
\documentclass{amsart}
\usepackage{amssymb,amsmath,latexsym,xypic}


\DeclareMathOperator{\Pf}{\mathbf{Pf}}
\DeclareMathOperator{\PSim}{\mathbf{PSim}}
\DeclareMathOperator{\Sim}{\mathbf{Sim}}
\DeclareMathOperator{\PSimprp}{PSim}
\DeclareMathOperator{\Simprp}{Sim}
\DeclareMathOperator{\Sprp}{Sp}
\DeclareMathOperator{\GSp}{\mathbf{GSp}}
\DeclareMathOperator{\GSprp}{GSp}
\DeclareMathOperator{\PGSp}{\mathbf{PGSp}}
\DeclareMathOperator{\PGSprp}{PGSp}
\DeclareMathOperator{\GU}{\mathbf{GU}}
\DeclareMathOperator{\SU}{\mathbf{SU}}
\DeclareMathOperator{\GUprp}{GU}
\DeclareMathOperator{\PGU}{\mathbf{PGU}}
\DeclareMathOperator{\PGUprp}{PGU}
\DeclareMathOperator{\GO}{\mathbf{GO}}
\DeclareMathOperator{\OO}{\mathbf{O}}
\DeclareMathOperator{\GOprp}{GO}
\DeclareMathOperator{\PGO}{\mathbf{PGO}}
\DeclareMathOperator{\PGOprp}{PGO}
\DeclareMathOperator{\Gm}{\mathbf{G}_{m}}
\DeclareMathOperator{\GmK}{\mathbf{G}_{m,\mathit{K}}}
\DeclareMathOperator{\AbGrps}{\mathbf{AbGrps}}
\DeclareMathOperator{\W}{\mathbf{W}}
\DeclareMathOperator{\HH}{\mathbf{H}}
\DeclareMathOperator{\Sets}{\mathbf{Sets}}
\DeclareMathOperator{\Spin}{\mathbf{Spin}}
\DeclareMathOperator{\Id}{Id}
\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\SSym}{SSym}
\DeclareMathOperator{\Nrd}{Nrd}
\DeclareMathOperator{\Nrp}{Nrp}
\DeclareMathOperator{\Trd}{Trd}
\DeclareMathOperator{\Int}{Int}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\disc}{disc}
\DeclareMathOperator{\Skew}{Skew}
\DeclareMathOperator{\Hyp}{Hyp}
\newcommand{\Ck}{\mathfrak{C}_k}
\newcommand{\FF}{\mathbf{F}}
\newcommand{\Herm}{\mathbf{Herm}}
\newcommand{\Quad}{\mathbf{Quad}}
\DeclareMathOperator{\zz}{\mathbb{Z}}
\DeclareMathOperator{\qq}{\mathbb{Q}}
\DeclareMathOperator{\rr}{\mathbb{R}}
\DeclareMathOperator{\cc}{\mathbb{C}}
\DeclareMathOperator{\cores}{Cor}
\DeclareMathOperator{\Br}{Br}
\DeclareMathOperator{\SB}{SB}
\DeclareMathOperator{\ind}{ind}
\DeclareMathOperator{\Res}{Res}
\DeclareMathOperator{\carac}{char}
\newtheorem{prop}{Proposition}
\newtheorem{thm}[prop]{Theorem}
\newtheorem{lem}[prop]{Lemma}
\newtheorem{coro}[prop]{Corollary}


\def\ov#1{{\overline{#1}}}
\def\uv#1{{\underline{#1}}}

\theoremstyle{definition}

\newtheorem{defn}{Definition}
\newtheorem{ex}[prop]{Example}
\newtheorem{exs}[prop]{Examples}
\newtheorem{rem}[prop]{Remark}
\newtheorem{exo}[prop]{Exercise}


\parskip=5pt
\parindent=0pt

\title[Finiteness of $R$-equivalence groups]{Finiteness of $R$-equivalence groups of some adjoint classical groups of type $^2D_3$}
\author{Gr\'egory Berhuy}
\date{\today}
\address{University of Southampton\\
School of Mathematics\\
Highfield\\
SO17 1BJ Southampton\\
United Kingdom}
\email{berhuy@soton.ac.uk}


\begin{document}

\begin{abstract}
Let $F$ be a field of charateristic different from $2$. We construct families of adjoint groups $G$ of type $^2D_3$ defined over $F$ (but not over $k$) such that $G(F)/R$ is finite for various fields $F$ which are finitely generated over their prime subfield. We also construct families of examples of such groups $G$ for which $G(F)/R\simeq \zz/2\zz$ when $F=k(t)$, and $k$ is (almost) arbitrary. This gives the first examples of adjoint groups $G$ which are not quasi-split nor defined over a global field, such that $G(F)/R$ is a non-trivial finite group.
\end{abstract}

\maketitle


%===========================================
%            Introduction
%==========================================


\section*{Introduction}

For an algebraic group $G$ defined over a field $F$, let
$G(F)/R$ be the group of
$R$-equivalence classes introduced by Manin in \cite{Ma}. The
algebraic group $G$ is called $R$-trivial if $G(L)/R=1$ for every field
extension $L/F$. It was established by Colliot-Th\'el\`ene and
Sansuc in \cite{CTS} (see also \cite[Proposition~1]{Me})
that the group $G$ is $R$-trivial if the variety of $G$ is stably
rational. Moreover, in \cite{CTS}, the following question was
raised:


{\bf Question: }Let $F$ be a field which is finitely generated
over its prime subfield, and let $G$ be a connected linear algebraic group
defined over $F$. Assume that $F$ is perfect or $G$ is reductive.
Is $G(F)/R$ finite ?


The question was answered positively by Colliot-Th\'{e}l\`{e}ne and
Sansuc if $G$ is quasi-split (cf. Proposition 14, {\it loc.cit})
and by Gille for any reductive group $G$ defined over a global
field in \cite{Gi1}.
Lemma II.1.1 c) of \cite{Gi1} immediately implies that this
question has a positive answer if $F$ is a rational extension of
a global field $k$ and $G$ is defined over $k$.
Various examples of classical adjoint groups which are not
$R$-trivial were constructed in \cite{BMT} or \cite{Gi2},\cite{Me}. Throughout this paper, we will assume that $F$ is a field of characteristic different from $2$ and we will focus on absolutely simple adjoint groups
of type $^2D_3$.
If $F/k$ is a finitely generated field extension, we construct an infinite family of adjoint groups $G$ of type $^2D_3$ defined over $F$ such that $G(F)/R$ is finite as soon as $H^3_{nr}(F/k,\mu_2)$ is finite. If $F=k(t)$, where $k$ is an arbitrary field, we will also give a family of examples of such groups for which $G(F)/R\simeq \zz/2\zz$. This gives the first examples of adjoint groups $G$ such that $G(F)/R$ which are not quasi-split nor defined over a global field, such that $G(F)/R$ is a non-trivial finite group.

\section{Unramified cohomology}


Let $X$ be a smooth proper irreducible variety defined over $k$. 
We denote by $X^{(1)}$ the set of points of codimension $1$ in $X$. The ring ${\mathcal{O}}_{X,x}$ is then a discrete valuation ring. We will denote by $\upsilon_x$ the corresponding discrete valuation and by $\pi_x$ a local parameter. We have a residue map 
$$\partial_x:H^n(k(X),\mu_2)\to H^{n-1}(\kappa(x),\mu_2),$$
where $\kappa(x)$ denotes the residue field ${\mathcal{O}}_{X,x}/(\pi_x)$. If $u\in{\mathcal{O}}_{X,x}$, we will denote by $\bar{u}$ its image in $\kappa(x)$.

The residue of a cohomology class $\alpha\in H^n(k(X),\mu_2)$ can be computed as follows: 
denote by $k(X)_x$ the completion of $k(X)$ with respect to the valuation on ${\mathcal{O}}_{X,x}$. Then $\pi_x$ is also a local parameter for the unique discrete valuation on $k(X)_x$ extending $\upsilon_x$, and we have an injection $H^n(\kappa(x),\mu_2)\hookrightarrow H^n(k(X)_x,\mu_2)$. Then we have a decomposition
$$\Res_{k(X)_x/k(X)}(\alpha)=\alpha_0+(\pi_x)\cup \alpha_1,$$ 

for some uniquely determined $\alpha_i\in H^{n-i}(\kappa(x),\mu_2)$. We then have the equality $\partial_x(\alpha)=\alpha_1$.
In particular, for every $a_1,\cdots,a_{n-1},b_1,\cdots,b_{n-1}\in {\mathcal{O}}_{X,x}^\times$, we have 
$$\partial_x((\pi_x)\cup(a_1)\cdots\cup(a_{n-1}))=(\ov{a}_1)\cdots\cup(\ov{a}_{n-1})$$
$$\partial_x((b_1)\cdots\cup(b_{n}))=0$$


We say that $\alpha\in H^n(k(X),\mu_2)$ is {\it unramified at $x$} if $\partial_x(\alpha)=0$. In this case, the class $\alpha_0$ is called {\it the specialization of $\alpha$ at $x$}, and is denoted by $s_x(\alpha)$. It does not depend on the choice of $\pi_x$.  
If $\partial_x(\alpha)\neq 0$, we say that $\alpha$ is {\it ramified at $x$}, and that $x$ is {\it a pole} of $\alpha$.
It is well-known that the set of poles of $\alpha$ is finite.
 The {\it unramified cohomology group} $H^n_{nr}(k(X),\mu_2)$ is the subgroup of $H^n(k(X),\mu_2)$ consisting of classes which are unramified at every $x\in X^{(1)}$. 
It is a birational invariant of $X$. In particular, if $X$ is a rational variety, then the restriction map induces an isomorphism $H^n(k,\mu_2)\simeq H^n_{nr}(k(X),\mu_2).$
Therefore if $F/k$ is a finitely generated extension, we can define the group of unramified elements $H^{n}_{nr}(F/k,\mu_2)$ by 
$$H^{n}_{nr}(F/k,\mu_2)=H^{n}_{nr}(k(X),\mu_2),$$ where $X$ is any irreducible smooth proper model of $F/k$.  We refer to \cite{CT} for more details.

Notice that for any finitely generated field extension $F/k$, the elements lying in the image of $\Res_{F/k}:H^n(k,\mu_2)\to H^n(F,\mu_2)$ are unramified. Such elements are called {\it constant}. Notice also that if $\alpha\in H^n(F,\mu_2)$ is constant, then we have $s_x(\alpha)=\Res_{\kappa(x)/k}(\alpha)$ for all $x\in X^{(1)}$.
 
\section{$R$-equivalence groups of adjoint groups of type $^2 D_3$}

\subsection{A result of Merkurjev}
%------------------------------
%   Merkurjev's result
%------------------------------

In this section, we recall Merkurjev's computation of the group of $R$-equivalence classes of some absolutely simple adjoint classical groups of type $^2D_3$ (cf. \cite{Me}). 
Let $(A,\sigma)$ be a $F$-central simple algebra of degree $6$
with an orthogonal involution, so we can write $A=M_3(Q)$, where
$Q$ is a quaternion $F$-algebra, and let $\PGO^+(A,\sigma)$ be the connected component of $\PGO(A,\sigma)$, the group-scheme of projective similitudes of $(A,\sigma)$.

Assume that $A$ is not split, $\disc(\sigma)\in F^\times/F^{\times 2}$ is not trivial,
and that the Clifford algebra $C(A,\sigma)$ has index
$2$. If $L=F(\sqrt{\disc(\sigma)})$ then $A_L$ (or
equivalently $Q_L$) is split. Hence we can write $Q\simeq(\disc(\sigma),\alpha)$, for some $\alpha\in F^\times$.
Let $1,i,j,ij$ be the corresponding standard basis for $Q$, and let $\gamma$ be
the canonical (symplectic) involution on $Q$.
The involution $\sigma$ is adjoint to a skew-hermitian form
$(V,h)$ over $(Q,\gamma)$, where $V$ is a right $Q$-vector space
of dimension $3$.

The skew-hermitian form $h$ represents $xi$ for some $x\in
F^\times$, so we can write $h=h'\perp \langle xi\rangle$ for some
skew-hermitian form $(V',h')$ over $(Q,\gamma)$ of trivial discriminant,
where $V$ is a right $Q$-vector space of dimension $2$.

Set $(A',\sigma'):=(\mathrm{End}_Q(V'),\sigma_{h'})$. Then
$C(A',\sigma')=Q_1\times Q_2$, for some quaternion $F$-algebras
$Q_1$ and $Q_2$ satisfying $Q_1\otimes Q_2=Q$ in
$\mathrm{Br}(F)$.
Moreover, $(Q_1)_L\simeq (Q_2)_L$ and $C(A,\sigma)=(Q_i)_L$ in
$\mathrm{Br}(L)$ (so $(Q_i)_L$ is not split for $i=1,2$).


\begin{prop}\label{Req}
Under the previous notation, we have the following group isomorphism:
$$\PGO^+(A,\sigma)(F)/R\simeq N_{L/F}(L^\times)\cap\Nrd(Q_1^\times)\cdot\Nrd(Q_2^\times)/
 N_{L/F}(L^\times)\cap\Nrd(Q_i^\times)$$
\end{prop}

For a proof of all these facts, see \cite[Section 3]{Me}.
Notice that in \cite{Me}, Merkurjev described more generally the group $G(F)/R$, when $G$ is an absolutely simple adjoint classical group defined over $F$. 

\subsection{Finiteness of some $R$-equivalence groups}

\subsubsection{Some useful lemmas}

We will assume that $(A,\sigma)$ is as in the previous section.
We start to investigate the finiteness of $\PGO^+(A,\sigma)(F)/R$. Keeping the notation above, we will identify this group to $$N_{L/F}(L^\times)\cap\Nrd(Q_1^\times)\cdot\Nrd(Q_2^\times)/
 N_{L/F}(L^\times)\cap\Nrd(Q_i^\times)$$

If $\lambda\in N_{L/F}(L^\times)\cap\Nrd(Q_1^\times)\cdot\Nrd(Q_2^\times)$, we will denote by $[\lambda]$ its class modulo $N_{L/F}(L^\times)\cap\Nrd(Q_i^\times)$. We start with an easy lemma:

\begin{lem}\label{inj}
Let $F$ be any field of characteristic different from $2$.
Then the map $$\varphi:\PGO^+(A,\sigma)(F)/R\to H^3(F,\mu_2),[\lambda]\mapsto (\lambda)\cup [Q_1]$$ is a well-defined injective group homomorphism.
\end{lem}

\begin{proof}
Since $(\Nrd_{Q_1}(Q_1^\times))\cup [Q_1]=0$, this map is a well-defined group homomorphism .
If $\lambda\in N_{L/F}(L^\times)\cap\Nrd_{Q_1}(Q_1^\times)\cdot\Nrd_{Q_2}(Q_2^\times)$ satisfies $(\lambda)\cup [Q_1]=0$, then  
$\lambda\in \Nrd_{Q_1}(Q_1^\times)$ by a well-known theorem of Merkurjev \cite{Me2}, so $[\lambda]=1$. 
\end{proof}

\begin{rem}\label{imphi}
In view of this lemma, we just have to investigate the finiteness of the image of $\varphi$.
\end{rem}
We now assume until the end that $X$ is a smooth irreducible proper model of $F$ defined over $k$.

\begin{lem}\label{imres}
Assume that $Q_1$ and $Q_2$ have no common pole, and let $x\in X^{(1)}$. Then 
$$\partial_x((\lambda)\cup [Q_1])=\left\lbrace\begin{array}{ll}0 & \mbox{ if $x$ is not a pole of $[Q_1]$ or $[Q_2]$ }\cr 0 \mbox{ or } s_x[Q_2] & \mbox{ if $x$ is a pole of $[Q_1]$ }
\cr 0 \mbox{ or } s_x[Q_1] & \mbox{ if $x$ is a pole of $[Q_2]$  }\end{array}\right.$$
\end{lem}

\begin{proof}

Notice that since $\lambda=N_{L/F}(z)$ for some $z\in L^\times$ and that
$(Q_1)_L\simeq (Q_2)_L$, we get
$$(\lambda)\cup [Q_1]=\mathrm{Cor}_{L/F}((z)\cup [Q_1]_L)=\mathrm{Cor}_{L/F}((z)\cup [Q_2]_L)=(\lambda)\cup
[Q_2].$$ 

Let $x\in X^{(1)}$, and assume first that $[Q_1]$ and $[Q_2]$ are both unramified at $x$.
If $(\lambda)$ is unramified at $x$, then $(\lambda)\cup[Q_1]$ is also unramified at $x$, that is $\partial_x((\lambda)\cup [Q_1])=0$. If $(\lambda)$ is ramified at $x$, then write $\lambda=\lambda_1\lambda_2,\lambda_i\in\Nrd_{Q_i}(Q_i^\times)$. Then $(\lambda_1)$ or $(\lambda_2)$ is ramified at $x$, since $\partial_x((\lambda))=\partial_x((\lambda_1))+\partial_x((\lambda_2))$ and $\partial_x((\lambda_i))\in\mathbb{Z}/2\mathbb{Z}$.
If $(\lambda_2)$ is unramified at $x$, then $\partial_x((\lambda)\cup [Q_1])=\partial_x((\lambda_2)\cup [Q_1])=0$.
Now assume that $(\lambda_2)$ is ramified at $x$, so $(\lambda_1)$ is unramified at $x$.
Since $[Q_2]$ is unramified at $x$ as well, then
$\partial_x((\lambda)\cup [Q_1])=\partial_x((\lambda)\cup [Q_2])=\partial_x((\lambda_1)\cup [Q_2])=0$.
Hence $\partial_x((\lambda)\cup [Q_1])=0$ if $x$ is not a pole of $[Q_1]$ or $[Q_2]$.

Now assume that $x$ is a pole of $[Q_1]$, so $[Q_2]$ is unramified at $x$ by assumption.
If $(\lambda)$ is unramified at $x$ then $\partial_x((\lambda)\cup [Q_1])=\partial_x((\lambda)\cup [Q_2])=0$. If $(\lambda)$ is ramified at $x$, then
$\partial_x((\lambda)\cup [Q_1])=\partial_x((\lambda)\cup [Q_2])=s_x([Q_2])$.
If $x$ is a pole of $[Q_2]$, then similar computations show that $\partial_x((\lambda)\cup [Q_1])=0$ or $s_x([Q_1])$.

\end{proof}

\subsubsection{The case where $H^3_{nr}(F/k,\mu_2)$ is finite}

\begin{prop}
Assume that $[Q_1]$ and $[Q_2]$ have no common pole.
If $H^3_{nr}(F/k,\mu_2)$ is finite, then $\PGO^+(A,\sigma)(F)/R$ is finite.
\end{prop}

\begin{proof}
By assumption, the kernel of the map $$(\partial_x)_{x\in X^{(1)}}:{\rm Im}(\varphi)\to \prod_{x\in X^{(1)}}H^2(\kappa(x),\mu_2)$$ is finite. By the previous lemma, its image is finite as well, so we are done by Remark \ref{imphi}.

\end{proof}


\begin{exs}
The group $H^3_{nr}(F/k,\mu_2)$ is finite in the following cases (and therefore the previous proposition may be applied):

$1)$ $H^3(k,\mu_2)$ is finite and $X$ is a smooth conic over $k$

$2)$ $k$ is a finite field and $X$ is a smooth proper variety of dimension $2$ over $k$ 

$3)$ $k$ is either a local field (i.e. a finite extension of $\qq_p$), $\rr$ or $\cc$ and $X$ is a proper smooth geometrically irreducible curve over $k$

$4)$ $k$ is a number field and $X$ is a smooth proper geometrically irreducible curve over $k$.
 
\end{exs}

Case $1)$ readily follows from Proposition 3 and Proposition A.1 of \cite{KRS}. 
Case $2)$ follows from Theorem 0.8 of \cite{Ka}. Now let us consider Case $3)$: if $k$ is a local field, it follows from Corollary 2.9. of \cite{Ka}. If $k=\rr$, it follows from a result of Colliot-Th\'{e}l\`{e}ne and Parimala (see \cite{CTPa}). Finally, if $k=\cc$, then $k(X)$ has cohomological dimension at most $1$ and therefore $H^3(k(X),\mu_2)=0$.
In case $4)$, it readily follows from Theorem 0.8 of \cite{Ka} that we have an injective homomorphism $$H^3_{nr}(k(X),\mu_2)\hookrightarrow \prod_{\upsilon\in P(k)}H^3_{nr}(k_\upsilon(X),\mu_2),$$
where $P(k)$ denotes the set of all places of $k$. By Corollary 2.9 of \cite{Ka}, $H^3_{nr}(k_\upsilon(X),\mu_2)$ is zero if $X$ has good reduction with respect to $\upsilon$. Since $X$ has good reduction with respect to all but finitely many places, it follows from Case $3)$ that $H^3_{nr}(k(X),\mu_2)=H^3_{nr}(F/k,\mu_2)$ is finite.  

The reader may find more finiteness results for $H^3_{nr}(F/k,\mu_2)$ in \cite{CT}.

\subsubsection{The case where $H^3_{nr}(F/k,\mu_2)\simeq H^3(k,\mu_2)$}

We give here another family of examples. 
Keeping notation of the previous sections, we will assume that $Q_1$ and $Q_2$ have no common poles.
We then set $$S_1=\{ x\in X \vert x \mbox{ is a pole of }Q_2 \mbox{ such that } s_x([Q_1])\neq 0 \}$$ 
$$S_2=\{ x\in X \vert x \mbox{ is a pole of }Q_1 \mbox{ such that } s_x([Q_2])\neq 0 \}$$ 

\begin{prop}
Assume that $[Q_1]$ and $[Q_2]$ have no common pole, and let $n_i$ be the number of elements of $S_i$.
Assume that $H^3_{nr}(F/k,\mu_2)\simeq H^3(k,\mu_2)$ (e.g. $F/k$ is rational) and that there exists $x_0\in X^{(1)}$ satisfying the following conditions:

$1)$ One of the class $[Q_i]$ is unramified at $x_0$ and the corresponding specialization is zero

$2)$ The restriction map $\Res_{\kappa(x_0)/k}:H^3(k,\mu_2)\to H^3(\kappa(x_0),\mu_2)$ is injective.

Then $\PGO^+(A,\sigma)(F)/R$ is finite, and its cardinality is at most $2^{n_1+n_2}$. 
\end{prop}

\begin{proof}
Without any loss of generality, we may assume for example that $[Q_1]$ is unramified at $x_0\in X^{(1)}$ and that $s_{x_0}([Q_1])=0$. Assume that $(\lambda)\cup [Q_1]\in {\rm Im}(\varphi)$ lies in the kernel of the map $$(\partial_x)_{x\in X^{(1)}}:{\rm Im}(\varphi)\to \prod_{x\in X^{(1)}}H^2(\kappa(x),\mu_2)$$ By assumption $(\lambda)\cup [Q_1]$ is constant, so we have $$s_x((\lambda)\cup [Q_1])=\Res_{\kappa(x)/k}((\lambda)\cup [Q_1]) \mbox{ for all }x\in X^{(1)}$$

Since $\partial_{x_0}([Q_1])=s_{x_0}([Q_1])=0$, we have $\Res_{k(X)_{x_0}/k(X)}((\lambda)\cup [Q_1])=0$, and therefore $s_{x_0}((\lambda)\cup [Q_1])=\Res_{\kappa(x_0)/k}((\lambda)\cup [Q_1])=0$.
Since the restriction map $\Res_{\kappa(x_0)/k}:H^3(k,\mu_2)\to H^3(\kappa(x_0),\mu_2)$ is injective, we get $(\lambda)\cup [Q_1]=0$.
Therefore $[\lambda]=1\in\PGO^+(A,\sigma)(F)/R$ by Lemma \ref{inj}.
It follows that we have an injection $$\PGO^+(A,\sigma)(F)/R\hookrightarrow \prod_{x\in X^{(1)}}H^2(\kappa(x),\mu_2)$$ The use of Lemma \ref{imres} leads to the conclusion. 
\end{proof} 

Let us now consider the case where $F=k(t)$, where $t$ is an indeterminate over $k$, so one may take $X=\mathbb{A}^1_k$. A point $x$ of $\mathbb{A}^1_k$ of codimension $1$ then corresponds to a unique monic irreducible polynomial $\pi\ k[t]$ and $\kappa(x)\simeq k[t]/(\pi)$. In this case, we will say that a cohomology class is (un)ramified at $\pi$, and $\partial_x$ and $s_x$ will be respectively denoted by $\partial_\pi$ and $s_\pi$. 
If $\pi$ has odd degree, a classical restriction-corestriction argument show that the restriction map $H^3(k,\mu_2)\to H^3(k[t]/(\pi),\mu_2)$ is injective. Hence, from the previous proposition, we obtain:

\begin{coro}\label{corokt}
Let $F=k(t)$ and assume that $[Q_1]$ and $[Q_2]$ have no common pole. Let $n_i$ be the number of elements of $S_i$.
Assume that there exists a monic irreducible polynomial $\pi\in k[t]$ of odd degree such that one of the class $[Q_i]$ is unramified at $\pi$ and the corresponding specialization is zero.
Then $\PGO^+(A,\sigma)(F)/R$ is finite, and its cardinality is at most $2^{n_1+n_2}$. 
\end{coro}

Using this corollary, it is easy to construct an infinite family of non quasi-split adjoint groups $G$ of type $^2 D_3$ defined over $k(t)$ (but not over $k$) such that $G(k(t))/R$ is finite for an (almost) arbitrary field $k$. 

\begin{ex}

Let $k$ be a field of characteristic different from $2$ and let $F=k(t)$.
Let $a,\alpha\in k^\times$ and let $\pi\in k[t]$ be a monic irreducible
polynomial satisfying the following conditions:

$1)$ $(-1)\cup (a)\cup (\alpha)=0$

$2)$ The quaternion $k$-algebra $(a,\alpha)$ is not split over
$\kappa(\pi)$ (In particular $(a,\alpha)$ is not split
over $k$, and therefore is not split over $F$, and $\alpha\notin k^{\times 2}$).

$3)$ There exists $b\in k$ such that $\pi(b)$ is a non-zero norm in $k(\sqrt{\alpha})$.

Let $Q_1=(a,\alpha)\otimes_k F,Q_2=(\pi,\alpha),Q=(a\pi,\alpha)$ and
$L=F(\sqrt{a\pi})$. Let $1,i,j,ij$ be the standard basis of $Q$
and $\gamma$ its canonical involution. Notice that $Q$ is a division algebra, since $\partial_{\pi}([Q])=\Res_{\kappa(\pi)/k}(\alpha)\neq 0$ (otherwise $(a,\alpha)$ would be split over $\kappa(\pi)$).

Let $\sigma$ be the involution on $A=M_3(Q)$ adjoint to the
skew-hermitian form $\langle j,-aj,i\rangle$ over $(Q,\gamma)$. The skew-hermitian form $h':=\langle j,-aj\rangle$ has trivial discriminant and the corresponding adjoint involution $\sigma'$ on $A':=M_2(Q)$ can be written 
$$\sigma'\simeq \sigma_{\langle 1,-a\rangle}\otimes\rho,$$ where $\rho$ is the orthogonal involution on $Q$ defined by $$\rho(1)=1,\rho(i)=i \mbox{ and }\rho(j)=-j$$ It is then easy to check that $C(A',\sigma')=Q_1\times Q_2$,
using the formulas describing Clifford algebras of tensor products
of involutions (see\cite{KMRT}, p.150 for example or \cite{Tao}), and the fact that $\disc(\rho)=\alpha\in F^\times/F^{\times 2}$.

{\bf Claim: }$\PGO^+(A,\sigma)(F)/R\simeq \mathbb{Z}/2\mathbb{Z}$ .

Indeed, $[Q_1]$ has no pole and $[Q_2]$ has exactly one pole. Notice also that $\pi$ is not a scalar multiple of $t-b$, since $\pi(b)\neq 0$ by assumption. Hence $[Q_2]$ is unramified at $t-b$. Moreover we have $s_{t-b}([Q_2])=(\pi(b))\cup(\alpha)=0$ by assumption. By Corollary \ref{corokt}, we then get that $\vert \PGO^+(A,\sigma)(F)/R\vert \leq 2$.
Now it is enough to find a non trivial-class in $\PGO^+(A,\sigma)(F)/R$. First of all, we clearly have $-a\pi\in N_{L/F}(L^\times)$. Moreover, since $(-1)\cup (a)\cup (\alpha)=0$, we have $-1\in\Nrd_{Q_1}(Q_1^\times)$, so $a=(-1)\cdot(-a)\in\Nrd(Q_1^\times)$.
Since $-\pi\in\Nrd_{Q_2}(Q_2^\times)$, we get $-a\pi=a\cdot(-\pi)\in
N_{L/F}(L^\times)\cap\Nrd(Q_1^\times)\cdot\Nrd(Q_2^\times)$.
It remains to show that the $R$-equivalence class of $-a\pi$ is not trivial.
For, it suffices to prove that $\varphi([-a\pi])\neq 0$; this is
the case since $\partial_\pi((-a\pi)\cup
[Q_1]))=(a,\alpha)_{\kappa(\pi)}\neq 0$.

\end{ex}

\begin{rem}
The group $\PGO^+(A,\sigma)$ obtained is not quasi-split since $Q$ is a division algebra. Moreover, it is not defined over $k$. Otherwise $[Q]$ would be unramified at $\pi$, which is not the case as we have seen above.  
To obtain concrete examples, one may take for $k$ any field such that $-1\in k^{\times 2}$ such there exists a non split quaternion algebra $(a,\alpha)$ over $k$, and for $\pi$ any arbitrary monic irreducible polynomial of odd degree satisfying $\pi(0)=1$.
\end{rem}

{\bf Acknowledgements: }We would like to thank P. Gille and J.-L. Colliot-Th\'{e}l\`{e}ne for stimulating conversations, and the anonymous referee for his constructive comments and suggestions.
%==========================
%       References
%==========================
\begin{thebibliography}{4444}

\bibitem{BMT} \textsc{G. Berhuy, M. Monsurr\`{o}, J.-P. Tignol}, \emph{Cohomological invariants and R-triviality of adjoint classical groups. }Math. Z. {\bf 248}, 313--323 (2004)

\bibitem{CT} \textsc{J.-L. Colliot-Th\'{e}l\`{e}ne}, \emph{Birational invariants, purity and the Gersten conjecture}, in  K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras, AMS Summer Research Institute, Santa Barbara 1992, ed. W. Jacob and A. Rosenberg, Proceedings of Symposia in Pure Mathematics {\bf 58}, Part I, 1--64 (1995) 

\bibitem{CTPa} \textsc{J.-L. Colliot-Th\'{e}l\`{e}ne, R. Parimala}, \emph{Real components of algebraic varieties and \'{e}tale cohomology. }Invent. Math. {\bf 101}, 81--99 (1990) 

\bibitem{CTS} \textsc{J.-L. Colliot-Th\'{e}l\`{e}ne, J.-J
Sansuc}, \emph{La R-\'{e}quivalence sur les tores. }Ann. Sci.
ENS, 4\`{e}me s\'{e}rie. {\bf 10}, 175--230 (1997)

\bibitem{Gi1} \textsc{P. Gille}, \emph{La R-\'{e}quivalence sur les
groupes r\'{e}ductifs d\'{e}finis sur un corps global. }Pub.
Math. I.H.\'{E}.S. {\bf 86}, 199--235 (1997)

\bibitem{Gi2} \textsc{P. Gille}, \emph{Examples of non-rational
varieties of adjoint groups. }J. Algebra {\bf 193}, 728--747 (1997)

\bibitem{KRS} \textsc{B. Kahn, M. Rost, R. Sujatha}, \emph{Unramified cohomology of quadrics I}.  Amer. J. Math.
{\bf 120}, no. 4, 841--891 (1998)

\bibitem{Ka}\textsc{K. Kato}, \emph{A Hasse principle for two-dimensional global fields (with an appendix by Jean-Louis Colliot-Th\'{e}l\`{e}ne). }J. Reine Angew. Math. {\bf 366}, 142--183. (1986)

\bibitem{KMRT}\textsc{M.-A. Knus, A. Merkurjev, M. Rost, J.-P. Tignol},
\emph{The Book of Involutions}. Amer. Math. Soc. Coll. Pub. \textbf{44},
AMS, Providence, RI (1998)

\bibitem{Ma}\textsc{Y. Manin}, \emph{Cubic forms}.
North-Holland, Amsterdam, 1974.

\bibitem{Me} \textsc{A.S. Merkurjev}, \emph{R-equivalence and
rationality problem for semisimple adjoint classical groups.
}Pub. Math. I.H.\'{E}.S. {\bf 46}, 189--213 (1996)

\bibitem{Me2} \textsc{A.S. Merkurjev}, \emph{Certain $K$-cohomology groups of Severi-Brauer varieties, }in  K-Theory and Algebraic Geometry: Connections with Quadratic Forms and Division Algebras, AMS Summer Research Institute, Santa Barbara 1992, ed. W. Jacob and A. Rosenberg, Proceedings of Symposia in Pure Mathematics {\bf 58}, Part II, 319--331 (1995)  

\bibitem{Tao} \textsc{D. Tao}, \emph{The generalized even Clifford
algebra. }J. Algebra {\bf 172}, No 1, 184--204 (1995)

\end{thebibliography}
\end{document}

