
%$L/K$ with the nontrivial automorphism $\bar\ $ and we obtain:
%\end{question}


A substantial part of this work is devoted to these questions.
In certain situations, we are able to give precise answers. 
We will especially deal with the following cases: 
\begin{itemize}
\item $D/E$ is a quadratic extension of fields (cf. Remark~\ref{rem-u-inv-quad},
Proposition~\ref{reel}, Proposition~\ref{u-invariant-biquadratic}).
\\

\item $D$ is a quaternion algebra and $E$ is a maximal subfield of $D$
(cf. Corollary~\ref{principal}, Remark~\ref{rem-u-inv-quad}, 
Corollary \ref{lbound}).\\

\item more generally if
there exist two invertible elements $\lambda$ and $\mu$ in $D$
such that $\lambda\mu=-\mu\lambda$, $\sigma(\lambda)=-\lambda$,
$\sigma(\mu)=-\mu$ and $K(\lambda)$ is a quadratic extension of $K$
where $K$ is the centre of $D$. Let $E$ be the centralizer 
of $K(\lambda)$ in $D$
(cf. Proposition~\ref{ppp}, Remark~\ref{rppp},
Proposition~\ref{down}).
%This particular situation of $D$ and $E$ is studied by
%Bayer-Fluckiger and Parimala in \cite{bayer-parimala}. 
\\
%See 
%\cite[Lemma~3.1.1]{bayer-parimala} for the link between the existence of $\lambda$ and
%$\mu$. 

\item $E$ is the fixed field of the restriction of $\sigma$ to the centre of
$D$ (cf. Proposition~\ref{bound}).\\

\item $D$ is the tensor product $E\otimes_kL$ where $k$ is the fixed field of 
$\sigma|_K$, $K$ is the centre of $D$ and $L/k$ is a field extension 
(cf. Proposition~\ref{tensori}, Proposition~\ref{tensorii}, 
Remark~\ref{remark-tensor}). 
\end{itemize}

In particular the Question~\ref{question1} has an affirmative answer when $E$
is the fixed field of $\sigma|_K$ where $K$ is the centre of $D$.
By contrast, the finiteness of $\u(D,\sigma,\varepsilon)$ does not imply,
in general, that of $\u(k)$, 
for example if $D=(-1,-1)_{\mathbb{R}}$ is the algebra of
Hamiltonian quaternions and $\sigma$ is its canonical involution
 we have $\u(D,\sigma,-1)=1$ and $\u(\mathbb{R})=\infty$.
In fact we prove that if $\dim_KD$ is a power of $2$, and if 
both $\u(D,\sigma,\varepsilon)$ and $\u(D,\sigma,-\varepsilon)$ are finite 
then $\u(K,\sigma|_K)<\infty$ (cf. Theorem~\ref{nonreal}). 
In particular if $\sigma$ is of the first kind, then $\u(K)$ is finite 
if and only if $\u(D,\sigma,1)$ and $\u(D,\sigma,-1)$ are finite.
 
The principal ideas to prove 
these results are to use 
some exact sequences of Witt groups, mainly the exact 
sequence of Milnor-Husemoller \cite[Ch.10, 1.2]{scharlau},
Lewis  \cite{lewis2} and Parimala-Sridharan-Suresh 
\cite[Appendix 2]{bayer-parimala} and to use 
the results of Leep
\cite{leep} and Pfister \cite{pfister89} on  
systems of quadratic and hermitian forms.
We also use a variation of the 
exact sequence of Parimala-Sridharan-Suresh discussed in \cite{suite}.

On one occasion, we need to compare the hermitian $\u$-invariant of a field 
with that of some of its subfields. We do this by using an exact sequence 
of Witt groups of biquadratic extensions. We present this exact sequence in
\S\ref{biquadratiques}, (see Theorem~\ref{suite-biquadratic}).
This exact sequence  
may be regarded as a particular case of a general exact sequence
of $L$-groups and projective Witt groups due to Ranicki,
see \cite{ranicki1} and \cite[p.242]{ranicki2}.

Another natural question is to ask about possible values for the $\u$-invariant 
for some particular type of division algebra with involution. We 
consider the case of a quadratic extension 
$L/K$ together with the nontrivial automorphism $\bar\ $ 
and we prove that $\u(L,\bar\ )\neq 3,5,7$  (cf. Proposition~\ref{357}).

This result is a hermitian analogue of  a classical theorem 
in the theory of quadratic forms which states that the $\u$-invariant
cannot be equal to $3$, $5$ and $7$ 
 (cf. \cite[Ch.2, 16.2]{scharlau}). Our proof is of course similar
to the proof of this theorem.

One could also define the notion of $\u$-invariant for central simple
algebras with involution. 
But thanks to Morita theory, the problem can be translated to division 
algebras with involution, so we are only interested in this situation. 

As David Lewis pointed out to me, one can also define the notion of the 
$\u$-invariant for $G$-equivariant forms, i.e., the forms stable under 
the action of a finite group $G$ and with the same ideas
one can obtain some results that are similar to those stated here.

Patrick Morandi pointed out to me that by using a version of a theorem
of Springer for hermitian forms obtained by Larmour \cite{larmour},
one can calculate the $\u$-invariant of a valued division algebra
over a henselian valued field, in terms of the $\u$-invariant of
its residue division algebra.  

Finally in section \S\ref{sec-an},
we give some applications of these results to describe the Tits index of some 
classical groups over $\qp(t)$ with $p\neq 2$. 
We recall that according to a generalized Witt decomposition theorem (\cite{tits}), 
semisimple algebraic groups defined over an arbitrary field $K$ are determined
up to isomorphism by their $K^{sep}$ class, their anisotropic kernel and their
Tits index, 
see Tits \cite{tits} or 
Springer \cite[Ch. 17]{springer} for these notions.
Describing the Tits index is therefore important. For the groups of classical
type, these indices can be described in terms of algebras with involution.
For finite fields, the field of real numbers, number fields and $p$-adic fields,
all possible indices are known (see \cite{tits}). 
We try to describe these indices over $\qp(t)$, $p\neq 2$
(cf. 
\S\ref{sec-an}).

In consequence of the results due to Saltman
\cite{saltman} 
on the structure of the group $\br_2(\qp(t))$,
the only division algebras over $\qp(t)$ with an involution of the first kind
are split ones, quaternions and biquaternions. 
For split case,  by a result due to Parimala and Suresh 
\cite{parimala-suresh} we have 
$8\leqslant\u(\qp(t))\leqslant 10$ (an earlier result due to  
Hoffmann and Van Geel \cite{hoffmann-vangeel} states that 
$8\leqslant\u(\qp(t))\leqslant 22$).  
By using these results and by
examining anisotropic hermitian and skew hermitian forms over
quaternion algebras and biquaternions algebras over $\qp(t)$
we obtain some information about possible indices.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The $\u$-invariant of a division algebra with involution}
Let $K$ be a field of characteristic different from $2$. For 
a division algebra $D$ over $K$, a $K/k$-involution $\sigma$ on $D$
is an involution where $k$ is the fixed field of $\sigma|_K$. 

Let us denote by $S^{\ve}(D,\sigma)$ the semigroup of isometry classes 
of $\ve$-hermitian forms over $(D,\sigma)$ and 
by $W^{\ve}(D,\sigma)$ 
the Witt group of $\ve$-hermitian forms over $(D,\sigma)$. 

We refer to \cite{scharlau} and \cite{boi} for basic notions about 
quadratic and hermitian forms and algebras with involution.

Every division algebra over a field $K$ considered in this paper is
implicitly assumed to be $K$-central.  

 Let $D$ be a division algebra over a field $K$ with an involution
$\sigma$. Let $\ve\in K$ with $\ve\sigma(\ve)=1$.
A system $h=(h_1,\cdots,h_r):V\times V\rightarrow D^r$ of $r$, $\ve$-hermitian forms
over a right $D$-vector space $V$ is called 
{\it anisotropic} if $x\in V$, $h(x,x)=0$
implies that $x=0$.

We consider the $\u$-invariant in the sense of Kaplansky:
 
 \begin{dff}{\rm  (\cite[Ch. 9, Definition 2.4]{pfister}) 

$\u_r(D,\sigma,\varepsilon)= \sup\{\dim_D V; \text{ there exists an
anisotropic $\ve$-hermitian map}\\  
\text{ \hspace{4.5cm}$h:V\times V\rightarrow D^r$} \}$


Let us simplify the notation by writing $\u(D,\sigma,\varepsilon)$ 
instead of $\mathop{\mathrm{u}_1}(D,\sigma,\varepsilon)$ and  
by  $\u(D,\sigma)$ instead of  $\mathop{u}(D,\sigma,1)$. }
 \end{dff}

We recall that according to a result due to Leep \cite{leep},
the system $\u$-invariant $\u_r=\u(k)$ satisfies
$\u_r\leqslant r\u_1+\u_{r-1}$ and $\u_r\leqslant\frac{r(r+1)}{2}\u_1.$
This result has been generalized by Pfister \cite{pfister89} to 
the system $\u$-invariant of $\ve$-hermitian forms over a division 
algebra with involution and also over skew fields with involution  
not necessarily finite-dimensional over their centres. 

As a first observation, we have:

\begin{prop}\label{sadeh}
Let  $D$ be a division algebra over a field $K$.
Let $\sigma$ and $\tau$ be two involutions on $D$ 
with the same restriction to $K$: $\sigma|_K=\tau|_K$. 
Let $\ve\in K$ with $\ve\sigma(\ve)=1$.
\begin{itemize}
\item[\rmm{1}] If $\sigma$ and $\tau$ are of the second kind and 
 $\ve'\in K$ with $\ve'\tau(\ve')=1$ then
$\u(D,\sigma,\ve)$ and $\u(D,\tau,\ve')$ do not depend on the choice
of $\ve$ and $\ve'$; moreover $\u(D,\sigma,\ve)=\u(D,\tau,\ve')$.  
\item[\rmm{2}]
 If $\sigma$ and $\tau $ are of the first kind then we have: 
$\u(D,\sigma,\ve)=\u(D,\tau,\ve)$ if $\sigma$ and $\tau$ have the same type,
otherwise we have $\u(D,\sigma,\ve)=\u(D,\tau,-\ve)$.
\end{itemize}
 \end{prop}

 \begin{proof}
For (1), there exists an element $b\in D^*$ such that $\sigma=\intt(b)\circ\tau$. 
We have $b\tau(b)^{-1}=\lambda\in K^*$. We obtain then 
$\lambda\tau(\lambda)=1$.
Let $\lambda'=\lambda^{-1}\ve'\ve^{-1}$. We have $\lambda'\tau(\lambda')=1$.
It follows from Hilbert 90 that there exists $\mu\in K^*$ such that
$\mu\tau(\mu^{-1})=\lambda'$.

If we take $c=\mu b$, then 
$\sigma=\intt(b)\circ\tau=\intt(c)\circ\tau$. 
Now the correspondence 
$\varphi\mapsto c^{-1}\varphi$ gives a bijection between the semigroups
$S^{\ve}(D,\sigma)$ and $S^{\ve'}(D,\tau)$. This  bijection preserves 
isometry, orthogonal sum and dimension.
So we conclude that $\u(D,\sigma,\ve)=\u(D,\tau,\ve')$. 

For (2), we use the same argument. There exists $b\in D^*$ such that
 $\sigma=\intt{(b)}\circ\tau$. We have
$\lambda:=b\tau(b)^{-1}\in K^*$. Moreover $\lambda=1$ if $\sigma$
and $\tau$ are of the same type and $\lambda=-1$ if $\sigma$ and
$\tau$ are of different type. The correspondence 
$\varphi\mapsto b^{-1}\varphi$ from $S^{\ve}(D,\sigma)$ to
 $S^{\lambda\ve}(D,\tau)$ gives $\u(D,\sigma,\ve)=\u(D,\tau,\lambda\ve)$.  
 ~\qed\end{proof}

\begin{remark}
{\rm 
As pointed out to me by the Referee, the preceding 
proposition actually says that for a given division algebra $D$, 
there are three possible $\u$-invariants, let us say a unitary, an orthogonal and
a symplectic one; namely for any $\sigma$ and $\ve$, the $\u$-invariant of 
$(D,\sigma,\ve)$ coincides with one of them, depending on the type of $\sigma$ 
and the value of $\ve$. Note that for unitary case, the preceding proposition states that 
for a given $D$, $\u(D,\sigma,\ve)$ depends only to the restriction of $\sigma$ to 
the centre of $D$. This leads us to introduce the notation $\u^+(D)$ for the orthogonal 
$\u$-invariant, $\u^-(D)$ for the symplectic $\u$-invariant 
%and $\u^{k}(D)$  for unitary $\u$-invariant, 
%where $k$ is the fixed field of $\sigma|_K$ 
(this was also suggested to me by Karim Becher). In this way, if 
$\tau_1$ is a symplectic involution and $\tau_2$ is an orthogonal involution on 
$D$ we have $\u^+(D)=\u(D,\tau_1,-1)=\u(D,\tau_2,1)$ and 
$\u^-(D)=\u(D,\tau_1,1)=\u(D,\tau_2,-1)$. 
This point of view might lead to some simplification 
in the presentation of some parts of this paper,
%could enlighten some
%parts of this paper, 
cf. for instance Corollary \ref{principal} and 
Proposition \ref{bound} or Theorem~\ref{nonreal}.
}\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Going up results}\label{sec-going-up}
Let $D$ be a division algebra over a field $K$ with an involution
$\sigma$. 
We suppose that
there exist two invertible elements $\lambda$ and $\mu$ in $D$
such that $\lambda\mu=-\mu\lambda$, $\sigma(\lambda)=-\lambda$,
$\sigma(\mu)=-\mu$ and $K(\lambda)$ is a quadratic extension of $K$.
%This situation is studied by 
%Bayer-Fluckiger and Parimala in \cite{bayer-parimala}.

Let
$\wt{D}$ be the centralizer of $L=K(\lambda)$ in $D$. According to 
\cite[Appendix~2]{bayer-parimala}, we have 
$\mu\wt{D}\mu^{-1}=\wt{D}$, $\mu^2\in\wt{D}$, $\mu^2\in\wt{D}$ and 
$D=\wt{D}\oplus\mu\wt{D}$.  On $\wt{D}$ we 
have two natural involutions $\sigma_1=\sigma|_D$ and 
$\sigma_2=\intt(\mu^{-1})\circ\sigma_1$.
We have 
$\deg\wt{D}=\frac{1}{2}\deg D$. The involution $\sigma_1$ is always of the second kind.
The involution $\sigma_2$ is of the same kind as $\sigma$ but of different type
if $\sigma$ is of the first kind.  
See \cite[\S3.1]{bayer-parimala} for more details. 

Let $\pi_i:D\rightarrow\wt{D}$ be the $L$-linear projections 
$\pi_1(\alpha+\mu\beta)=\alpha$ and $\pi_2(\alpha+\mu\beta)=\beta$. 
If $h:V\times V\rightarrow D$ is a $\ve$-hermitian space over $(D,\sigma)$, then 
$h_i:V\times V\rightarrow\wt{D}$ is defined by $h_i(x,y)=\pi_i(h(x,y))$. It is easily 
verified that $h_1$ is an $\ve$-hermitian space over $(\wt{D},\sigma_1)$ and
$h_2$ is an $-\ve$-hermitian space over $(\wt{D},\sigma_2)$. 
See \cite[Appendix~2]{bayer-parimala} for more details.
 
We prove the following proposition which plays an important role in this paper. 


\begin{prop}\label{ppp}
  Let $D$ be a division algebra over a field $K$ with a $K/k$-involution 
$\sigma$. 
  Suppose that there exist $\lambda$, $\mu\in D^*$ such that
  $\sigma(\lambda)=-\lambda$, $\sigma(\mu)=-\mu$, $\lambda\mu=-\mu\lambda$ and
  $L=K(\lambda)$ is a quadratic extension of $K$.
 Let $\wt{D}$ be the centralizer of $L$ in $D$,
 $\sigma_1=\sigma|_{\wt{D}}$ and
$\sigma_2=\intt(\mu^{-1})\circ\sigma_1$ and $\ve\in K$ with $\ve\sigma(\ve)=1$.
Then we have:\\
$$\u(D,\sigma,\varepsilon)\leqslant
  \frac{1}{2}\u(\wt{D},\sigma_2,-\varepsilon)+
  \u(\wt{D},\sigma_1,\varepsilon).$$
 \end{prop}
 \begin{proof}
Let $\pi_1$ and $\pi_2$ be the projections from $D$ to 
$\wt{D}$ induced by the decomposition $D=\wt{D}\oplus\mu\wt{D}$, i.e,
$\pi_i(d_1+\mu d_2)=d_i$, for $i=1,2$.
Let $(V,h)$ be a nondegenerate $\varepsilon$-hermitian space over $(D,\sigma)$ and
  $h_1=\pi_1h$, $h_2=\pi_2h$. We have
  $\dim_{\wt{D}}(h_1)=\dim_{\wt{D}}(h_2)=2\dim_D(h).$ 
 If 
  $\dim_{\wt{D}}(h_2)\geqslant \u(\wt{D},
  \sigma_2,-\varepsilon)+2m-1$ for some positive integer
$m\geqslant 1$, then $h_2$  contains an orthogonal sum of $m$ 
hyperbolic planes.
Consequently $h_2$ is totally isotropic over a  $\wt{D}$-vector subspace $W$
of $V$ of dimension $m.$
If moreover $m\geqslant \u(\wt{D},\sigma_1,\varepsilon)+1$ then  $h_1$ is 
isotropic over $W$. In this way, in order that $h$ be isotropic,
it is sufficient to have 
  $2n=\dim_{\wt{D}}(h_2)\geqslant \u(\wt{D},\sigma_2,-\varepsilon)+
  2\u(\wt{D},\sigma_1,\varepsilon)+1$. This is equivalent to\\
\centerline{
$\u(D,\sigma,\varepsilon)\leqslant \frac{1}{2}
\u(\wt{D},\sigma_2,-\varepsilon)+
  \u(\wt{D},\sigma_1,\varepsilon).\ \ \ \  \qed$}
%~\qed
\end{proof}
\begin{remark}\label{rppp}{\rm
In the proof of the previous proposition, one may interchange the role of
$h_1$ et $h_2$, 
in this way we obtain:
$$\u(D,\sigma,\varepsilon)\leqslant \u(\wt{D},\sigma_2,-\varepsilon)+
 \frac{1}{2} \u(\wt{D},\sigma_1,\varepsilon).$$}
\end{remark}

\begin{remark}{\rm
One may also give an alternative proof of the previous result by using
an exact sequence of Parimala-Sridharan-Suresh. See the proof
of Proposition~\ref{down} which uses this idea.}
\end{remark}

 \begin{cor}\label{principal}
Let $Q=(a,b)_K$ be a quaternion division algebra over a field $K$. 
Let $\bar\ $ be the canonical involution  of $Q$ and
$\hat{\ }$ an orthogonal involution of $Q$ and let 
$L=K(\sqrt{a})\subset Q$ which is stable under $\bar\ $,
  then we have:  
\[
\begin{array}{l} 
  \u(Q,\hat{\ })=\u(Q,\bar\ ,-1)\leqslant\min\{\frac{1}{2}\u(L)+\u(L,\bar\ ), \u(L)+\frac{1}{2}\u(L,\bar\ )\},\\
  \u(Q,\bar\ )=\u(Q,\hat{\ },-1)\leqslant\frac{1}{2}\u(L,\bar\ ).
\end{array}
\]
\end{cor} 

\begin{remark}\label{rem-u-inv-quad}{\rm
Let $L/K$ be a quadratic extension 
and let $\bar\ $ be its nontrivial automorphism. 
We have the bound $\u(L,\bar\ )\leqslant \frac{1}{2}{\u(K)}$ 
because to every anisotropic hermitian form over $(L,\bar\ )$ of dimension 
$n$, one can associate an anisotropic quadratic form over $K$ of
dimension $2n$. 
In the same way, if $Q$ is a quaternion algebra over a field $K$ with 
the canonical involution $\bar\ $, we have $\u(Q,\bar\ )\leqslant\frac{1}{4}\u(K)$.
See also Proposition~\ref{bound} which states a more general result.
}\end{remark}

 \begin{prop}\label{bound}  
  Let $D$ be a division algebra of degree $m$ over its centre $K$ with a
 $K/k$-involution $\sigma$ and let $\ve\in K$ with $\ve\sigma(\ve)=1$. Then : 
\begin{equation}{\label{bound-equation}}
\u(D,\sigma,\varepsilon)\leqslant\frac{r(r+1)}{2m^2[K:k]}\u(k)
\end{equation}
where
  $r$ is the dimension of $k$-vector space of $\varepsilon$-hermitian elements
of $D$. 
In particular, if $\u(k)$ is finite then so is 
$\u(D,\sigma,\varepsilon)$.
 \end{prop}

\begin{proof}
Let $(V,h)$ be an anisotropic $\varepsilon$-hermitian space over $(D,\sigma)$ of 
dimension $n$. Take $D^\varepsilon$ the $k$-vector space of  $\varepsilon$-hermitian
elements of $D$, in other words:
  $$D^\varepsilon=\{x\in D: \sigma(x)=\varepsilon x\}.$$ 
  It is well known that  
$D=D^{+1}\oplus D^{-1}$ and $\dim_k(D^{+1})=\dim_k(D^{-1})=m^2$ when
  $\sigma$ is of the second kind and $\dim_k(D^{+1})=\frac{1}{2}m(m+1)$ or 
  $\frac{1}{2}m(m-1)$ when $\sigma$ is of the first kind. 
Let
  $\{e_1,\cdots,e_r\}$ be a $k$-basis of  $D^\varepsilon$ and 
$\{f_1,\cdots,f_s\}$ a $k$-basis of  $D^{-\varepsilon}$. 
One can write $h$ in the form: 
  \[
  \begin{array}{lll}
  h(x,y)&=&\varphi_1(x,y)e_1+\cdots+\varphi_r(x,y)e_r\\
  &+&\psi_1(x,y)f_1+\cdots+\psi_s(x,y)f_s
  \end{array}
  \]
where $\varphi_1,\cdots,\varphi_r$ are symmetric bilinear forms and
 $\psi_1,\cdots,\psi_s$ are skew symmetric bilinear forms over $k$
  (the forms $\varphi_i$ ($1\leqslant i\leqslant r$) 
and $\psi_i$ ($1\leqslant i\leqslant s$) are possibly degenerate).
 So $\psi_i(x,x)=0$ for all $x\in V$. We deduce that 
  $$h(x,x)=\varphi_1(x,x)e_1+\cdots+\varphi_r(x,x)e_r.$$
As $h$  is anisotropic, $\varphi_1,\cdots,\varphi_r$ have no common isotropic vector.
Now by using a result due to Leep 
(cf.~\cite[Ch 9, 2.1]{pfister} 
or \cite[Ch.2, 16.5]{scharlau}) we obtain: 
$$m^2[K:k]n=\dim_k(V)\leqslant {\u}_r(k)
  \leqslant\frac{r(r+1)}{2}\u(k).$$ 
  Therefore $n\leqslant \frac{r(r+1)}{2m^2[K:k]}\u(k)$ which implies 
the claimed inequality
(\ref{bound-equation}). 
 \qed
\end{proof}

\begin{remark}{\rm
The previous result is a finiteness statement. For many situations, one may
have better estimates for $\u(D,\sigma,\ve)$. For example
for quadratic extensions and quaternion algebras, see
Proposition~\ref{ppp}, Remark \ref{rppp}, Corollary \ref{principal},
Proposition~\ref{reel}, Proposition~\ref{tensori}, Proposition~\ref{tensorii}, 
Remark \ref{remark-tensor}. 
Nevertheless this bound is optimal for  $m=1$ or  ($m=2$ and $\sigma$ 
symplectic).}
\end{remark}


\begin{prop}\label{tensori}
 Let  $D$ be a division algebra over a field $K$ with a $K/k$-involution $\sigma$.
Let $L/k$ be an extension of degree $n$ and suppose that $D\otimes_kL$  is also
a division algebra. Then 
\begin{equation}{\label{tensori-eq}}
  \u(D\otimes_kL,\sigma\otimes \id,\varepsilon)\leqslant \frac{n+1}{2}
  \u(D,\sigma,\varepsilon).
\end{equation}
In particular the finiteness of 
$\u(D,\sigma,\varepsilon)$ implies that of 
$\u(D\otimes_kL,\sigma\otimes \id,\varepsilon)$.
 \end{prop}

\begin{proof}
Let  $(V,\varphi)$ be an $\ve$-hermitian space over 
$(D\otimes_kL,\sigma\otimes\id)$. We choose a $k$-basis $\{e_1,\cdots,e_n\}$ of
 $L$.
We can write 
  $\varphi(x,y)=(\varphi_1(x,y)\otimes e_1))
+\cdots+(\varphi_n(x,y)\otimes e_n)$
  where $\varphi_1,\cdots,\varphi_n$ are $\varepsilon$-hermitian forms
(possibly degenerate) over $(D,\sigma)$. If  $\varphi$ is anisotropic and
 $\dim(\varphi)=m$, then   $\varphi_1,\cdots,\varphi_n$ have no common isotropic
vector, therefore 
 $mn=\dim(\varphi_i)\leqslant \u_n(D,\sigma,\varepsilon)$ so:
  $$\u(D\otimes_kL,\sigma\otimes
  \id,\varepsilon)\leqslant\frac{1}{n}\mathrm{u}_n(D,\sigma,\varepsilon).$$
According to a result due to Pfister (cf. \cite[Ch.9, 2.5]{pfister}
or \cite{pfister89}) we have: 
${\u}_n(D,\sigma,\varepsilon)\leqslant\frac{n(n+1)}{2}
u(D,\sigma,\varepsilon)$. This implies 
(\ref{tensori-eq}).
\qed
\end{proof}

\begin{remark} {\rm
If in the previous statement we take $D=k$, then we
retrieve Leep's estimate  $\u(L)\leqslant\frac{n+1}{2}\u(k)$.}
\end{remark}

\begin{prop}\label{tensorii}
Let  $D$  be a division algebra over a field  $K$ with a $K/k$-involution $\sigma$.
Let $L/k$ be a quadratic extension and let $\bar{\ }:L\longrightarrow L$ be the
nontrivial $k$-automorphism of $L$. Suppose that $D\otimes_kL$ is a division 
algebra. Then we have:
$$\u(D\otimes_kL,\sigma\otimes\bar{\ },\varepsilon)\leqslant
\frac{1}{2}\u(D,\sigma,-\varepsilon)+\u(D,\sigma,\varepsilon).$$ 
 \end{prop}

\begin{proof}
Let $L=k(\xi)$ with $\xi^2\in k$ and $\bar{\xi}=-\xi$. Let $(V,\varphi)$ 
be an $\varepsilon$-hermitian space over  $(D\otimes_kL,\sigma\otimes\bar{\ })$. 
We can write $\varphi$ in the form 
$$\varphi(x,y)=\varphi_1(x,y)\otimes1+\varphi_2(x,y)\otimes\xi,$$ 
where $\varphi_1$ is an $\varepsilon$-hermitian form and $\varphi_2$ 
is a $-\varepsilon$-hermitian form over
$(D,\sigma)$.  By repeating the argument given in the proof of
Proposition~\ref{ppp} we conclude the result.~
~\qed\end{proof}

\begin{remark}\label{remark-tensor}{\rm 
In the proof of Proposition~\ref{tensorii}, one can interchange the role of
 $\varphi_1$ and $\varphi_2$, in this way
we obtain:
$$\u(D\otimes_kL,\sigma\otimes\bar{\ },\varepsilon)\leqslant
\u(D,\sigma,-\varepsilon)+\frac{1}{2}\u(D,\sigma,\varepsilon).$$ }
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section
{An exact sequence of Witt groups for biquadratic extensions}\label{biquadratiques}
Let  $L=K_1\otimes_kK_2/k$ be a field extension of degree $4$ where
$K_1/k$ and $K_2/k$ are two quadratic extensions with nontrivial automorphisms
$\tau_1$ et $\tau_2$ (resp.). Suppose that
 $K_2=k(\lambda)$ with $\lambda\in K_2$, $\lambda^2\in k$ and
 $\tau_2(\lambda)=-\lambda$.
Every element  $\alpha$ of $L$ can be uniquely written in the form
$\alpha_1\otimes 1+\alpha_2\otimes\lambda$ where 
$\alpha_1,\ \alpha_2\in K_1$. We consider two projections:
\[
\begin{array}{ll}
\pi_1:L\longrightarrow K_1  & \pi_2:L\longrightarrow K_1\\
\alpha\mapsto \alpha_1                &  \alpha\mapsto \alpha_2 
\end{array}
\]
For every nondegenerate hermitian space $(V,h)$ over
$(K_1\otimes_kK_2,\tau_1\otimes\tau_2)$ (resp. over
$(K_1\otimes_kK_2,\tau_1\otimes\id)$), we associate the hermitian space 
$(V,\pi_1h)$ (resp. $(V,\pi_2h)$) over $(K_1,\tau_1)$ defined by
\[
\begin{array}{l}
(\pi_1h)(x,y)=\pi_1(h(x,y)),\ x,\ y\in V\\
(\pi_2h)(x,y)=\pi_2(h(x,y)),\ x,\ y\in V
\end{array}
\]
It is easy to check that 
$\pi_1h$ and $\pi_2h$ are nondegenerate
hermitian forms over $(K_1,\tau_1)$.

For every nondegenerate hermitian space $(W,f)$ over
$(K_1,\tau_1)$, we associate the hermitian spaces
$(W\otimes_kK_2,\rho_1f)$ over $(K_1\otimes_kK_2,\tau_1\otimes\id)$
and $(W\otimes_kK_2,\rho_2f)$ over
$(K_1\otimes_kK_2,\tau_1\otimes\tau_2)$ by
\[
\begin{array}{ll}
(\rho_1f)(x\otimes\alpha,y\otimes\beta)=
f(x,y)\otimes\alpha\beta, & x,\ y\in W;\ \  
\alpha,\ \beta\in K_2\\
(\rho_2f)(x\otimes\alpha,y\otimes\beta)=
f(x,y)\otimes\tau_2(\alpha)\beta, & x,\ y\in W;\ \  
\alpha,\ \beta\in K_2
\end{array}
\]

\begin{prop}{\label{4.1}}
{\rm(1)}
Let $(V,h)$ be a nondegenerate hermitian form of dimension $1$ over
$(K_1\otimes_kK_2,\tau_1\otimes\tau_2)$ with $h\simeq \langle d\rangle$ where
$d\in K_1\otimes_kK_2$ and $(\tau_1\otimes\tau_2)(d)=d$. 
Then $\pi_1h$ is isometric to 
\[
  \left(
  \begin{array}{cc}
  d_1 & d_2\lambda^2 \\
  -d_2\lambda^2 &-d_1\lambda^2
  \end{array}
  \right)
  \]
where $d_1=\pi_1(d)$ and $d_2=\pi_2(d)$.\\
{\rm(2)} 
Let $(V,h)$ be a nondegenerate hermitian space of dimension
$1$ over $(K_1\otimes_kK_2,\tau_1\otimes\id)$ with $h\simeq \langle d\rangle$
where $d\in K_1\otimes_kK_2$ and $(\tau_1\otimes\id)(d)=d$.
Then $\pi_2h$ is isometric to
\[
  \left(
  \begin{array}{cc}
  d_2 & d_1 \\
  d_1 &d_2\lambda^2
  \end{array}
  \right)
  \]
where $d_1=\pi_1(d)$ and $ d_2=\pi_2(d)$.\\
{\rm(3)} For the one dimensional form $f=\langle a\rangle$
over  $(K_1,\tau_1)$ we have
\[
\begin{array}{ll} 
\rho_1(f)\simeq\langle a\otimes1\rangle, &
\rho_2(f)\simeq\langle a\otimes1\rangle.
\end{array}
\]
\end{prop}

\begin{proof}
{\rm(1)}
Let $0\neq x\in V$ with $h(x,x)=d$. We have a basis $\{x,x\lambda\}$ 
for the $K_1$-vector space $V$.
In this basis we have: 
\[
\begin{array}{ll}
(\pi_1h)(x,x)=d_1, 
& (\pi_1h)(x,x\lambda)=d_2\lambda^2\\
(\pi_1h)(x\lambda,x)=-d_2\lambda^2 &
(\pi_1h)(x\lambda,x\lambda)=-d_1\lambda^2
\end{array}
\] 
These relations imply the isometry we are looking for.
The proof of {\rm(2)} is similar and {\rm(3)} is obvious. 
~\qed\end{proof}

\begin{prop}\label{calculs-biquad}
{\rm(1)}
Let $f$ be an anisotropic hermitian form over
$(K_1,\tau_1)$ such that $\rho_1 f$ is isotropic.
Then $f$ contains a subform isometric to  $\pi_1(\langle d\rangle)$ 
where $\langle d\rangle$ is a one dimensional form over $(K_1\otimes K_2,\tau_1\otimes\tau_2)$.\\
{\rm(2)} Let $h$ be an anisotropic hermitian form over 
$(K_1\otimes_kK_2,\tau_1\otimes\id)$ such that
 $\pi_2 h$ is isotropic , then there exists a one dimensional form
$\langle a\rangle$ over $(K_1,\tau_1)$ such that $f$ contains a subform
isometric to $\rho_1(\langle a\rangle)$.\\
{\rm(3)} Let $f$ be an isotropic hermitian form over $(K_1,\tau_1)$
such that $\rho_2 f$ is isotropic. Then  $f$ 
contains a subform isometric to $\pi_2(\langle d\rangle)$ where $\langle d\rangle$ 
is a one dimensional hermitian form over 
$(K_1\otimes_kK_2,\tau_1\otimes\id)$.
\end{prop}

\begin{proof}
{\rm(1)} Let $v=x_1\otimes1+y_1\otimes\lambda\neq 0$ be an isotropic vector
for $\rho_1(f)$, i.e., $\rho_1(f)(v,v)=0$. 
This relation implies that 
\begin{equation}
 \left \lbrace
\begin{array}{l}
f(x_1,x_1)+f(y_1,y_1)\lambda^2=0\\
f(x_1,y_1)+f(y_1,x_1)=0
 \end{array} \right.
\end{equation}
The vectors $x_1$ and $y_1$ are linearly independent over
 $K_1$. 
In fact if $x_1=\alpha y_1$ for some $\alpha\in K_1$, 
the previous system gives
 \[\left \lbrace
\begin{array}{l}
\alpha\tau_1(\alpha)+\lambda^2=0\\
\tau_1(\alpha)+\alpha=0
 \end{array} \right.
\]
which implies that $\alpha^2=\lambda^2$ so
$K_1\simeq K_2$, contradiction because $K_1\otimes_kK_2$ 
is a field. 
Now consider the $K_1$-vector space $W$
generated by $x_1$ and $y_1$.
For $d_1=f(y_1,y_1)$ and $d_2=f(y_1,x_1)\lambda^{-2}$,
the representing matrix of $f|_W$ in the basis
 $\{y_1,x_1\}$ is
\[
  \left(
  \begin{array}{cc}
  d_1 & d_2\lambda^2 \\
  -d_2\lambda^2 &-d_1\lambda^2
  \end{array}
  \right)
  \]
According to Proposition~\ref{4.1}, for a hermitian element 
$d=d_1\otimes1+d_2\otimes\lambda$
with respect to $(\tau_1\otimes\tau_2)$
we have $$\pi_1(\langle d\rangle)\simeq
\left(
  \begin{array}{cc}
  d_1 & d_2\lambda^2 \\
  -d_2\lambda^2 &-d_1\lambda^2
  \end{array}
  \right)$$ 
so $f$ contains a subform isometric to $\pi_1(\langle d\rangle)$.\\
{\rm(2)} Let $x\neq 0$ be an isotropic vector for $\pi_2h$, i.e.,
 $(\pi_2h)(x,x)=0.$
This relation implies that 
$h(x,x)=a\otimes 1\in K_1\otimes_kK_2 $ for some $a\in K_1$.
Since $h$ is anisotropic, we have $a\neq 0$. We deduce then that $h$ 
contains a subform isometric to  $\rho_1(\langle a\rangle)$.\\
{\rm(3)} The argument is similar to that of {\rm(1)}. 
Let $v=x_1\otimes1+y_1\otimes\lambda\neq 0$ be an anisotropic vector
for $\rho_2h$, i.e., $(\rho_2h)(v,v)=0$. 
This relation implies that \[\left \lbrace
\begin{array}{l}
f(x_1,x_1)-f(y_1,y_1)\lambda^2=0\\
f(x_1,y_1)-f(y_1,x_1)=0
 \end{array} \right.
\]
The vectors $x_1$ and $y_1$ are linearly independent over
$K_1$. Let $W$ be the $K_1$-vector space generated by
$x_1$ and $y_1$. 
The matrix of the form $f|_W$ with respect to the basis 
$\{y_1,x_1\}$ is
\[
  \left(
  \begin{array}{cc}
  d_2 & d_1 \\
  d_1 &d_2\lambda^2
  \end{array}
  \right)
  \]  
According to Proposition~\ref{4.1}, for the one dimensional hermitian form 
$\langle d\rangle$ over $(K_1\otimes_kK_2,\tau_1\otimes\id)$ where
$d=d_1\otimes1+d_2\otimes\lambda$ we have:
$$\pi_2(\langle d\rangle)\simeq
 \left(
  \begin{array}{cc}
  d_2 & d_1 \\
  d_1 &d_2\lambda^2
  \end{array}
  \right)$$ which completes the proof.
~\qed\end{proof}

\begin{thm}{\label{suite-biquadratic}}
We have the following exact sequence of Witt groups:\\
$$\begin{array}{c}
W(K_1\otimes_kK_2,\tau_1\otimes\tau_2)
\overset{\pi_1}\rightarrow W(K_1,\tau_1)
\overset{\rho_1}\rightarrow 
W(K_1\otimes_kK_2,\tau_1\otimes\id)
\\\\ 
\overset{\pi_2}\rightarrow W(K_1,\tau_1) 
\overset{\rho_2}\rightarrow 
W(K_1\otimes_kK_2,\tau_1\otimes\tau_2)
\end{array}$$
\end{thm}
\begin{proof}
Thanks to the previous proposition, it is enough to verify that
this sequence is a complex. 
Let $\langle d\rangle$ be a one dimensional hermitian form over 
$(K_1\otimes_kK_2,\tau_1\otimes\tau_2)$ where
$d=d_1\otimes1+d_2\otimes\lambda$
where $d_1,\ d_2\in K_1$. 
According to Proposition~\ref{4.1} we have
$$\pi_1(\langle d\rangle)\simeq
 \left(
  \begin{array}{cc}
  d_1 & d_2\lambda^2 \\
  -d_2\lambda^2 &-d_1\lambda^2
  \end{array}
  \right).$$ We obtain then
$$\rho_1\pi_1(\langle d\rangle)\simeq
 \left(
  \begin{array}{cc}
  d_1\otimes1 & d_2\lambda^2\otimes1 \\
  -d_2\lambda^2\otimes1 &-d_1\lambda^2\otimes1
  \end{array}
  \right)$$
A simple calculation shows that
$$v=\left[
  \begin{array}{l}
1\otimes1\\
1\otimes\lambda^{-1}
  \end{array}
\right]$$ is an isotropic vector for 
$\rho_1\pi_1(\langle d\rangle)$. 
Consequently we have $\rho_1\pi_1=0$. 

In order to show that $\pi_2\rho_1=0$, we consider a one dimensional  form
$\langle a\rangle$ over $(K_1,\tau_1)$. 
We have $\rho_1(\langle a\rangle)\simeq \langle a\otimes 1\rangle$. So 
$$\pi_2\rho_1(\langle a\rangle)\simeq
 \left(
  \begin{array}{cc}
  0 & a \\
  a & 0
  \end{array}
  \right)$$ 
which is a hyperbolic form.

In order to show that  $\rho_2\pi_2=0$, we consider  a one dimensional form
$\langle d\rangle$ over $(K_1\otimes_kK_2,\tau_1\otimes\id)$ where
$d=d_1\otimes1+d_1\otimes\lambda$ is a hermitian element with respect
to  $(\tau_1\otimes\id)$  with $d_1,\ d_2\in K_1$. We have: 
$$\rho_2\pi_2(\langle d\rangle)\simeq
 \left(
  \begin{array}{cc}
  d_2\otimes1 & d_1\otimes1 \\
  d_1\otimes1 &d_2\lambda^2\otimes1
  \end{array}
  \right).$$
This form is hyperbolic because
$$v=\left[
  \begin{array}{l}
1\otimes1\\
1\otimes\lambda^{-1}
  \end{array}
\right]$$
is an isotropic vector.
~\qed\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Going down results}\label{section-descente}
\begin{prop}\label{reel}
Let $L/K$ be a quadratic extension 
and let $\bar\ $ be its nontrivial automorphism.
Then $\u(L)$ and $\u(L,\bar\ )$ are finite if and only if
$\u(K)$ is finite,
moreover we have 
$\u(K)\leqslant 2\u(L,\bar\ )+\u(L)$
and $\u(K)\leqslant \u(L,\bar\ )+2\u(L)$.
\end{prop}
\begin{proof}
We write $L=K(\sqrt{a})$ where $a\in K^*$.
Let $q$ be an anisotropic form of dimension $n$ over $K$. 
Consider the following exact sequence of Witt groups due to Milnor-Husemoller
\cite[Ch.10, 1.2]{scharlau}:
$$0\rightarrow W(L,\bar\ )\overset{\pi}\rightarrow
W(K)\overset{r^*}\rightarrow W(L)$$
In this exact sequence,  $\pi$ is the transfer map induced by the projection
$\pi:L\rightarrow K,\ x+y\sqrt{a}\mapsto x$ and $r^*$ is the restriction map.
If $r^*(q)$ is anisotropic, then $q$ contains a subform isometric to 
$\langle b,-ab\rangle$ for some $b\in K$. 
We can then write (by induction) 
$$q\simeq(\langle 1,-a\rangle\otimes q_1)\oplus q_2$$ where $q_1$, $q_2$ are two 
nondegenerate quadratic forms over $K$ 
so that $r^*(q_2)$ is anisotropic.
Let $\varphi$ be the hermitian form over $(L,\bar\ )$
induced by $q_1$. We have $\pi(\varphi)\simeq\langle 1,-a\rangle\otimes q_1$.
Therefore we have a Witt $r^*$-decomposition, i.e., 
there exists an orthogonal decomposition 
$q\simeq q'\perp q''$ where $q'\simeq\pi(\varphi)$ 
for some nondegenerate hermitian form
$\varphi$ over $(L,\bar\ )$ (in particular $r^*(q')$ is hyperbolic) and 
$r^*(q'')$ is anisotropic.
We may suppose that $\varphi$ is anisotropic, therefore:
$\dim(q)=\dim(\pi(\varphi))+\dim(q'')=2\dim(\varphi)+\dim(q'')
\leqslant 2\u(L,\bar\ )+\u(L)$ which implies the result. 
If we use the same argument with the following exact sequence of Witt groups
due to Lewis \cite{lewis2}:
$$W(L)\overset{s_*}\rightarrow
W(K)\overset{r^*}\rightarrow W(L,\bar\ ),$$
we obtain $\u(K)\leqslant \u(L,\bar\ )+2\u(L)$.
In this exact sequence, $s_*$ is the Scharlau transfer map and
$r^*$ is the restriction map.

Now suppose that we have $\u(K)<\infty$. According to a result due to Elman and Lam
(cf.~\cite[theorem~4.3]{elman-lam}) we have
$\u(L)\leqslant \frac{3}{2}\u(K)<\infty$. The finiteness of $\u(L,\bar\ )$ is
easy to check. In fact if $\varphi=\langle a_1,\cdots,a_n\rangle$ is a form over $(L,\bar\ )$
  where $a_i\in K$, its trace form is isometric to 
  $\psi=\langle a_1,\cdots,a_n,-aa_1,\cdots,-aa_n\rangle$. 
Isotropy of  $\varphi$ and $\psi$ are equivalent. The finiteness of  $\u(K)$ 
implies therefore that of $\u(L,\bar\ )$.
~\qed\end{proof}

\begin{remark}{\rm
One can regard the estimates of Proposition~\ref{reel} as an improvement, in some situations,
of the estimate $\u(K)\leqslant 4\u(L)$ for a non formally real field $K$ due 
to Elman 
\cite[Theorem~3.1(iii)]{elman}, because one can find many examples where
$$\min\{2\u(L,\bar\ )+\u(L)\ ,\ \u(L,\bar\ )+2\u(L) \}<4\u(L).$$ 
For example if $K$ is a $p$-adic field then $\u(K)=\u(L)=4$ and
$\u(L,\bar\ )\leqslant 2$. 
Our bound is independent of the formally real nature of $L$ or $K$. To our best
knowledge, the going down result of Elman, is the best one for 
arbitrary fields.
}\end{remark}

In the sequel, we need the following lemma which is an
immediate consequence of a theorem of Springer. 
\begin{lemme}
Let $L/K$ be a quadratic extension and let $\tau$ be its 
nontrivial automorphism. Let $M/K$ be an extension of odd degree
and let $\varphi$ be an anisotropic hermitian form over 
$(L,\tau)$. Then $\varphi$ remains anisotropic over 
$(L\otimes_KM,\tau\otimes\id )$. 
\end{lemme}
\begin{proof}
For a hermitian space $(V,\varphi)$ over $(L,\tau)$,
we denote its trace form by  $tr(\varphi)$, defined by 
$tr(\varphi)(x,y)=tr_{L/K}(\varphi(x,y))$ for every $x,\ y\in V$.
We have $tr(\varphi|_{(L\otimes_KM)})\simeq tr(\varphi)|_M$.
If $\varphi|_{(L\otimes_KM)}$ is isotropic, then  
$tr(\varphi)|_M$ is also isotropic. The strong version of Springer's theorem
implies that $tr(\varphi)$ is isotropic. Consequently $\varphi$ is isotropic.
~\qed\end{proof}

\begin{cor}{\label{u-invariant-odd-springer}}
With the notation of the previous lemma we have: 
$$\u(L,\tau)\leqslant\u(L\otimes_KM,\tau\otimes\id).$$
\end{cor}

Let $D$ be a division algebra over a field $K$ 
with an involution $\sigma$. Suppose that 
there exist 
$\lambda,\ \mu\in D^*$ such that $\sigma(\lambda)=-\lambda$,
$\sigma(\mu)=-\mu$, $\lambda\mu=-\mu\lambda$ and $L=K(\lambda)$ is
a quadratic extension of $K$. In this situation,
we have the following exact sequence of Witt groups due to
Parimala, Sridharan and Suresh \cite[Appendix 2]{bayer-parimala}:
\begin{equation}{\label{pss}}W^\varepsilon(D,\sigma)\overset{\pi_1^{\ve}}{\rightarrow}W^\varepsilon(\wt{D},\sigma_1)\overset{\rho_1^{\ve}}
   {\rightarrow}W^{-\varepsilon}(D,\sigma)
\overset{\pi_2^{-\ve}}{\rightarrow}W^\varepsilon(\wt{D},\sigma_2)
\end{equation}
In this sequence, the map $\pi_1^{\ve}$ and $\pi_2^{-\ve}$  
are transfers induced by the projections
$\pi_1$ 
and $\pi_2$ defined in 
\S\ref{sec-going-up}.
The map $\rho_1^{\ve}$ is a restriction map defined by the multiplication by
$\lambda$ and the usual restriction map $r^*$:
$$\xymatrix{
W^\varepsilon(\wt{D},\sigma_1)\ar[rr]^{\rho_1}\ar[dr]^{\times\lambda}&&
W^{-\varepsilon}(D,\sigma)\\
&W^{-\varepsilon}(\wt{D},\sigma_1)\ar[ur]^{r^*}&
}$$
See \cite[Appendix~2]{bayer-parimala} for more details. 
These maps also induce homomorphisms between semigroups of isometry 
classes of hermitian forms: 

\begin{equation}{\label{pss-like}}S^\varepsilon(D,\sigma)\overset{\pi_1^{\ve}}{\rightarrow}S^\varepsilon(\wt{D},\sigma_1)\overset{\rho_1^{\ve}}
   {\rightarrow}S^{-\varepsilon}(D,\sigma)
\overset{\pi_2^{-\ve}}{\rightarrow}S^\varepsilon(\wt{D},\sigma_2)
\end{equation}

Now we can reformulate the exact sequence of (\ref{pss}) in the following way:

\begin{prop}{\label{rho1-decomp}}
Let $\varphi\in S^{\ve}(\wt{D},\sigma_1)$ be an anisotropic form. Then
$\varphi$ has a Witt $\rho_1^{\ve}$-decomposition, i.e., there exists
an orthogonal decomposition $\varphi\simeq\varphi_1\oplus\varphi_2$ such that
$\rho_1^\ve(\varphi_1)$ is hyperbolic and $\rho_1^{\ve}(\varphi_2)$ is 
anisotropic. Moreover there exists $\psi\in S^{\ve}(D,\sigma)$ such that
$\varphi_1\simeq\pi_1^{\ve}(\psi)$.
\end{prop} 

\begin{proof}
If $\rho_1^{\ve}(\varphi)$ is anisotropic, we take $\varphi_2=\varphi$. If
$\rho_1^{\ve}(\varphi)$ is isotropic then $\varphi$ contains a subform
$\varphi_0$ which comes from $S^{\ve}(D,\sigma)$, i.e., 
$\varphi_0\simeq\pi_1^{\ve}(\psi_0)$ for some $\psi_0\in S^{\ve}(D,\sigma)$;
see the proof of (\ref{pss}) in \cite[Appendix 2]{bayer-parimala}
where this has been implicitly proved, 
see also \cite[4.4]{suite}.
We have then an orthogonal decomposition $\varphi\simeq\varphi_0\oplus\varphi'$ for some
$\varphi'\in  S^{\ve}(\wt{D},\sigma_1)$. 
As $\rho_1^{\ve}(\varphi_0)$ is hyperbolic and $\dim \varphi'<\dim \varphi$
we can use induction on $\dim\varphi$ to finish the proof.
 ~\qed\end{proof}

We have also the following exact sequence of Witt groups:
\begin{equation}{\label{suite-3-terme}}
   W^{-\varepsilon}(D,\sigma)\overset{\pi_2^{-\ve}}{\rightarrow}
W^\varepsilon(\wt{D},\sigma_2)\overset{\rho_2^{\ve}}
   {\rightarrow}W^{-\varepsilon}(D,\sigma)
 \end{equation}
which is a variation of the exact sequence of Parimala, Sridharan and Suresh, 
see \cite{suite}. 
In this sequence, $\rho_2^{\ve}$ is the composition of the multiplication by 
$-\lambda$ and the usual restriction map $r^*$ and the multiplication by 
$-\mu$:
$$
\xymatrix{
W^\varepsilon(\wt{D},\sigma_2)\ar[r]^{\rho_2}\ar[d]^{-\lambda}
& W^{-\varepsilon}(D,\sigma)\\
W^\varepsilon(\wt{D},\sigma_2)\ar[r]_{r^*}& 
W^{\varepsilon}(D,\intt(\mu^{-1})\circ\sigma)\ar[u]_{\times\mu}
}
$$
We state a Witt-decomposition-like result for the exact sequence of
(\ref{suite-3-terme}):

\begin{prop}\label{rho2-decomp}
Let $\varphi\in S^{\ve}(\wt{D},\sigma_2)$ be an anisotropic form. Then
$\varphi$ has a Witt $\rho_2^{\ve}$-decomposition, i.e., there exists
an orthogonal decomposition $\varphi\simeq\varphi_1\oplus\varphi_2$ such that
$\rho_2^\ve(\varphi_1)$ is hyperbolic and $\rho_2^{\ve}(\varphi_2)$ is 
anisotropic. Moreover there exists $\psi\in S^{-\ve}(D,\sigma)$ such that
$\varphi_1\simeq\pi_2^{-\ve}(\psi)$.
\end{prop}

\begin{proof}
We use the same argument as in Proposition~\ref{rho1-decomp}.
 If
$\rho_2^{\ve}(\varphi)$ is isotropic then $\varphi$ contains a subform
$\varphi_0$ which comes from $S^{-\ve}(D,\sigma)$, i.e., 
$\varphi_0\simeq\pi_2^{\ve}(\psi_0)$ for some $\psi_0\in S^{-\ve}(D,\sigma)$;
see  \cite[4.4]{suite}.
~\qed\end{proof}

\begin{prop}\label{down}
Let $D$, $\wt{D}$, $\sigma_1$, $\sigma$ 
and $\sigma_2$ 
  be as in Proposition~\ref{ppp}, then we have:\\
{\rm(1)}
  $\u(\wt{D},\sigma_1,\varepsilon)\leqslant \u(D,\sigma,-\varepsilon)
  +2\u(D,\sigma,\varepsilon).$\\
 {\rm(2)} $\u(\wt{D},\sigma_2,\ve)\leqslant
3\u(D,\sigma,-\ve)$.
 \end{prop}
 
\begin{proof} {\rm(1)} 
  Let  $\varphi$ be an anisotropic $\varepsilon$-hermitian form over 
 $(\wt{D},\sigma_1)$. 
According to Proposition~\ref{rho1-decomp}, there exists an 
  $\varepsilon$-hermitian form $\psi$ over $(D,\sigma)$ such that 
  $\varphi\simeq\pi_1(\psi)\oplus\varphi'$ for some form 
  $\varphi'$ over $(\wt{D},\sigma_1)$ such that $\rho(\varphi')$  is 
anisotropic.
As $\varphi$ is anisotropic, so is  $\psi$, therefore
  $\dim(\psi)\leqslant \u(D,\sigma,\varepsilon).$  As 
  $\rho(\varphi')$ is anisotropic, $\dim(\varphi')=\dim\rho(\varphi')
  \leqslant \u(D,\sigma,-\varepsilon)$, consequently
  $\dim(\varphi)=\dim(\pi_1(\psi))+\dim(\varphi')=
2\dim(\psi)+\dim(\varphi')
  \leqslant 2\u(D,\sigma,\varepsilon)+\u(D,\sigma,-\varepsilon)$ . 

For {\rm(2)} we 
apply the same argument by using Proposition~\ref{rho2-decomp}.
~\qed\end{proof}

\begin{cor}\label{lbound}
With the notation of Corollary~\ref{principal} we have:\\
$(1)$ $\u(L,\bar\ )\leqslant 2\u(Q,\bar\ )+\u(Q,\hat{\ })=2\u^-(Q)+\u^+(Q)$\\
$(2)$ $\u(L)\leqslant 3\u(Q,\hat{\ })=3\u^+(Q)$
\end{cor}

\begin{prop}{\label{biquad-decomp}}
With the notation of \S\ref{biquadratiques}, let 
$\varphi\in S(K_1,\tau_1)$ be an aniso\-tropic form.
Then $\varphi$ has a Witt $\rho_1$-decomposition, i.e., there exists
an orthogonal decomposition $\varphi\simeq\varphi_1\oplus\varphi_2$ such that
$\rho_1(\varphi_1)$ is hyperbolic and $\rho_1(\varphi_2)$ is 
anisotropic. Moreover there exists 
$\psi\in  S(K_1\otimes_kK_2,\tau_1\otimes\tau_2)$ such that
$\varphi_1\simeq\pi_1(\psi)$.
\end{prop}

\begin{proof}
If $\rho_1(\varphi)$ is isotropic then according to 
Proposition~\ref{calculs-biquad},
$\varphi$ contains a subform
$\varphi_0$ which comes from $S(K_1\otimes_kK_2,\tau_1\otimes\tau_2) $, i.e., 
$\varphi_0\simeq\pi_1(\psi_0)$ for some 
$\psi_0\in S(K_1\otimes_kK_2,\tau_1\otimes\tau_2) $, we can then use an
induction argument as in Proposition~\ref{rho1-decomp}
~\qed\end{proof}

\begin{prop}{\label{u-invariant-biquadratic}}
With the notation of Theorem~\ref{suite-biquadratic}, we have 
$$\u(K_1,\tau_1)\leqslant 2\u(K_1\otimes_kK_2,\tau_1\otimes\id)
+\u(K_1\otimes_kK_2,\tau_1\otimes\tau_2).$$ 
\end{prop}

\begin{proof}
The proof is similar to that of Proposition~\ref{down};
we use Proposition~\ref{biquad-decomp}.
~\qed\end{proof}


\begin{remark}{\rm
Using \cite[4.4]{suite}, and Proposition~\ref{calculs-biquad}, one 
can state similar Witt-decomposition-like results for other maps
involved in these exact sequences. For example 
by a Witt $\pi_1^{\ve}$-decomposition result one can give an 
alternative proof of Proposition~\ref{ppp}.}
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A Finiteness result}
In the article \cite{bayer-parimala}, one can find
several useful results about extensions of odd degree and their
connections with substructures of codimension $2$ of central simple
algebras with involution, see 
(\cite{bayer-parimala}, Lemma 3.1.1, Lemma 3.3.1, Lemma 3.3.2, Lemma 3.3.3).
%%% FROM PROBLEM:
%>From these results we can derive the following proposition which has
From these results we can derive the following proposition which has
been proved in \cite{torsion}.  

\begin{prop}{\label{odd}}
Let $D$ be a noncommutative $K$-division algebra and $\sigma$ a
$K/k$-involution on $D$. Suppose that the degree of $D$ is a $2$-power.
Then there
exists an extension $M/k$ of odd degree such that $D_M=D\otimes_kM$ contains
the elements $\lambda$, $\mu$ such that $\tau(\lambda)=-\lambda$ and
$\tau(\mu)=-\mu$ and $\lambda\mu=-\mu\lambda$ and $[F(\lambda):F]=2$ where
$F=KM=K\otimes_kM$ and $\tau$ is the involution $\sigma\otimes\id$ when
 $\sigma$ is of second
kind or $\sigma$ is symplectic and
$D$ is a quaternion algebra otherwise $\tau=\intt(\mu)\circ(\sigma\otimes\id)$.
\end{prop}

\begin{thm}\label{nonreal}
 Let $D$ be a division algebra of dimension
a power of $2$ over its centre $K$ with a $K/k$-involution $\sigma$. 
Suppose that $\u(D,\sigma,\varepsilon)<\infty$
   for $\varepsilon=1$ 
and $\varepsilon=-1$, 
then $\u(K,\sigma|_K)<\infty.$ 
In particular if both $\u^+(D)$ and $\u^-(D)$ are finite then $\u(K)$
is finite too.
 \end{thm}

 \begin{proof}
  We prove this result by induction on  $\dim_K(D)$. 
For $\dim_K(D)=1$ the conclusion is evident.
Suppose that $\dim_K(D)>1$.
 
First suppose that  $\sigma$ is of the second kind. According to 
Proposition~\ref{odd}, there exists an extension $M/k$ of odd degree such that
$D_M=D\otimes_kM$ contains the elements $\mu$ and $\lambda$ such that
 $\tau(\lambda)=-\lambda$, $\tau(\mu)=-\mu$ and 
$\lambda\mu=-\mu\lambda$
 for $\tau=\sigma\otimes \id$ and $F(\lambda)/F$ is a quadratic extension 
where $F=K\otimes_kM$.
As $M/k$ is an extension of odd degree, $E=D_M$ is a division algebra.
According to Proposition~\ref{tensori} we have $\u(E,\tau,\ve)<\infty$ and
 $\u(E,\tau,-\ve)<\infty$. 
By applying Proposition~\ref{down} we obtain:  
$\u(\wt{E},\tau_1,\pm\ve)<\infty$
and $\u(\wt{E},\tau_2,\pm\ve)<\infty$ where $\wt{E}=C_E(F(\lambda))$.
As  $\tau$ is of the second kind,
so are $\tau_1$ and $\tau_2$. By the induction hypothesis, we have
$\u(L,\tau_1|_L)<\infty$ and
$\u(L,\tau_2|_L)<\infty$ where $L=F(\lambda)$
is the centre of $\wt{E}$. Let $F'$ be the fixed field of $\tau_2|_L$.
We have
\[
\begin{array}{c}
(L,\tau_1|_L)\simeq(F\otimes_MF',\tau|_F\otimes\tau')\\
(L,\tau_2|_L)\simeq(F\otimes_MF',\tau|_F\otimes\id)
\end{array}\]
where $\tau'$ is the nontrivial automorphism of $F'/M$.
Proposition~\ref {u-invariant-biquadratic}
implies that $\u(F,\tau|_F)<\infty$ and from 
Corollary~\ref{u-invariant-odd-springer} we deduce that
$\u(K,\sigma|_K)\leqslant\u(F,\tau|_F)<\infty$.

Now consider the case where $\sigma$ is of the first kind.
If  $D$ is a quaternion algebra and $\sigma$ is its canonical involution, then
there exist $\lambda,\ \mu\in D$  with $\sigma(\lambda)=-\lambda$,
$\sigma(\mu)=-\mu$, $\lambda\mu=-\mu\lambda$
et $[K(\lambda):K]=2$. 

Otherwise there exists an extension $M/K$ of odd degree 
such that  $E=D\otimes_KM$ contains $\lambda$ and $\mu$ with  
$\tau(\lambda)=-\lambda$, $\tau(\mu)=-\mu$, $\lambda\mu=-\mu\lambda$ and
$[M(\lambda):M]=2$ where 
$\tau=\intt(\mu)\circ(\sigma\otimes\id)$ 
(cf. Proposition~\ref{odd}).
Them involution $\tau$ is of the first kind but of a different 
type from that of $\sigma$. In any case take:
 \[\tau=\left \lbrace
\begin{array}{ll}
\sigma & \dim_K(D)=4,\  \sigma \ \rm{symplectic} \\
\intt(\mu)\circ(\sigma\otimes\id) & {\rm otherwise}
 \end{array} \right.
\]
As $M/K$ is an extension of odd degree, $E$ is a division algebra.
According to Proposition~\ref{tensori} we have $\u(E,\tau,\ve)<\infty$ and
 $\u(E,\tau,-\ve)<\infty$ (note that according to Proposition \ref{sadeh},
the finiteness hypothesis (which is for $\sigma$) is still valid for $\tau$). 
By applying Proposition~\ref{down} we obtain:  
$\u(\wt{E},\tau_1,\pm\ve)<\infty$
and $\u(\wt{E},\tau_2,\pm\ve)<\infty$ where $\wt{E}=C_E(M(\lambda))$. The 
involution $\tau_1$ is unitary and by the first part of the proof the condition 
$\u(\wt{E},\tau_1,\pm\ve)<\infty$ implies that  
 $\u(L,\tau_1|_L)<\infty$ where $L=M(\lambda)$. 
The involution $\tau_2$ is of the first kind. 
The condition  $\u(\wt{E},\tau_2,\pm\ve)<\infty$ states
that both $\u^-(\wt{E})$ and $\u^+(\wt{E})$ are finite. So we conclude 
by induction that $\u(L)=\u(L,\tau_2|_L)<\infty$.
Here $\tau_1|_L$ is the
nontrivial automorphism of  $L/M$. Now  Proposition~\ref{reel} implies that
$\u(M)<\infty$ and from the strong version of Springer's theorem 
we deduce that $\u(K)\leqslant\u(M)<\infty$.
 ~\qed\end{proof}

\begin{remark}{\rm
Note that in preceding theorem, according to Proposition \ref{sadeh} one of the two hypotheses 
$\u(D,\sigma,\ve)<\infty$ and
$\u(D,\sigma,-\ve)<\infty$ is actually enough in the unitary case. }
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Values of the $\u$-invariant, a particular case}\label{valeurs}
Let  $L/K$ be a quadratic extension and let $\bar\ $ be its nontrivial automorphism.
The signed discriminant $\d$ defines a map from $W(L,\bar\ )$ 
to $K^*/N(L^*)$. Unfortunately this map is not a homomorphism.
But for forms $\varphi$ and $\psi$ of even dimension we have
$\d(\varphi\oplus\psi)=\d(\varphi).\d(\psi)$.
Let $I(L,\bar\ )\subset W(L,\bar\ )$ be the classes of all nondegenerate hermitian 
forms over  $(L,\bar\ )$. The group $W(L,\bar\ )$ has a natural ring structure.

\begin{prop}\label{I/I^2} 
\rmm{1}
The map $e_1:I(L,\bar\ )\rightarrow K^*/N(L^*)$
defined by $e_1(\varphi)=\d(\varphi)$
is a surjective homomorphism.\\
\rmm{2} $\ker e_1=I^2(L,\bar\ )$.\\
\rmm{3} Via  $e_1$, the group $I/I^2$ is isomorphic to $K^*/N(L^*)$.
\end{prop}

\begin{proof}
(1) The map $e_1$ is surjective because 
$\d(\langle 1,-a\rangle)=aN(L^*)$.\\
(2) The group $I=I(L,\bar\ )$ is generated by hermitian forms $\langle a,b\rangle$ where
 $a,\ b\in K^*$.
Thus $I^2$ is generated by the hermitian  forms:\\
$$\varphi=\langle a,b\rangle\otimes\langle c,d\rangle=\langle ac,ad,bc,bd\rangle$$
We have $\d(\varphi)=1$. So  $I^2\subset\ker e_1.$
Conversely suppose that $\varphi\in I$ with $e_1(\varphi)=1$. 
The form  $\varphi$ is represented by
$\langle a_1,\cdots,a_{2n}\rangle$ 
with $n\geqslant 1$. 

For $n=1$, we have $\varphi=\langle a_1,a_2\rangle$ with 
$-a_1a_2\in N(L^*)$. So $\varphi\simeq\langle a_1,-a_1\rangle$, therefore
$\varphi$ is hyperbolic and $\varphi=0$ in  $W(L,\bar\ )$.    

Now suppose that $n\geq 2$. We can write:
$\varphi=\langle a_1,a_2,a_3\rangle\oplus\langle a_4,\cdots,a_{2n}\rangle$. So
$\varphi\sim
\langle a_1,a_2,a_3,a_1a_2a_3\rangle\oplus\langle -a_1a_2a_3,a_4,\cdots,a_{2n}\rangle.$
We have  $$\langle a_1,a_2,a_3,a_1a_2a_3\rangle\simeq \langle a_1,a_2\rangle\otimes \langle 1,a_1a_3\rangle\in
I^2.$$
 The dimension of $\varphi'=\langle -a_1a_2a_3,a_4,\cdots,a_{2n}\rangle$
is $2(n-1)$ and $\d\varphi'=1$. By induction we obtain $\varphi'\in I^2$.\\
(3) is deduced from (1) and (2).
~\qed\end{proof}

Let $D$ be a division algebra over a field $K$ with a $K/k$-involution
$\tau$. Here we call a hermitian form $\varphi$ over $(D,\tau)$ 
an $n$-fold Pfister form if 
$\varphi$ is the restriction of an $n$-fold Pfister form
$q$ over $k$ to  $D$. This notion appears in
\cite{lewis88} for quaternion algebras.
 A hermitian form $\varphi$ induced by the
$n$-fold Pfister form $q=\langle \langle a_1,\cdots,a_n\rangle\rangle$, is still denoted by 
$\varphi=\langle \langle a_1,\cdots,a_n\rangle\rangle$. 

\begin{dff}\label{universelle}{\rm
Let $D$ be a division algebra over a field $K$ with a $K/k$-involution $\tau$ and
let $\ve$ be  an element of $K$ with $\ve\tau(\ve)=1$.
An $\ve$-hermitian form $\varphi$ over $(D,\tau)$ is called {\it universal} if 
$\varphi$ represents all nonzero $\ve$-hermitian elements of $D$.}
\end{dff}

\begin{prop}{\label{357}}
Let $L/K$ be a quadratic extension and let $\bar\ $ be the nontrivial automorphism
of  $L/K$. Then we have $\u(L,\bar\ )\neq 3,5,7$.
\end{prop}

\begin{proof}
Suppose that  $\u(L,\bar\ )<4$. Every $2$-fold hermitian Pfister form
 $\langle \langle a,b\rangle\rangle$ is hyperbolic. The hermitian form $\langle 1,a,b\rangle$ is a hermitian
neighbor of $\langle \langle a,b\rangle\rangle$  and therefore it is isotropic.
We deduce that every hermitian form $\langle a,b,c\rangle$ of dimension $3$ over 
$(L,\bar\ )$ is isotropic, so we have $\u(L,\bar\ )\leqslant 2$.

Suppose that $\u(L,\bar\ )<8$. We conclude that every  $3$-fold hermitian Pfister
form $\langle \langle a,b,-c\rangle\rangle$ is hyperbolic. Thus for every $a,\ b,\ c\in K^*$ we have:
$$\langle \langle a,b\rangle\rangle\simeq c\langle \langle a,b\rangle\rangle$$
Every form in $I^2=I^2(L,\bar\ )$ is an orthogonal sum of the forms
 $\langle \langle a_i,b_i\rangle\rangle$.
As for every hyperbolic plane $\H$ and  $c\in K^*$  we have $\H\simeq c\H$,
the Witt cancellation  theorem implies that $\varphi\simeq c\varphi$ for every
 $\varphi\in I^2$.
In particular $\varphi$ is universal  over
 $(L,\bar\ )$ (in the sense of 
Definition~\ref{universelle}).

Now suppose that  $u=\u(L,\bar\ )=5$ or $7$. Let $\varphi$ be an anisotropic
 hermitian form of
dimension $u$. In particular $\varphi$ represents its discriminant 
$d=\d(\varphi)$. We have then $\varphi=\psi\oplus \langle d\rangle$
where $\psi$ is a form of dimension $4$ or $6$ and $\d(\psi)=1$. 
As $\psi\in I$, Proposition~\ref{I/I^2} implies that $\psi\in I^2$. 
We have already shown that $\psi$ is universal, consequently
$\varphi$ is isotropic which is a contradiction to the choice of $\varphi$.~  
~\qed\end{proof}

\begin{remark}{\rm 
In a similar way, if $D=(a,b)_K$ is a quaternion division algebra
with the canonical involution $\bar\ $, then $\u(D,\bar\ )\neq 3,5,7$.
However the value $3$ for the $\u$-invariant $\u(D,\bar\ ,-1)$ is possible,
for example if $K$ is a $p$-adic field then $\u(D,\bar\ ,-1)=3$.
}
\end{remark}

%\begin{question}
%Let  $D=(a,b)_K$ be a quaternion division algebra over a field 
%$K$ and let $\bar\ $ be the canonical involution of $D$. 
%What are the possible values of $\u(D,\bar\ ,-1)$?
%\end{question}


\begin{prop}\label{u=pair}
Let $L_0$ be a field with $\u(L_0)=n$. Let $L=L_0((x))$ be the field of
Laurent series over $L_0$ and let $\sigma$ be the $L_0$-automorphism 
of $L$ induced by $x\mapsto -x$. Then $\u(L,\sigma)=n$.
\end{prop}

\begin{proof}
Let $K$ be the fixed field of $\sigma$. We have  $K=L_0((x^2))$ so
$\u(K)=2n$. Consequently $\u(L,\sigma)\leqslant n$
(cf. Remark~\ref{rem-u-inv-quad}).
Let $q=\langle a_1,\cdots,a_n\rangle$  be an anisotropic quadratic form of dimension \
$n$ over $L_0$. 
The restriction of $q$ to $L$ is anisotropic. In fact the isotropy of 
$q|_{(L,\sigma)}$ is equivalent to that of the quadratic form 
$q\oplus x^2 q=\langle a_1,\cdots,a_n,-x^2a_1,\cdots,-x^2a_n\rangle$ over $K$. 
The anisotropy of this form is equivalent to that of $q$ over $L_0$.
~\qed\end{proof}

\begin{remark}{\rm 
Proposition~\ref{u=pair} state in particular that
the possible values for the hermitian $\u$-invariant of  
commutative fields contain the possible values
of the usual $\u$-invariant.
}
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Classical groups over $\qp(t)$, $p\neq 2$ 
} \label{sec-an}
We refer to \cite{tits} and \cite{springer} for basic notions about
Tits's indices.
In the symbol $^gX_{n,r}^t$, where $X=A,\ B,\ C,\ D$, the integers $n$ and $r$
are respectively the absolute and relative rank of the considered
classical group $G$, $g$ denotes the order of the quotient of the Galois 
group $\Gamma=Gal(k^{sep}/k)$ which operates effectively on the
Dynkin diagram. In case the diagram has no nontrivial automorphism, $g$ is
necessarily $1$. If $g=1$, $G$ is called of {\it inner} type, otherwise
$G$ is called of {\it outer} type. The integer $t$ is the degree of 
a certain division algebra which occurs in the definition of the considered group.   
If $t$ or $g$ are omitted in the symbol, they are necessarily $1$.
\\

{\centerline {\bf Type $A_n$}}
\begin{lemme}\label{ulbar4}
Let $k$ be a function field of a $p$-adic field with 
$p\neq 2$. Let $L/k$ be a quadratic extension and let $\bar\ $
be the nontrivial automorphism of $L/k$. Then $\u(L,\bar\ )\leqslant 4$.
\end{lemme}

\begin{proof}
Let $\varphi\simeq\langle a_1,\cdots,a_5\rangle$ be a hermitian form of dimension $5$ 
over $(L,\bar\ )$, where $a_i\in k$, $i=1,\cdots,5$. Then, $\varphi$ is isotropic
if and only if the quadratic form 
$q=\langle a_1,\cdots,a_5,-aa_1,\cdots,-aa_5\rangle$ is isotropic over $k$
where $L=k(\sqrt{a})$. 
But this form is isotropic over $k$, because its Hasse invariant is
$c(q)=(-a,d)$ where $d=\det(\varphi)$, which has index $\leqslant 2$ in the
Brauer group of $k$ and according to the Theorem~4.6 of \cite{parimala-suresh}, $q$ is isotropic.
We have in particular  $\u(L,\bar\ )\leqslant 4$.\qed
\end{proof}

\begin{prop}
Let $k=\qp(t)$ with $p\neq 2$.  The index $^1A_{n,r}^{(d)}$
occurs over $k$ for every positive integers $d$, $r$ and
$n$ satisfying $rd=n+1$.
\end{prop}

\begin{proof}
According to \cite{tits} or \cite[17.1.3]{springer}, 
  $^1A_{n,r}^{(d)}$ occurs over $k$ if and only if there exists a
division algebra $D$ over  $k$ of degree $d$ with $rd=n+1$. 
The existence of such algebras comes from the fact that 
there exist division algebras
of arbitrary degree over $\qp$. 
~\qed\end{proof}

\begin{prop} Let $k=\qp(t)$ with $p\neq 2$.
\begin{itemize}
\item[\rmm{1}] If the index $^2A_{n,r}^{(1)}$ occurs over $k$, 
then $n+1-2r\in\{0,1,2,3,4\}$.
All the anisotropic indices $^2A_{1,0}^{(1)}$, $^2A_{2,0}^{(1)}$
$^2A_{3,0}^{(1)}$ and  $^2A_{4,0}^{(1)}$ occur over $k$.
\item[\rmm{2}] If the index  $^2A_{n,r}^{(2)}$ occurs over $k$ then
$ n+1-4r\in\{0,2,4,6\}$.
\end{itemize}
\end{prop}

\begin{proof}
According to \cite{tits} or \cite[17.1.6]{springer}, 
$^2A_{n,r}^{(d)}$ occurs over 
$k$ if and only if there exist a quadratic extension
$E/k$, a division algebra $D$ over $E$ and an involution $\sigma$ on 
$D$ of the second kind such that $k$ is the fixed field of $\sigma|_E$ and
a nondegenerate hermitian form $h$ over $(D,\sigma)$ of dimension
 $d^{-1}(n+1)$ and of Witt index $r$.

(1) For $d=1$, $D=E$. 
According to Lemma~\ref{ulbar4} we have $\u(E,\sigma)\leqslant 4$.
We conclude that  
$0\leqslant n+1-2r\leqslant 4$. As for $E=\qp(t)(\sqrt{t})$, we have
$\u(E,\sigma)=4$, the indices  $^2A_{1,0}^{(1)}$, $^2A_{2,0}^{(1)}$
$^2A_{3,0}^{(1)}$ and $^2A_{4,0}^{(1)}$ occur over $k$.

(2) For $d=2$, $D$ is a quaternion algebra over $E$. 
Thank to Proposition~\ref{bound} we have $\u(D,\sigma)\leqslant 3$. We have then 
$0\leqslant \frac{n+1}{2}-2r\leqslant 3$.
~\qed\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\centerline{\bf Type $B_n$}}
\begin{prop}\label{bn-qpt}
Let $k=\qp(t)$ with $p\neq 2$. If the index $B_{n,r}$ occurs over
$k$ then we have:
$n-4\leqslant r\leqslant n$. 
\end{prop}

\begin{proof}
According to \cite{tits} or \cite[17.2.3]{springer},  $B_{n,r}$
can occur over $k$ if and only if there exists a nondegenerate 
quadratic form $q$ of dimension $2n+1$ and of Witt index $r$. 
Let $q_a$ be the unique anisotropic part of $q$ up to isometry.
We have:
\begin{equation}\label{2n+1-2r}
\dim q_a=2n+1-2r
\end{equation}
According to \cite{parimala-suresh}, the dimension of $q_a$
cannot exceed $10$. We obtain then  $0\leqslant 2n-2r+1\leqslant 10$
and so $n-4\leqslant r\leqslant n$.
~\qed\end{proof}

\begin{prop}
The anisotropic indices  $B_{1,0}$, $B_{2,0}$ and $B_{3,0}$ can occur over
$k=\qp(t)$ with $p\neq 2$. 
\end{prop}

\begin{proof}
According to (\ref{2n+1-2r}), it is enough to find anisotropic quadratic forms
of dimension $3,\ 5$ and $7$ (resp.), which is possible because
$\u(\qp(t))\geqslant 8$.
~\qed\end{proof}

\begin{remark}{\rm
According to (\ref{2n+1-2r}), the existence of  $B_{4,0}$
over $\qp(t)$ is equivalent to 
$\u(\qp(t)) \geqslant 9$.   
According to a conjecture, one believes that
$\u(\qp(t))=8$ (cf. \cite[Chapter 5, 2.5]{pfister}, \cite{hoffmann-vangeel}
and \cite{parimala-suresh}).}
\end{remark}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\centerline{\bf Type $C_n$}}
\begin{prop}
Let $k=\qp(t)$ with $p\neq 2$. If the index $C_{n,r}^{(d)}$ occurs over $k$
then  we have  $d\in\{1,2,4\}$. For $d=1$ we have
$n=r$. For $d=2$ we have $n-2r\in\{0,1,2\}$. For $d=4$, we have
$n-4r\in\{0,2,4,6,8,10\}$.
\end{prop}

\begin{proof}
According to \cite{tits} or \cite[17.2.10]{springer}, 
$C_{n,r}^{(d)}$ can occur over $k$ if and only if there exists a division
algebra $D$ over $k$ of degree $d$ with an orthogonal 
involution $\sigma$ and a skew hermitian form $h$ over $(D,\sigma)$ 
of dimension $2d^{-1}n$ and of Witt index $r$. 
As $D$ has an involution of the first kind, $D$
lies in the $2$-torsion of the Brauer group 
$\br_2(k)$
(cf. \cite[Ch. 8, 8.4]{scharlau}). 
According to a result due to Saltman \cite{saltman},
we know that $d\in\{1,2,4\}$.
Let $h_a$ be the anisotropic part of $h$. We have
\begin{equation}\label{2d^{-1}n-2r}
\dim h_a=2d^{-1}n-2r
\end{equation}

If $d=1$, $h$ is alternating and in this case  we have $r=n$.

If $d=2$, $D$ is a quaternion algebra over $k$. According to
Corollary~\ref{principal} and Lemma~\ref{ulbar4}
we have  
$\u(D,\sigma,-1)=\u(D,\bar\ )\leqslant\frac{1}{2}{\u(L,\bar\ )}\leqslant 2$
 where $\bar\ $ is the canonical 
involution of $D$ and $L$ is a maximal subfield of $D$ stable under $\bar\ $. 
We deduce that $0\leqslant \dim h_a\leqslant 2$. 
Now (\ref{2d^{-1}n-2r}) implies that $n-2r\in\{0,1,2\}$.

If $d=4$, according to a result due to Albert 
(cf. \cite[16.1]{boi}),
 $D$ is isomorphic to a biquaternion algebra. 
We write $D=D_1\otimes D_2$ where $D_1$ and $D_2$
are quaternion algebras. According to Proposition~\ref{sadeh} we have
$\u(D,\sigma,-1)=\u(D_1\otimes D_2,\bar\ \otimes\bar\ ,-1)$
where $\bar\ $ (resp. $\bar\ $) is the canonical involution of $D_1$ (resp. $D_2$).
Let $L$ be a maximal subfield of $D_2$ stable under $\bar\ $.
Thanks to Proposition~\ref{ppp} we have:
\[
\begin{array}{ll}
\u(D_1\otimes D_2,\bar\ \otimes\bar\,-1) & \leqslant \u(D_1\otimes L,\bar\ 
\otimes\id)+\frac{1}{2}\u(D_1\otimes L,\bar\ \otimes\bar\ ,-1)
\end{array}
\]

Thanks to Proposition \ref{bound} we have 
$\u(D_1\otimes L,\bar\ \otimes\id)\leqslant\frac{1}{4}\u(L)$. According to 
\cite{parimala-suresh}, $\u(L)\leqslant 10$. We conclude then 
$\u(D_1\otimes L,\bar\ \otimes\id)\leqslant 2$. 
Now by using 
Proposition~\ref{tensorii} and Proposition~\ref{ppp} we obtain 
$$\u(D_1\otimes L,\bar\ \otimes\bar\ ,-1)\leqslant 6.$$
We obtain then $\u(D_1\otimes D_2,\bar\ \otimes\bar\ ,-1)\leqslant 5$.
We conclude that $0\leqslant\dim h_a\leqslant 5$ and 
(\ref{2d^{-1}n-2r}) implies that $n/2-2r\in\{0,1,\cdots,5\}$.
~\qed\end{proof}

\begin{prop}
The anisotropic indices $C_{1,0}^{(2)}$, $C_{2,0}^{(2)}$ and
$C_{2,0}^{(4)}$ occur over $k=\qp(t)$ with $p\neq 2$.
\end{prop}

\begin{proof}
According to (\ref{2d^{-1}n-2r}), it is enough to find a skew
hermitian form of dimension $2$ over $(D,\sigma)$ where 
$D$ is a 
suitable
quaternion 
division algebra over $\qp(t)$ and  
$\sigma$ is an orthogonal involution of $D$. 
This choice is possible because $\u(D,\sigma,-1)=2$ 
for
$D=(-p,u)_{\qp(t)}$ where $u\in\mathbb{Z}_p^*\backslash{\mathbb{Z}_p^*}^2$.
In fact by Proposition \ref{bound}, $\u(D,\sigma,-1)\leqslant\frac{1}{4}\u(\qp(t))$. 
We obtain then $\u(D,\sigma,-1)\leqslant 2$, because 
$\u(\qp(t))\leqslant 10$ according to \cite{parimala-suresh}.
So it is enough to construct an anisotropic skew hermitian form of dimension 
$2$ over $(D,\sigma)$ or equivalently 
an anisotropic hermitian form of 
dimension $2$ over $(D,\bar\ )$ where $\bar\ $ is the canonical involution of $D$.
We may take for example  
the hermitian form  $\langle1,t\rangle$.
~\qed
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
{\centerline{\bf Type $D_n$ inner}}~
\begin{lemme}\label{h+th}
Let $D$ be a division algebra over a field $K$ with a $K/k$-involu\-tion
$\sigma$. Consider the division algebra $D(t)=k(t)\otimes_kD$ with the involution
$\widehat{\sigma}=\id\otimes\sigma$.
Let $(V,h_V)$ and $(W,h_W)$ be two anisotropic $\ve$-hermitian spaces over 
$(D,\sigma)$. Let $V(t)=k(t)\otimes_kV$,  
$W(t)=k(t)\otimes_kW$, $\widehat{h}_V$ and $\widehat{h}_W$ the restrictions of
$h_V$ and $h_W$ (resp.) to $D(t)$. Then the hermitian form 
$\widehat{h}_V\oplus t\widehat{h}_W$ is anisotropic over $(D(t),\widehat{\sigma})$.
In particular $\u(D(t),\widehat{\sigma},\ve)\geqslant 2\u(D,\sigma,\ve)$ where
$\ve\in K$ satisfies $\ve\sigma(\ve)=1$.
\end{lemme}

\begin{proof}
Let $x_1\oplus x_2\in V(t)\oplus W(t)$ be a nonzero anisotropic vector for
$\widehat{h}_V\oplus t\widehat{h}_W$, i.e.,
$\widehat{h}_V\oplus t\widehat{h}_W(x_1\oplus x_2,x_1\oplus x_2)=0$.
This relation implies that:

\begin{equation}\label{x1x2}
\widehat{h}_V(x_1,x_1)+t\widehat{h}_W(x_2,x_2)=0
\end{equation}

By using the embedding $V(t)\subset V((t))$ we may suppose that 
$x_1=\Sigma_{i=N}^{\infty}v_it^i$ and $x_2=\Sigma_{i=M}^{\infty}w_it^i$
where $v_i\in V$, $w_i\in W$, $v_N\neq 0$ and $w_M\neq 0$.
We consider two cases: $N\leqslant M$ and $N>M$. If $N\leqslant M$,
 (\ref{x1x2}) implies that $h_V(v_M,v_M)=0$ which is a contradiction 
because $h_V$ is anisotropic. If $N>M$, (\ref{x1x2}) implies that  
$h_W(w_N,w_N)=0$ which is a contradiction because
$h_W$ is anisotropic.
~\qed\end{proof}

\begin{prop}\label{dn-qpt1}
Let $k=\qp(t)$ with $p\neq 2$. Then if the index $^1D_{n,r}^{(d)}$
occurs over  $k$ then $d\in\{1,2,4\}$. Moreover:
\begin{itemize}
\item[\rmm{1}] For $d=1$ we have $n-r\in\{0,2,3,4,5\}$
\item[\rmm{2}] For $d=2$ we have $n-2r\in\{0,2,3,4,5,6,7,8,9\}$
\item[\rmm{3}] For $d=4$ we have $n-4r\in\{0,2,4,\cdots,20\}$.
\end{itemize}
\end{prop}

\begin{proof}
According to \cite{tits} or \cite[17.3.13]{springer},
$^1D_{n,r}^{(d)}$  occurs over $k$ if and only if there exist a
division algebra $D$ over $k$ of degree $d$
with an orthogonal involution $\sigma$ and a nondegenerate hermitian form
$h$ of dimension $2nd^{-1}$ with trivial discriminant and of Witt index $r$,
and moreover $d\geqslant 1$, $rd\leqslant n$ and $n\neq rd+1$.
As $D$ has an involution of the first kind, according to a result of Albert 
(cf.  \cite[Ch. 8, 8.4]{scharlau}),  $D$ lies in the $2$-torsion
of $\br(k)$. By using a result of Saltman  \cite{saltman}, 
we obtain $d\in\{1,2,4\}$.
Let $h_a$ be the anisotropic part of $h$. We have: 
\begin{equation}\label{2dn-r}
\begin{array}{ll}
\dim h_a=2d^{-1}n-2r,& \d h_a=1
\end{array}
\end{equation}

(1) If $d=1$, we have $D=k$ and $h_a$ is a symmetric bilinear form over
 $k$ and $\dim h_a$ is even. By using a result due to Parimala-Suresh 
\cite{parimala-suresh} the dimension of $h_a$ cannot  exceed $10$,
we obtain then $2n-2r=\dim h_a\in\{0,2,4,6,8,10\}$. 

The case $\dim h_a=2$ is impossible because
in this case $\d h_a=1$ implies the isotropy of 
$h_a$. Consequently $2n-2r=2$ is impossible. 

The case $\dim h_a=4$ is possible, we can take the anisotropic form  
$h_a=\langle 1,p,-u,-pu\rangle$ which has trivial discriminant where
$u\in{\mathbb{Z}_p}^*\backslash{{\mathbb{Z}_p}^*}^2$.
This form is anisotropic over $\qp(t)$.
The case $2n-2r=4$ is then possible. 

The case $\dim h_a=6$ is possible. Consider the biquaternion division algebra
$(u,t)\otimes_k(t+1,p)$.
The fact that this algebra is a division algebra can be found in 
an appendix of the Saltman's paper \cite{saltman} due to W. Jacob 
and
J.-P. Tignol.
Let $h_a=\langle u,t,-ut,-(t+1),-p,(t+1)p\rangle$ be the Albert form of this algebra.
This form is anisotropic and has trivial discriminant. 
Consequently $2n-2r=6$ is possible.

The case $\dim h_a=8$ is also possible. We can take the anisotropic form 
$h_a=\langle 1,t\rangle\otimes\langle 1,p,-u,-pu\rangle$. 
The anisotropy of $h_a$ can be deduced for example from
Lemma~\ref{h+th}.

(2) If $d=2$, $D$ is a quaternion division algebra over $k$. 
Now (\ref{2dn-r}) implies that $\d h=\d h_a=1$. 
We write $D=(a,b)_k$, the quaternion division algebra over $k$ generated by
$i,\ j$ with $i^2=a\in k^*$, $j^2=b\in k^*$, $ij=-ji$.
Let $L=k(i)=k(\sqrt{a})\subset D$.
According to Proposition~\ref{sadeh}, $\u(D,\sigma)=\u(D,\bar\ ,-1)$ where
$\bar\ $ is the canonical involution of $D$.

As $L$ is the function field of a  $p$-adic field $($$p\neq 2$$)$, 
we have $\u(L)\leqslant 10$ (cf. \cite[4.5]{parimala-suresh}).
According to Lemma~\ref{ulbar4}, $\u(L,\bar\ |_L)\leqslant 4$.
Now by applying Corollary~\ref{principal} we obtain:
$$\u(D,\bar\ ,-1)\leqslant\frac{10+8}{2}=9.$$ 
Consequently $\dim h_a\leqslant 9$. We have then $n-2r\in\{0,1,2,\cdots,9\}$.

The case $\dim h_a=1$ is impossible because the discriminant of every
skew hermitian form over $D$ of dimension $1$ is different from $1$.
Consequently the case $n-2r=1$ is impossible.

The case $\dim h_a=2$ is possible. To construct, consider 
an arbitrary nondegenerate skew hermitian form 
$h_0$ of dimension $1$ over $(D_0,\sigma_0)$ where $D_0$ is the
unique quaternion division algebra over
 $\qp$ and $\sigma_0$ is its canonical involution. Consider the skew hermitian form 

\begin{equation}\label{hh0}
h=h_0\oplus t h_0
\end{equation}

According to Lemma~\ref{h+th}, $h$ is anisotropic, moreover its discriminant is trivial.
We conclude that the case $n-2r=2$ is possible. 

It is well known that, there exists an anisotropic skew hermitian form
of dimension $3$ over $D_0$
(cf. \cite[Ch. 10, 3.6]{scharlau}, \cite{ramanathan} or  \cite{tsukamoto}).
The restriction of this form to $(D,\sigma)$ is anisotropic.
Consequently the case $\dim h_a=3$ or $n-2r=3$ is possible.

The cases $\dim h_a=4\text{ or }6$ are similar; we take an anisotropic 
skew hermitian form $h_0$ over $(D_0,\sigma_0)$ of dimension $2$ or $3$
and we consider the anisotropic form $h$ defined in (\ref{hh0}). 

For the case $\dim h_a=5$, we first consider a subform of dimension 
$5$ 
of the anisotropic skew hermitian form of dimension $6$ over
$(D,\sigma)$ that we constructed above. By multiplying this form
by its discriminant, we obtain an anisotropic form of 
dimension $5$ with trivial discriminant. Therefore 
$n-2r=5$ is possible.

(3) If $d=4$, $D$ is a biquaternion algebra. We write 
$D=D_1\otimes D_2$ where $D_1$ and $D_2$ are two quaternion algebras.
We have $\u(D,\sigma)=\u(D_1\otimes D_2,\bar\ \otimes\bar\ )$ where
$\bar\ $ (resp. $\bar\ $) is the canonical involution of $D_1$ (resp.  $D_2$). 
Let $L$ be a maximal subfield of $D_2$ stable under $\bar\ $.
 According to Proposition~\ref{ppp} and Proposition~\ref{tensorii} we have
\[
\begin{array}{ll}
\u(D_1\otimes D_2,\bar\ \otimes\bar\ ) & \leqslant \frac{1}{2}\u(D_1\otimes L,\bar\ 
\otimes\id,-1)+\u(D_1\otimes L,\bar\ \otimes\bar\ )\\\\
& \leqslant\frac{9}{2}+6=\frac{21}{2},
\end{array}
\]
We obtain then $\u(D,\sigma)\leqslant 10$. Now 
(\ref{2dn-r}) implies that $0\leqslant n/2-2r\leqslant 10.\qed$
%~\qed
\end{proof}

\begin{cor}
The anisotropic indices $^1D_{n,0}^{(1)}$ for $2\leqslant n\leqslant 4$
and $^1D_{n,0}^{(2)}$ for $2\leqslant n\leqslant 6$
occur over $k=\qp(t)$ with $p\neq 2$.
\end{cor}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
{\centerline{\bf Type $D_n$ outer}}
\begin{prop}\label{dn-qpt2}

Let $k=\qp(t)$ with $p\neq 2$. If the index $^2D_{n,r}^{(d)}$
occurs over $k$ then $d\in\{1,2,4\}$. Moreover:
\begin{itemize}
\item[\rmm{1}] For $d=1$ we have $n-r\in\{0,1,2,3,4,5\}$
\item[\rmm{2}] For $d=2$ we have $n-2r\in\{0,1,2,3,5,6,7,8,9\}$
\item[\rmm{3}] For $d=4$ we have $n-4r\in\{0,2,\cdots,20\}$.
\end{itemize}
\end{prop}

\begin{proof}
We use the notation of the proof of Proposition~\ref{dn-qpt1}. 
The criterion for the existence of $^2D_{n,r}^{(d)}$ is the same 
as for $^1D_{n,r}^{(d)}$ except $\d h\neq 1$ (cf. \cite{tits}).
Therefore we have $d\in\{1,2,4\}$.

(1) For $d=1$, $\dim h_a=2$ is possible; it is enough to choose the 
anisotropic form $\langle 1,p\rangle$. Consequently $n-r=1$ is possible.

The case $\dim h_a=4$ is possible. In fact 
it is enough to choose a subform of dimension $4$ of the anisotropic
form $\langle 1,t\rangle\otimes\langle 1,p,-u,-pu\rangle$ with nontrivial discriminant. We can choose
for example $\langle p,-u,-pu,t\rangle$. Consequently $n-r=2$ is possible.

 For $\dim h_a=6$ we can choose $\langle p,-u,-pu,tp,-tu,-tpu\rangle$. Consequently
$n-r=3$ is possible. 

(2) For $d=2$, $D$ is a quaternion algebra over $k$.
We have that $\dim h_a\leqslant 9$ as in the proof of 
Proposition~\ref{dn-qpt1}.

For the case $\dim h_a=1$, it is enough to consider 
a skew hermitian form of dimension $1$ which has necessarily
nontrivial discriminant. Consequently the case $n-2r=1$ is possible.
 
For the case $\dim h_a=2$, we consider two arbitrary skew symmetric elements
of $D_0=(-p,u)_{\qp}$ with different reduced norm modulo ${\qp^*}^2$. 
We denote these elements by $a$ and $b$. 
Now consider the skew hermitian form $h=\langle a,tb\rangle$ over $D=D_0\otimes_{\qp}\qp(t)$.
The form $h$ is anisotropic form with nontrivial discriminant.
Consequently $n-2r=2$ is possible.

The case $\dim h_a=3$ is possible. 
In fact, there exists a skew hermitian form $h_0$ of dimension $2$ 
with discriminant equal to $c\neq 1\in \qp^*/{\qp^*}^2$
(cf.  \cite[Ch. 10, 3.6]{scharlau}, \cite{ramanathan} or \cite{tsukamoto}).
We also consider a skew hermitian form $h_0'$ 
over $(D_0,\sigma_0)$ of dimension $1$ and with the discriminant equal
to $c'\neq 1\in  \qp^*/{\qp^*}^2$. We may suppose that $c'\neq c$
(this choice is possible because $\card (\qp^*/{\qp^*}^2)=4)$. 
Thanks to  Lemma~\ref{h+th}, the form $h_a=h_0\oplus th_0'$ is an anisotropic
skew hermitian form over $(D,\sigma)$ of dimension $3$ and with
nontrivial discriminant. Consequently  $n-2r=3$ is also possible.
 
For $\dim h_a=5$, we consider an anisotropic skew hermitian form $h_0$
over $(D_0,\sigma_0)$ of dimension $3$ and with trivial discriminant
and an anisotropic skew hermitian form $h_0'$ over $(D_0,\sigma_0)$
of dimension $2$ and with nontrivial discriminant (According to 
\cite[Ch. 10, 3.6]{scharlau}, \cite{ramanathan} or \cite{tsukamoto}
 these choices are possible). 
Now consider the form $h=h_0\oplus th_0'$ over $(D,\sigma)$
which is of dimension $5$ and with nontrivial discriminant
Consequently $n-2r=5$ is possible.

Suppose that 
$h\simeq\langle a_1,a_2,\cdots,a_5\rangle$ 
 is an anisotropic skew hermitian form of dimension $5$ and with
nontrivial discriminant constructed as in the preceding paragraph.
At least one of the forms $\langle a_{i_1},\cdots,a_{i4}\rangle$ has
nontrivial discriminant where $1\leqslant i_1<i_2<i_3<i_4\leqslant 5$.
Consequently $\dim h_a=n-2r=4$ is possible.

(3) For $d=4$, $D$ is a biquaternion algebra.
As in the proof of Proposition~\ref{dn-qpt1} (3), we obtain $n/2-2r\leqslant 10$.
~\qed\end{proof}

\begin{cor}
The anisotropic indices $^2D_{n,0}^{(1)}$ for $2\leqslant n\leqslant 3$
and $^2D_{n,0}^{(2)}$ for $1\leqslant n\leqslant 5$
occur over $k=\qp(t)$ with $p\neq 2$.
\end{cor}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\bf Acknowledgements}
\end{center}
I would like to thank my supervisor Eva Bayer-Fluckiger 
for suggesting the topic of this paper
and for her guidance. I would also like to thank
David Lewis and the members of the department of Mathematics of
University College Dublin for their kind hospitality
during a predoctoral stay of the author at UCD. 
I would also like to thank Karim Becher for many helpful suggestions.
I would also like to thank the Referee for
careful reading of the manuscript and for numerous helpful suggestions 
that improved the final version of this paper. 
This research has been carried out in the framework of
the European research network HPRN-CT-2002-00287  
Algebraic K-Theory, Linear Algebraic Groups and Related Structures.
%\end{acknowledgement}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\end{thebibliography}
\noindent \sc Laboratoire de Math�matiques,
Universit� de Franche-Comt�,
25030 Besan\c{c}on Cedex,
France\\
Department of Mathematics, University College Dublin,
Belfield, Dublin 4, Ireland.\\
{\tt mohammad.mahmoudi@ucd.ie} 
\end{document}

% end of file template.tex


