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\begin{document}


\title{Serre's conjecture II: a survey}

\author{Philippe Gille}


\date{}
\smallskip

 \maketitle

\bigskip

%\thanks{{\tt Version of \today.}}





\bigskip

 \begin{small}
\noindent {\bf Abstract\footnote{{\tt  Version of \today.}}:} Our goal is to provide an up to date survey of Serre's conjecture II (1962) on the vanishing of
Galois cohomology for simply connected semisimple groups defined over a field of 
cohomological dimension $\leq 2$. 
\bigskip

\noindent {\bf Keywords:} Galois cohomology, linear algebraic groups.

\noindent {\bf MSC: 20G05}


%\noindent
%{\bf Keywords: }

\end{small}



%{\small \tableofcontents }


 \vskip14mm 



%\date{December 2008}



%\vskip14mm


\section{Introduction}


 Serre's original conjecture II (1962) states that the Galois cohomology set $H^1(k,G)$
vanishes for a semisimple simply connected  algebraic group $G$ defined over a perfect field $k$ of cohomological dimension 
$\cd(k) \leq 2$ \cite[\S 4.1]{Se0} \cite[II.3.1]{Se1}.
In other words,  that all $G$-torsors (or principal homogeneous spaces) over $\Spec(k)$ are trivial.
%Possibly, this conjecture has been done with analogy with the following result of Topology (??).

For example, if $A$ is a central simple algebra defined over a field $k$ and $c \in k^\times$,
the subvariety
$$
X_c:=\{ \nrd(y)=c \} \, \subset \GL_1(A)  
$$
of elements of reduced norm $c$ is a torsor under the special linear group
$G=\SL_1(A)$ which is semisimple and simply connected.
If $\cd(k) \leq 2$, we expect then that this $G$-torsor is trivial, i.e.
 $X_c(k) \not= \emptyset$. By considering all scalars $c$, we expect then that the reduced norm map
$A^\times \to k^\times$ is surjective.


For imaginary number fields, the surjectivity of the reduced norm map goes back to Eichler in 1938 (see \cite[\S 5.4]{Kn}).
For function fields of complex surfaces, this follows from the Tsen-Lang theorem given that
the reduced norm is a homogeneous form of degree $\deg(A)$ in $\deg(A)^2$-indeterminates
\cite[II.4.5]{Se1}.
The general case of the surjectivity of  reduced norm maps
was established in 1981 by  Merkurjev and Suslin  \cite[th. 24.8]{Su}. As we shall explain below, this fact essentially 
characterizes fields of cohomological dimension $\leq 2$.


Throughout its history, the evidence for and progress towards establishing conjecture II has been gathered by either considering special classes of fields, or by looking at the implications that the conjecture would have on the classification of algebraic groups. We will explore both points of view in this survey. 

>From the groups point of view, the strongest evidence
for the validity of the conjecture is given by the description of the classical groups  established in 1995 by Bayer  and Parimala  \cite{BP}.
>From the point of view of fields,  we know that the  conjecture holds
in the case of imaginary number fields (Kneser \cite{Kn}, Harder \cite{Harder}, 
Chernousov \cite{C89}, see \cite[\S 6]{PR}),
and  more recently for function fields of complex surfaces.
For exceptional groups with no factors of type $E_8$,
the relevant reasonings and references are given in \cite{CTGP}. A general proof for all types using deformation methods was given in 2008 by  He-de Jong-Starr \cite{dJHS}.
This  result has a clear geometric meaning:
If $G/\C$ is a semisimple simply connected group and $X$ a smooth complex surface, then 
a  $G$-torsor over $X$ (or a $G$-bundle) is  locally trivial with respect to the Zariski topology (see \S \ref{cor-main}).


There are previous surveys on Galois cohomology discussing Serre's conjecture II and related topics.
Tits' lectures at Coll\`ege de France from 90-91 discuss 
the Hasse principle  and group classification \cite{T90}. Serre's Bourbaki seminar  deals among other 
things with  
progress and the status of  conjecture II as of  1994. For classical groups,  there is 
 Bayer's survey  \cite{Ba}.
For function fields of surfaces, see the surveys of 
Starr \cite{Starr} and Lieblich \cite{L}.

For exceptional groups (trialitarian, type $E_6$, $E_7$ and
$E_8$), the general conjecture is still open in spite of some considerable progress
\cite{C03}\cite{CTGP}\cite{CTOP}\cite{Gi01}. 



We finish the introduction by mentioning that Serre's conjecture II can be linked  with  analogous considerations in Topology
within Morel-Voevodky's theory \cite{MV}. Indeed, if $G$ is a semisimple simply connected complex group,
we know that $\pi_1( G)= \pi_2(G)=0$, hence $G$ is $2$--connected.
 Then for every $CW$-complex of dimension $\leq 2$,
the $G$-bundles over $X$ are trivial (cf. \cite[th. 11.34]{Sw}).

%\footnote{The same theme is found in the so-called Serre's conjecture:   Projective modules of finite type over a polynomial ring with coefficients in a field are free. This assertion, which was established independently by Quillen and Suslin, serves as the algebraic counterpart to the trivial topological fact that all vector bundles over (real or complex) affine space are trivial.} 


\bigskip

\noindent{\bf Acknowledgement.}
We thank Vladimir Chernousov, Skip Garibaldi, Fabien Morel, Arturo Pianzola and Jean-Pierre Tignol 
for useful comments which improved the exposition.

\medskip



\section{Fields of cohomological dimension $\leq 2$}

Let $k$ be a field and $l$ be a prime.
Recall that $k$ is of $l$--cohomological dimension at most $d$, written  $\cd_l(k) \leq d,$ if 
$H^i(k,A) =0$ for every finite $l$--primary Galois  module $A$ and for all  $i \geq d+1$.
We know that this assertion is equivalent to the vanishing of
$H^{d+1}(L, \Z/l\Z)$ for any  finite separable extension $L/k$.
Recall the following examples of fields of cohomological dimension $2$. 

\begin{examples} {\rm
 (1) Imaginary number fields;

\smallskip

\noindent (2) Function fields of complex surfaces;

\smallskip

\noindent (3)
 Merkurjev's tower of fields $F_\infty$, namely an extension 
of $\C(X_1,...,X_{2n})$ such that the $u$-invariant is $u(F_\infty)=2n$.
This means that  every $2n+1$-dimensional quadratic form over $F_\infty$
is isotropic but the form $\langle X_1,X_2, \cdots X_{2n}\rangle$ remains anisotropic over $F_\infty$.
Furthermore the tensor product of the quaternion algebras $(X_{2i-1}, X_{2i})$ for $i=1,...,n$
  is a division algebra over $F_\infty$ \cite{M0}\cite[th. 3]{M1}. 
}
\end{examples}

The third example shows that  central simple algebras and quadratic forms 
are not in general low dimensional objects. 
We have already mentioned the following characterization which uses 
 Merkurjev-Suslin's theorem \cite{MS}.


\begin{theorem}\label{MSjo} \cite[th. 24.8]{Su}
Let $l$ be an invertible prime in $k$.
The following are equivalent:

\begin{enumerate}
\item $\cd_l(k) \leq 2$.
\item For any finite separable extension $L/k$ and any $l$-primary central simple $L$--algebra
$A/L$, the reduced norm $\nrd: A^\times \to L^\times$ is surjective.
\item For any finite extension $L/k$ and any $l$-primary central simple $L$--algebra
$A/L$, the reduced norm $\nrd: A^\times \to L^\times$ is surjective.
\end{enumerate}
 
\end{theorem}

We have  added here the easy implication   $2) \Longrightarrow 3)$ which follows of 
by a familiar transfer argument.
We say that $k$ is of cohomological dimension  $\leq d$ if 
$k$ is of $l$--cohomological dimension $\cd_l(k) \leq d$ for all primes $l$.

If $k$ is of positive characteristic $p$, we always have $\cd_p(k) \leq 1$; this  explains the necessary change
in  the following analogous statement.


\begin{theorem}\label{MSp} \cite[th. 7]{Gi00}
Assume that ${\rm char}(k) =p >0$. The following are equivalent:

\begin{enumerate}
\item $H^3_{p}(L)=0$ for any finite separable extension $L/k$;
\item For any finite separable extension $L/k$ and any $l$-primary central
 simple $L$--algebra $A/L$, the reduced norm $\nrd: A^\times \to L^\times$ is surjective.
\end{enumerate}
 
\end{theorem}

Here  $H^3_{p}(k)$ is the cohomology group of Kato 
defined by means of logarithmic differential forms \cite{K}, see also \cite[\S 9]{GiS}.
We shall say that  $k$ is of separable $p$--dimension $\leq d$ if
$H^{d+1}_{p}(L)=0$ for all finite separable extension $L/k.$
This defines in the obvious way the separable dimension $\sd_p(k)$
of $k$. For $l \not=p$, we define 
 $\sd_l(k)=\cd_l(k)$.\footnote{Kato defined the $p$--dimension $\dim_p(k)$  as follows \cite{K}.
If $[k:k^p] = \infty$, define  $\dim_p(k)=\infty$.
If  $[k:k^p] = p^r < \infty$, 
  $\dim_p(k)=r$ if  $H^{r+1}_{p}(L)=0$ for  any finite  extension $L/k$,
and   $\dim_p(k)=r+1$ otherwise.}
If $k$ is perfect, then $H^i_p(L)=0$ for every finite extension $L/k$ and for every  $i \geq 2$.
Hence if $k$ is perfect and of cohomological dimension $\leq 2$,
$k$ is of separable  dimension $\leq 2$.



\begin{examples}{\rm
\noindent (1) The function field of a curve over a finite field is of separable dimension $2$.

\noindent (2) The function  field $k_0(S)$ of
a surface over an algebraically closed field $k_0$ of characteristic $p\geq 0$ 
is of separable dimension $2$.

\noindent 
(3) Given an arbitrary field $F$, Theorems \ref{MSjo} and \ref{MSp} provide a way to construct 
a ``generic'' field extension $E/F$ of separable dimension $2$, see Ducros \cite{D}.
}
\end{examples}



We can now state the strong form of Serre's conjecture II.
For each simply connected group $G$, Serre defined the set $S(G)$ of  primes in terms of
the Cartan-Killing type of $G$, cf.~\cite[\S 2.2]{Se2}. 
For  absolutely simple almost groups, the sets $S(G)$ are as follows.



\begin{table}[ht]
\centering
\caption{$S(G)$ for absolutely almost simple groups} \label{torsion.table}
\bigskip

\begin{tabular}{|c|c|} \hline
type&$S(G)$\\ \hline
$A_n$ ($n \ge 1$) & $2$ and the prime divisors on $n+1$\\
$B_n$ ($n \ge 3$),  $C_n$ ($n \ge 2$), $D_n$ (non trialitarian for $n = 4$)&2 \\
$G_2$&2 \\
trialitarian $D_4, F_4, E_6, E_7$&2, 3\\
$E_8$&2, 3, 5 \\ \hline
\end{tabular}

\end{table}



\begin{conjecture}\label{strong} 
 Let $G$ be a semisimple simply connected algebraic group.
Assume that $\sd_l(k) \leq 2$ for every prime $l \in S(G)$, then
$H^1(k,G)=0$.
\end{conjecture}



In the  original conjecture, $k$ was assumed perfect and of cohomological dimension $\leq 2$.
In characteristic $p>0$, Serre's strenghthened  question assumed furthermore that $[k:k^p] \leq p^2$
if $p$  belongs to  $S(G)$ 
\cite[\S 5.5]{Se2}.  In view of all known results, it would seem that there is no need for this assumption.


 
Conjecture \ref{strong} is indeed stronger than the original one. 
Theorems \ref{MSjo} and \ref{MSp} show that this strong version of the conjecture holds for groups of inner type $A,$ and that
 the hypothesis on $k$
are sharp.



\section{Link between the conjecture and  the classification of groups}

The classification of semisimple groups reduces essentially to that of
semisimple simply connected groups $G$ which are absolutely almost simple  \cite[\S 31.5]{KMRT}\cite{T66}.
This means that $G \times_k k_s$ is isomorphic to $\SL_{n,k_s}$, $\Spin_{2n+1, k_s}$, $\Sp_{2n, k_s}$,   $\Spin_{2n, k_s}$,...

Let $G/k$ be such a $k$--group and
let $G \to G_{ad}$ be the adjoint quotient of $G$.
Denote by  $G^{q}$ its quasi-split form and
by  $G^{q}_{ad}$ its adjoint quotient.
Then $G$ is  an inner twist of $G^{q}$, i.e. there exists a cocycle 
 $z \in  Z^1(k_s/k,G^q_{ad}(k_s))$ such that  $G \cong {_zG}^q$.
We  identify then  $G$ and ${_zG}^q$.

Converely, we know that there exists a unique class $\nu_G=[a] \in H^1(k,G_{ad})$
such that  $G^q \cong {_aG}$  \cite[31.6]{KMRT}.
We denote by $z^{op} \in  Z^1(k,{_zG_{ad}^q})$ the opposite cocycle of $z$, 
it is defined by $\sigma \mapsto z_\sigma^{-1} \in {_zG}(k_s)$.

We have $G^q \cong {_{z^{op}} \bigl({_zG}^q}\bigr)$.
Hence the image of $\nu_G$ under 
$H^1(k,G_{ad}) \simlgr   H^1(k, {_zG^q}_{ad})$ is nothing but $[z^{op}]$. 
We have  an exact sequence
$$
1 \to Z(G) \to G \to G_{ad} \to 1
$$
of $k$--algebraic groups with respect to the $fppf$-topology (faithfully flat of finite presentation, see \cite[III]{DG}
or \cite{Sh}).
This gives rise to an exact sequence of pointed sets \cite[app. B]{BFT}
$$
1 \to Z(G)(k) \to G(k) \to G_{ad}(k) \buildrel \varphi_G \over \lgr 
$$
$$
H^1_{fppf}(k,Z(G)) \to H^1_{fppf}(k,G)  \to H^1_{fppf}(k,G_{ad}) \buildrel \delta_G  \over \lgr  H^2_{fppf}(k,Z(G)).
$$
The homomorphism  $\varphi_G$ is called the characteristic map and
the mapping $\delta_G$ is the boundary. Since $G$  (resp. $G_{ad}$) are smooth,
the $fppf$-cohomology of $G$ (resp. $G_{ad}$) coincide with Galois cohomology \cite[XXIV.8]{SGA3}, 
i.e. we have a bijection  $H^1(k,G)  \simlgr  H^1_{fppf}(k,G)$.
Following  \cite[31.6]{KMRT}, one  defines  the Tits class $T_G$ of $G$ by the formula
$$
t_G= \, - \, \delta_G\bigl(   \nu_G  \bigr) \in  H^2_{fppf}(k,Z(G)).
$$
By the compatibility property under
 the torsion bijection  $\tau_z$ \cite[IV.4.2]{Gir}\footnote{Note  that $Z(G)=Z(G^{q})$ since 
$G^q_{ad}$ acts trivially on $Z(G^{q})$.} 
$$
\begin{CD}
H^1(k,G_{ad}) @>{\delta_G}>> H^2_{fppf}(k,Z(G)) \\
@V{\tau_z}V{\wr}V @V{ ? \, + \,  \delta_{G^q}([z])}V{\wr}V \\
H^1(k,G^q_{ad}) @>{\delta_{G_{ad}}}>> H^2_{fppf}(k,Z(G^q)) ,\\
\end{CD}
$$
we see that $t_G= \delta_{G^q}( [z])$ which is indeed Tits definition \cite[\S 1]{T90}.
%Recall that a strongly inner form of $G$ is 
%the twist of $G$ by a cocycle $a \in Z^1(k_s/k, G(k_s))$ \cite{T90}. 


\begin{proposition} 
Assume that $H^1(k,G)=1$.

\begin{enumerate}
\item The boundary map 
$H^1(k,G_{ad}) \to H^2_{fppf}(k, Z(G))$ has trivial kernel.

\item Let $G'$ be an inner  $k$--form of $G^{q}$.
Then $G$ and $G'$ are isomorphic if and only if $t_G=t_{G'}$. 

\end{enumerate}
\end{proposition}


\begin{proof}
(1) This follows  from the exact sequence above.

\smallskip

\noindent (2) Let $z' \in Z^1(k,G^q_{ad})$ be a cocycle such that
$G' \cong {_{z'}G}$. We assume that  $t_G=t_{G'}$.
Hence $\delta_{G^q}( [z])= \delta_{G^q}( [z']) \in H^2_{fppf}(k,Z(G^q))$.
The compatibility above shows that
$$
\tau_z^{-1}([z']) \in \ker\bigl( H^1(k,G_{ad}) \bigr)\to H^2_{fppf}(k,Z(G)) \bigr).
$$
By 1), we have $\tau_z^{-1}([z'])= [1] \in H^1(k,G_{ad})$, hence
$[z]=[z'] \in H^1(k,G_{ad})$.
Thus $G$ and $G'$ are $k$--isomorphic.
\end{proof}



Summarizing then, the validity of Serre's conjecture II implies that semisimple $k$-groups 
are classified by their quasi-split forms and their Tits classes.
For more precise results for classical groups, see Tignol-Lewis \cite{LT}.
The classification  is of special importance in view of the rationality question for  groups
(Chernousov-Platonov \cite{CP}, see also Merkurjev \cite{M3}), and consequently also 
 for the  Kneser-Tits problem (Gille, \cite{Gi07}).




\section{Approaches to  the conjecture} 

We would like to describe some of the methods that have been used to attack the conjecture to date, 
and their limitations. We should point out that separating each of the methods and looking at them individually is a bit artificial. In practice, most work is carried by simultaneously combining the different methods.



\subsection{Subgroup trick}
 
We illustrate how this method works by using  the following example due to Tits \cite{T92}. Let $G/k$ be the
split semisimple simply connected group of type  
$$
\mbox{$E_6 \qquad$}
\begin{picture}(65,45)
\put(00,02){\line(1,0){80}}
\put(40,02){\line(0,1){20}}
\put(00,02){\circle*{3}}
\put(20,02){\circle*{3}}
\put(40,02){\circle*{3}}
\put(60,02){\circle*{3}}
\put(80,02){\circle*{3}}
\put(40,22){\circle*{3}}
\put(-5,-11){$\alpha_1$}
\put(15,-11){$\alpha_3$}
\put(35,-11){$\alpha_4$}
\put(55,-11){$\alpha_5$}
\put(75,-11){$\alpha_6$}
\put(35,30){$\alpha_2$}
\end{picture}
\mbox{\qquad}
$$

\smallskip

\noindent Assume here that $k$ is infinite. Let $z \in Z^1(k_s/k,G),$ and consider the 
twisted group $G'={_zG}$. Since $t_{G'}=0$, the 27-dimensional standard representation of 
$G$ of highest weight $\overline \omega_6$ descends to $G'$ by \cite{T71}. We have then
a representation $\rho':   G' \to \GL(V)$. The point is that  
$G'$ has a dense orbit in the projective space $X=\P(V)$, so there exists a $k$-rational point $[x]$ 
in this orbit. The connected stabiliser $(G'_x)^0$ is then semisimple of type $F_4$ \cite[9.12]{Ga2}.
Assuming that Conjecture \ref{strong} holds for groups of type $F_4$, it follows that   
$(G'_x)^0$ is split. Hence $G'$ has relative rank $\geq 4$ and a glance on Tits tables \cite{T66}
tells us that $G'$ is split. It is then easy to conclude that $[z]=1 \in H^1(k,G)$.


The subgroup trick, and variations thereof, was fully investigated by Garibaldi in his Lens lectures \cite{Ga2}.
The underlying topic is that of prehomogeneous spaces, namely projective $G$-varieties with a dense orbit. 



Unfortunately, this  trick  works only in few cases.
Tits has shown that the general form of type $E_8$ is ``almost abelian''
namely that it has no non trivial (connected) reductive subgroups other than 
maximal tori  \cite{T92}. Garibaldi and the author have shown that the general trialitarian group
is almost abelian \cite{GaGi}.





% \subsection{Inductive process for quasi-split groups} 


\subsection{Rost invariant}
In this case, the idea is to derive Serre's conjecture II
from a more general setting. The Rost invariant \cite{GMS} generalizes 
the Arason invariant for $3$-fold Pfister form which attaches (in  characteristic $\not =2$) to a Pfister form
$\phi=\langle\langle a,b,c\rangle \rangle$ the cup-product $e_3(\phi)={(a) \cup (b) \cup (c)} 
\in H^3(k, \Z/2\Z)$.  We see it now as the cohomological invariant 
$H^1(k, \Spin_8) \to H^3(k, \Z/2\Z(2))$.
More generally, for $G/k$ simply connected and absolutely almost simple, there is a cohomological invariant
$$
r_k : H^1(k,G) \to H^3(k, \Q/\Z(2))
$$  
where the $p$-primary part has to be understood in  Kato's setting \cite{GMS}.
If this invariant has trivial kernel, then $H^1(k,G)=1$ for 
$G/k$ satisfying the hypothesis of Conjecture \ref{strong}.
This is the case for $\Spin_8$ by Arason's theorem, namely the fact that the invariant $e_3(\phi)$ determines $\phi$.


\subsection{Serre's injectivity question}

A special case of  a question raised by Serre in  1962 (\cite{Se0}, see also  \cite[\S 2.4]{Se2}) is the following.

\begin{question}\label{q-inj} Let $G/k$ be a connected linear algebraic group. Let $(k_i)_{i=1,..,r}$ be
a family of finite field extensions of $k$ such that $g.c.d.( [k_i:k])=1$. 
Is the kernel of  the map
$$
H^1(k,G) \to \prod\limits_{i=1,..,r}  H^1(k_i,G) 
$$
 trivial  ?
\end{question}


\begin{remarks}{\rm

\noindent (1) The hypothesis of  connectedness is necessary since there are counterexamples
with finite constant groups \cite{GGHZ}\cite{Pa}.
Let us mention here another counterexample concocted
by S. Garibaldi. Suppose that  $k$ supports central division algebras $B_i$ of degree $i$ for $i = 3, 5$. 
There are extensions $k_i$ of $k$ of degree $i$ that split $B_i$. The groups $\SL_1(B_3 \otimes_k  B_5 )$ and 
$\SL_1(B^{op}_3 \otimes_k B_5)$ are isomorphic over
$k_i$ for $i = 3, 5$ but they are not $k$-isomorphic.  So Serre's injectivity fails for the group
 $\Aut\bigl(\SL(B_3 \otimes B_5)\bigr)$, which has just two connected components. 
\smallskip


\noindent  (2) Question \ref{q-inj}  has been  generalized by Totaro  \cite[question 0.2]{To}.  See also
\cite{GaH}.
 

\smallskip

\noindent (3) If $k$ is of positive characteristic $p$, there exists a complete DVR $R$
with residue field $k$ and an $R$--group scheme
${\mathfrak G}$ with special fiber $G$ and such that the fraction field $K$ of $R$ is of characteristic zero.
An answer for ${\mathfrak G}_K$ to Serre's question yields 
an answer for $G$. 
A fortiori,  we can assume without lost of generality,  that the extensions $k_i/k$ are separable.
}
\end{remarks}

We shall rephrase  question \ref{q-inj} in terms of special fields.


\begin{definition} Let $l$ be a prime. We say that a field $k$ is $l$--special if
every finite separable extension of $k$ is of degree a power of $l$.
\end{definition}

 
The subfield $k_l$ of $k$ consisting of elements fixed by a $p$-Sylow subgroup of $\Gal(k_s/k)$ is $l$-special.  We call
 $k_l$ a \emph{co-$l$-closure} of $k$.
If we restrict  Serre's question for finite  separable extensions $k_i/k$
and consider all cases, it can be  rephased  by asking  whether the map
$$
H^1(k,G) \to \prod\limits_{l}  H^1(k_l,G) 
$$
has trivial kernel for $l$ running over the primes.
If  the  answer to this  question is in the affirmative, then conjecture II  becomes  a question
for $l$-special fields for primes $l$ in $S(G)$. 

Indeed there are very few cases for which  the answer to Serre's question is known:
unitary groups (Bayer-Lenstra \cite{BL}), 
%few other classical groups \cite{BS}, 
 groups of type $G_2$,
  quasi-split groups of
type  $D_4$, $F_4$, $E_6$, $E_7$  \cite{Gi97} \cite{C03} \cite{Ga1}.   


If we know that the Rost invariant has trivial kernel, then we easily deduce that the answer to Question 
\ref{q-inj} is yes. Thus we can answer Serre's question for groups of type $G_2$,
and quasi-split semisimple simply connected  groups of type
$D_4$, $F_4$, $E_6$ and $E_7$.



\section{Known cases in terms of groups}

\subsection{Classical groups}

Recall that a semisimple simply connected  group is classical if 
its factors are of type $A$, $B$, $C$ or  $D,$ and
there is no triality involved. 

\begin{theorem} \label{classic}
Let $G$ be a semisimple simply connected classical group 
which is absolutely almost simple.
 Then $H^1(k,G)=1$.
\end{theorem}

If $k$ is perfect or ${\rm char}(k) \not =2$, this is  the original Serre's conjecture II proven
 by Bayer-Parimala \cite{BP}.
The general case  is done in recent work by 
Berhuy-Frings-Tignol \cite{BFT}. Their proof is based on 
Weil's  presentation of classical  group in terms of unitary groups of  algebras with involutions \cite{W},
This proof  is characteristic free.  It provides  quite a different  approach to the conjecture than that present in  Bayer and Parimala's  work.

Possibly the most tricky case is that of outer groups of type $A$, namely
unitary groups of central simple algebras equipped with an involution 
of the second kind. It is enough to think  about the  number field case
using Landherr's theorem \cite[\S 5.5]{Kn} to see how complicated the case these outer groups is.  


\subsection{Quasi-split exceptional  groups}

For this type of groups, the best results to date have been obtained by investigating the Rost invariant.


\begin{theorem}\label{rost}  
Let $G/k$ be a quasi-split semisimple simply connected group of Cartan-Killing type $G_2$, $F_4$, $D_4$, $E_6$ or $E_7$. 
 Then the Rost invariant $H^1(k,G) \to H^3(k,\Q/\Z(2))$ has trivial kernel.
\end{theorem}


Note first that this kind of statement reduces to the characterictic zero case by a lifting argument \cite{Gi00}.
For the cases $G_2$ and $F_4$, see \cite{BP} or \cite{Se2}.
As pointed out by Garibaldi,  the $D_4$ case is done in the Book of Involutions but
not stated in this form. We need to know that a trialitarian algebra whose underlying  algebra
is split  arises  as the endomorphism of a twisted composition \cite[44.16]{KMRT}
and to use results on degree 3 invariants of twisted compositions ({\it ibid}, 40.16).
For type $E_6$ and $E_7$,  this is due independently to  Chernousov \cite{C03} and Garibadi \cite{Ga1}.

Thus Conjecture \ref{strong} holds for quasi-split groups of all types other than $E_8.$
The author has given  an independent proof  based on
Bruhat-Tits theory which is  quite different in spirit from the one outlined above \cite{Gi01}. 
For the split group of type  $E_8,$ which will be denoted by $E_8$, the Rost invariant has  in general a non trivial kernel 
(for the field of real numbers and also for suitable fields of
cohomological dimension $4$, see \cite[appendix]{GiGT}). In  characteristic $0$,
Semenov constructed recently a higher  invariant
$$
\ker\bigl[ H^1(k,E_8) \to H^3(k,\Q/\Z(2)) \bigr] \to H^5(k,\Z/2\Z)  
$$
which is non trivial since it is does not vanish for the field of real numbers \cite[\S 8]{Sm}.
Moreover, Semenov's invariant has trivial kernel for $2$-special fields.

By means of norm group  of varieties of Borel subgroups, the case of quasi-split groups is the
key ingredient  for proving the following.

\begin{theorem}\label{center} \cite[th. 6]{Gi01}
Let $G/k$ be a semisimple simply connected group which satisfies the hypothesis of Conjecture  \ref{strong}. 
Let $\mu \subset G$ be a finite central subgroup of $G$. Then the characteristic map
$$
(G/ \mu)(k) \to H^1_{fppf}(k,\mu)
$$  
is surjective.
\end{theorem}


The  flat cohomology (see \cite{Sh}, \cite[app. B]{BFT} or \cite{Gir}) is the right set up where to phrase the problem if the order of $\mu$
is not invertible in $k.$ If the order of $\mu$ i invertible in $k$ then the flat and usual Galois cohomology coincide.
By continuing the exact sequence of pointed sets 
$$
1 \to \mu(k) \to G(k) \to (G/ \mu)(k) \to H^1_{fppf}(k,\mu) \to H^1_{fppf}(k,G),
$$
we see that $H^1_{fppf}(k,\mu) \to H^1_{fppf}(k,G)$ is the trivial map.
In other words, the center of $G$  does not contribute to $H^1(k,G)$.\footnote{The reason why we can avoid  groups of type $E_8$ is because such groups have trivial center.}  



\subsection{Other  exceptional  groups}

\begin{theorem}\label{other} \cite{C03} \cite{Gi01} Let $G/k$ be a semisimple group satisfying
 the hypothesis of  Conjecture \ref{strong}.  
Then $H^1(k,G)=1$ in the following cases: 

\begin{enumerate}
\item  $G$ is trialitarian and its Allen algebra is of 
index $\leq 2$. 

\item $G$ is of quasi-split type $^1E_6$ or $^2E_6$ and its Tits   algebra is of 
index $\leq 3$. 

\item $G$ is of type $E_7$ and its Tits   algebra is of 
index $\leq 4$. 
\end{enumerate}

\end{theorem}

\smallskip

\noindent Furthermore the groups under consideration  are either quasi-split or isotropic  of Tits indices
$$
\begin{picture}(40,15)
\put(00,00){\line(1,0){30}}
\put(30,0){\oval(60,20)[l]}
\put(0,0){\circle*{3}}
\put(30,0){\circle*{3}}
\put(30,10){\circle*{3}}
\put(30,-10){\circle*{3}}
\put(00,0){\circle{10}}
\end{picture},
$$

$$
%\mbox{$^1E_{6,2}^{1,6}, \, ^2E_{6,2}^{1,6''} \quad$}
\quad 
\begin{picture}(80,30)
\put(00,02){\line(1,0){25}}
$$\put(65,02){\oval(80,10)[l]}
%\put(0,02){\circle*{3}}
%\put(25,02){\circle*{3}}
\put(0,02){\circle{10}}
\put(25,02){\circle{10}}
\put(65,-3){\circle*{3}}
\put(65,7){\circle*{3}}
\put(45,7){\circle*{3}}
\put(45,-3){\circle*{3}}
%\put(65,02){\oval(10,20)}
\put(0,02){\circle*{3}}
\put(25,02){\circle*{3}}
\put(-5,11){$\alpha_2$}
\put(20,11){$\alpha_4$}
\put(40,11){$\alpha_3$}
\put(62,17){$\alpha_1$}
\put(62,-15){$\alpha_6$}
\put(40,-13){$\alpha_5$}
\end{picture}
$$

$$
\begin{array}{ll}
\mbox{\rm a)}
\qquad 
\begin{picture}(105,45)
\put(00,02){\line(1,0){100}}
\put(60,02){\line(0,1){20}}
\put(00,02){\circle*{3}}
\put(20,02){\circle*{3}}
\put(40,02){\circle*{3}}
\put(60,02){\circle*{3}}
\put(80,02){\circle*{3}}
\put(100,02){\circle*{3}}
\put(60,22){\circle*{3}}
\put(20,02){\circle{10}}
\put(60,02){\circle{10}}
\put(80,02){\circle{10}}
\put(100,02){\circle{10}}
\put(-5,-11){$\alpha_7$}
\put(15,-11){$\alpha_6$}
\put(35,-11){$\alpha_5$}
\put(55,-11){$\alpha_4$}
\put(75,-11){$\alpha_3$}
\put(95,-11){$\alpha_1$}
\put(55,30){$\alpha_2$}
\end{picture}
&
\mbox{\qquad \rm  b)}
\qquad 
\begin{picture}(125,45)
\put(00,02){\line(1,0){100}}
\put(60,02){\line(0,1){20}}
\put(00,02){\circle*{3}}
\put(20,02){\circle*{3}}
\put(40,02){\circle*{3}}
\put(60,02){\circle*{3}}
\put(80,02){\circle*{3}}
\put(100,02){\circle*{3}}
\put(60,22){\circle*{3}}
%\put(00,02){\circle{10}}
%\put(20,02){\circle{10}}
\put(100,02){\circle{10}}
\put(-5,-11){$\alpha_7$}
\put(15,-11){$\alpha_6$}
\put(35,-11){$\alpha_5$}
\put(55,-11){$\alpha_4$}
\put(75,-11){$\alpha_3$}
\put(95,-11){$\alpha_1$}
\put(55,30){$\alpha_2$}
\end{picture}
\end{array}
$$

\vskip6mm

\noindent where case $a)$ (resp. $b)$) holds when the Tits algebra is of index $2$ (resp. $4$).
One more reason why other exceptional groups are difficult to deal with
is because they are anisotropic.

%\begin{corollary} \label{cor-other} \cite{CTGP}
%Let $G/k$ as in  Theorem \ref{other}. For every 
%separable finite field extension $L/k$,  assume that  every
%central simple $L$-algebras of period $2$  (resp. $3$)  is 
 % of index $\leq 2$ (resp. $3)$.
%If $G$ is trialitarian or of type $E_7$ (resp. $E_6$), then 
%$H^1(k,G)=1$.
%\end{corollary}

For function fields of surfaces, central simple algebras of period $2$ (resp.$3$) are
of index $2$ (resp. $3$)
 as pointed by Artin \cite{Ar}. Thus 
Theorem \ref{other} covers all groups of type $^{3,6}D_4$, $^{1,2}E_6$, and $E_7$ for these fields. 

In joint work with  with Colliot-Th\'el\`ene and Parimala \cite{CTGP},  we exploited Serre's conjecture II
to study  arithmetic properties of not necessarily simply connected groups. Our methods were inspired by Sansuc's paper \cite{Sa}
in the number field case. For more on this topic, see  the paper by  Borovoi and Kunyavski\v {\i} \cite{BKG}.  
 



\section{Known cases in terms of fields}

\subsection{$l$-special fields}

\noindent (a) If $l=2,3,5$ and $k$ is an $l$-special field of separable dimension $\leq 2$, Conjecture \ref{strong} holds for 
the split group of type $E_8$, see  \cite{C94} for $l=5$ and  \cite[\S III.2]{Gi01}. 

\medskip


\noindent (b) If $l=3$ and $k$ is an $l$-special field of characteristic $\neq 2$ and 
separable dimension $\leq 2$, then 
Conjecture \ref{strong} holds for trialitarian groups.
For $l=3$,  this follows of Theorem \ref{rost}.


\medskip

In both cases, a positive answer to Serre's injectivity question would provide Conjecture \ref{strong} for the groups under consideration.


\subsection{Complete valued fields}

Let $K$ be a henselian  valued field for a discrete valuation with perfect residue field
$\kappa$. A consequence of the Bruhat-Tits decomposition for Galois cohomology over complete fields
is the following. 


\begin{theorem} (Bruhat-Tits \cite[cor. 3.15]{BT3})
Assume that $\kappa$ is of cohomological dimension $\leq 1$.
Let $G/K$ be a simply connected semisimple group. 
Then $H^1(K,G)=1$.
\end{theorem}

Note that the hypotheses imply that $K$ is of separable dimension $\leq 2$.
Serre asked whether it can be generalized 
when assuming $[\kappa:\kappa^p] \leq p$ \cite[5.1]{Se2}.
The hypothesis $[\kappa:\kappa^p] \leq p$ alone is 
not enough  here because $K=\bF_p((x))((y))$ is of separable dimension $3$ 
and is complete with residue field
$\bF_p((x))$. 



But if $\kappa$ is separably closed and $[\kappa:\kappa^p] \leq p$, then $K$  
is of separable dimension $1$   and enough cases of the vanishing of $H^1(\kappa((x)), G)$ have been established
in view of the proof of Tits conjectures on unipotent subgroups \cite{Gi02}.
The general case, however,  is still open.







%This is actually a special case of Conjecture \ref{strong}
%since $K$ is then a field of separable dimension $2$.
%For $l \not =p$, this is \cite[]{GSM}.
%Indeed, if $K$ is of characteristic zero,
%Kato's filtration of  $H^3_p(K)$ shows that $H^3_p(K)=0$

%Indeed, if $K$ is of positive characteristic, $[K;K^p] \leq p^2$ and then $H^3_p(K')$.


Note also that the conjecture is proven for fraction fields of
 henselian two dimensional local rings  with algebraically closed
 residue field of characteristic zero, e.g. $\C[[x,y]]$, as shown in  \cite{CTGP}. For the  $E_8$ case, a key point is that 
the derived group of the absolute Galois group
is of cohomological dimension $1$ \cite[Th. 2.2]{CTOP}. 



\subsection{Global fields} The number field case is due to Kneser for classical groups \cite{Kn}, Harder for
exceptional groups of type other than  $E_8$ \cite[I, II]{Harder}, and Chernousov for type $E_8$ \cite{C89}, see also \cite{PR}.
The function field case due to Harder \cite[III]{Harder}. 



\subsection{Function fields}

He, de Jong and Starr have proven   Conjecture \ref{strong} for split groups over  function fields
in a uniform way in arbitrary characteristic.

\begin{theorem}  \label{main}  \cite[cor. 1.5]{dJHS}
Let $k$ be an algebraically closed field and let  $K$ be the function field of 
a quasi-projective smooth surface $S$. Let $G$ be a split semisimple simply connected group over $k$. Then $H^1(K,G)=1$.
\end{theorem}

Except $E_8$, all other cases of the conjecture were known by  case by case 
considerations \cite{CTGP}.
Hence Conjecture \ref{strong} is fully proven for function fields of surfaces.
The proof of Theorem \ref{main} is based on 
the existence of sections for fibrations in rationally simply connected varieties.\footnote{ Integral versions of this result have recently been studied. The case when the function field $\Bbb C (t_1, t_2)$ is replaced by the Laurent polynomial ring  $\Bbb C [t_1^{\pm 1}, t_2^{\pm 1}]$ appear in the study of infinite dimensional Lie algebras. In analogy with conjecture II it is natural to ask whether $H^1_{fppf}(\Bbb C [t_1^{\pm 1}, t_2^{\pm 1}],  \mathfrak{G})$ vanishes for all simple simply connected group schemes $\mathfrak{G}$ over $\Bbb C [t_1^{\pm 1}, t_2^{\pm 1}].$ An example of B. Margaux shows that the answer to this question is negative (see \cite{GP07} \S 3.6.) }


\begin{theorem} \label{step-main}  \cite[Th. 1.4]{dJHS}
Let $S/k$ as in Theorem \ref{main}. Let $X/S$ be a projective morphism whose
geometric generic fiber is a twisted flag variety. Assume that
$\Pic(X ) \to \Pic( X \times_K \overline K)$ is surjective.
Then $X \to S$ has a rational section. 
\end{theorem}

The assumption on the Picard group means that there is no ``Brauer obstruction''.
When applied to higher Severi-Brauer schemes, this statement yields as a corollary
de Jong's theorem ``period=index'' \cite{dJ} for central simple algebras over such fields; see also
\cite{CTdJ}. This is the first classification-free work described in this survey. 


\subsection{Why Theorem \ref{step-main} implies Theorem \ref{main}}
We take  this  opportunity to reproduce how this argument goes.



\begin{lemma} \label{panin} Set $G/F$ be a semisimple simply connected group 
over a field $F$. Let $E/F$ be a $G$-torsor.

\begin{enumerate}
\item $\Pic(E)=0$ and we have an exact sequence 
$$ 
0 \to \br(F) \to \br(E) \to \br(E \times_F F_s). 
$$
\item Let  $P$ be a $F$-parabolic subgroup of  $G$ and let 
$E/P$ the variety of  parabolic subgroups of the twisted $F$-group $E(G)$
having the same type than $P$.
Then $\Pic(P)=0$ and we have an exact sequence  $$
0 \to \br(F) \to \br(E/P) \to \br(E/P \times_F F_s)
$$   and 
an isomorphism
 $\Pic(E/P) \simlgr  \Pic(E/P \times_F F_s)^{\Gal(F_s/F)}$.
\end{enumerate}

\end{lemma}

\begin{proof} (1)
We have $H^1(F, (F_s)^\times)=0$ and $\Pic(E \times_F F_s)
\cong  \Pic(G \times_F F_s)=0$ since $G$ is simply connected \cite{FI}. The first  terms
of the Hochschild-Serre spectral sequence
$H^p(\Gal(F_s/F), H^q( E \times_F F_s, {\bf G}_m)) \Longrightarrow
H^{p+q}( E, {\bf G}_m)$ show that 
$\Pic(E)=0$ and that the sequence
 $0 \to \br(F) \to \br(E) \to \br(E \times_F F_s)$ is exact.

\smallskip

\noindent {$2)$} The fibration $G \to G/P$ is locally trivial for the Zariski topology (Borel-Tits). 
By a result of Sansuc applied to the fibration $G \to G/P$ 
 \cite[6.10.2]{Sa},\footnote{Strictly speaking, Sansuc's result is proved
under the assumption that $k$ be perfect or that the group of the fibration be reductive.
By d\'evissage, one can see that the result holds for fibrations under  groups that are extensions of 
reductive $k$-group by split  $k$--unipotent groups.}  we have a surjective map $\Pic(G) \to \Pic(P)$, hence 
$\Pic(P)=0$.
 By \cite[6.10.1]{Sa} applied to the fibration $E \to E/P$, there is
an exact sequence
$$
0  =  \Pic(P) \to \Br(E/P) \to \Br(E),
$$
hence the map  $\br(E/P) \to \br(E)$  is injective. We look  at the commutative diagram
$$
\begin{CD}
&& &&0  \\
&&&& @VVV \\
 0 @>>> \br(F) @>>>   \br(E/P) @>>> \br(E/P \times_F F_s) \\
&&\mid \mid && @VVV @VVV \\ 
 0 @>>>  \br(F) @>>>   \br(E) @>>> \br(E \times_F F_s) .\\
\end{CD}
$$
Since the bottom sequence is exact, we get by diagram chasing that the upper horizontal
sequence is exact as well. The second isomorphism $\Pic(E/P) \simlgr  {\Pic(E/P \times_F F_s)^{\Gal(F_s/F)}}$ comes from the 
Hochschild-Serre spectral sequence.
\end{proof}

For complete results on Picard and Brauer groups of twisted flag varieties, see Merkurjev-Tignol \cite[\S 2]{MT}.


\begin{proposition} \label{step-main2}  \cite[Th. 1.4]{dJHS}
Let $S/k$ as in Theorem \ref{main}. Let $G/K$ be a 
 simple simply connected $K$-group which is an inner form and let $P$ be 
 a $K$-parabolic subgroup of $G$.
Then the map $H^1(K,P) \to H^1(K,G)$ 
is bijective.
\end{proposition}


Proposition \ref{step-main2} implies Theorem \ref{main} by taking a Borel
subgroup of $G$ because $H^1(K,B)=1$.

\begin{proof} Injectivity is a general fact due to Borel-Tits (\cite{BoT}, th\'eor\`eme 4.13.a).
Let  $E/K$ be a $G$-torsor of class $[E] \in H^1(K,G)$.
Up to shrinking $S$, we can assume that $G/K$ extends to 
a semisimple group scheme $\GB/S$, $P/K$ extends to a $S$--parabolic subgroup scheme
$\PB/S,$ and that  $E/K$ extends to a $\GB$-torsor $\E/S$ \cite{Ma}. 
By \'etale descent, we can twist the $S$-group scheme  $\GB/S$
by inner automorphisms, namely we can define the $S$-group scheme
 $\E(\GB)/S$.
We define then  $\V/S:= \E/\PB$, i.e. the scheme of 
parabolic subgroup schemes of $\E( \GB) /S$ (\cite{SGA3}, exp. XXVI) of the same type 
than $P$.
The  morphism $\pi :\V \to X$ is projective, smooth  and with geometrically
integral fibers.
Set $V= \V \times_S K$, this is a generalised twisted flag variety.
Since  $G$ is assumed to be an inner form,
$\Pic(V \times_K K_s)$ is a trivial $\Gal(K_s/K)$-module.
By Lemma \ref{panin}.2, the map
$$
\Pic(V ) \to \Pic(V \times_K K_s) 
$$
is onto. Thus the composite map 
$\Pic(\V)  \to \Pic(V ) \to \Pic(V \times_K K_s)$ is onto. 
Theorem \ref{main} applies and shows that $V(K) \not= \emptyset$.
It means that the  torsor $E$ admits  a reduction to $P$
 (\cite{Se1}, \S I.5, proposition 37), that is 
$[E]\in {\rm Im}\bigl(  H^1(K,P) \to H^1(K,G) \bigr)$.
We conclude that the mapping   $H^1(K,P) \to H^1(K,G)$ is surjective.
\end{proof}



The  Grothendieck-Serre's conjecture on rationally trivial torsors has been proven
by Colliot-Th\'el\`ene and Ojanguren 
for  torsors over a semisimple group  defined over an algebraic closed field \cite{CTO}.
Thus He-de Jong-Starr's theorem has the following geometric application.


\begin{corollary}\label{cor-main} Let $S/k$ be a smooth quasi-projective surface.
Let $G/k$ be a (split)  semisimple simply connected group. Let $E/S$ be a $G$-torsor.
Then $E$ is locally trivial for the Zariski topology.
\end{corollary}




\section{Remaining cases and open questions}


$\bullet$ Provide  a classification free proof of 
the case of totally imaginary  number fields, at least in the quasi-split case.

\medskip

$\bullet$ The first remaining cases of Conjecture \ref{strong}  are that of trialitarian groups, 
groups of type $E_6$ over a $3$--special field, groups of type
$E_7$ over a $2$-special field and groups of type $E_8$.

\medskip



$\bullet$ Let $K$ be a function field  of surface over an algebraically closed field.
Are $K$--division algebras cyclic ?
Is it true that $\cd(K_{ab}) =1$ where $K_{ab}$ stands for the abelian closure of $K$ ?

\smallskip

In the global field case, class field theory  answers  both   questions positively.
The question on $K_{ab}$ is due to Bogomolov and  makes sense for arbitrary fields.
As noticed by Chernousov, Reichstein and the author, a positive answer would
provide a positive answer to Serre's conjecture II for groups of type $E_8$  \cite{CGR}. 

\medskip


$\bullet$  For the Kneser-Tits conjecture for perfect fields of cohomological dimension $\leq 2$,
there remains only the case of a group of Tits index \cite[\S 8.2]{Gi07} 
$$
\mbox{$E_{7,1}^{66} \qquad$}
\begin{picture}(105,45)
\put(00,02){\line(1,0){100}}
\put(60,02){\line(0,1){20}}
\put(00,02){\circle*{3}}
\put(20,02){\circle*{3}}
\put(40,02){\circle*{3}}
\put(60,02){\circle*{3}}
\put(80,02){\circle*{3}}
\put(100,02){\circle*{3}}
\put(60,22){\circle*{3}}
\put(20,02){\circle*{3}}
\put(60,02){\circle*{3}}
\put(80,02){\circle*{3}}
\put(100,02){\circle{10}}
\put(-5,-11){$\alpha_7$}
\put(15,-11){$\alpha_6$}
\put(35,-11){$\alpha_5$}
\put(55,-11){$\alpha_4$}
\put(75,-11){$\alpha_3$}
\put(95,-11){$\alpha_1$}
\put(55,30){$\alpha_2$}
\end{picture}
$$


\bigskip

\bigskip




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\bigskip


\noindent{\it Adress : DMA, UMR 8553 du CNRS, Ecole normale sup\'erieure, 45 rue d'Ulm}

\noindent{\it F-75005 Paris, France.} 

\end{document}


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\section{Fields of virtual cohomological dimension 2}

Colliot-Th\'el\`ene made the following conjecture which generalizes 
the Hasse principle \cite[Conjecture 2.11]{CT96}.


\begin{conjecture}\label{conjIII} Let $k$ be a field of characteristic zero 
such that ${\rm cd}_l(k (\sqrt{-1}) \leq 2$ for all torsion primes of $G$.
Let $G/k$ be a semisimple simply connected group. Then the map
$$
H^1(k,G) \to \prod\limits_{v} H^1(k_v,G)
$$
 is injective, where $v$ runs over the ordering of $k$ 
and $k_v$ stands for the real closure pf $k$ with respect to an ordering $v$.
\end{conjecture}

The condition ${\rm cd}_l(k (\sqrt{-1}) \leq 2$ is equivalent to require  that
the absolute Galois group of $k_l$ admits an open subgroup which is 
of cohomological dimension $\leq 2$.  This is the reason why we speak
about a field of virtual cohomological dimension $\leq 2$.


The  positive characteristic case is irrelevant here 
since the $l$-cohomological dimension does not change in that case by finite extension
 and there are no orderings by Artin-Schreier \cite[II.4.1]{Se1}.

Beyond the Hasse principle (Kneser \cite{Kn}, Harder \cite{Harder}, Chernousov), the main evidence is the following result
by Bayer-Parimala.


\begin{theorem} \cite{BP2} Under the hypothesis 
of conjecture \ref{conjIII}, assume that  $G$ is  classical 
or of type $G_2$ or $F_4$. 
Then Conjecture \ref{conjIII} holds for $G$.
\end{theorem}


The inner type $A$ is crucial. Precisely, given a central simple algebra $A$, 
 a scalar $c\in k^\times$ is a reduced norm if and only if
$c$ belongs to $\nrd(A \otimes_k k_v)$ for all
orderings $v$ or equivalently if $c$ is postive with respect to $v$
for all orderings $v$ such that $A \otimes_k k_s$ does not split.
This result is linked with the classification. A semisimple group $G/K$
is classified by his quasi-split form, its Tits algebras and the $G \times_K K_v$.
It is applied in hermitian forms and for study on of the groups \cite{KP}. 

For exceptional groups, we have the following result.

\begin{theorem} (Chernousov, \cite[th. 7.2]{C03})
 Let $G$ be a quasi-split semisimple group of Cartan-Killing type
$D_4$, $E_6$ or $E_7$ defined over a perfect field $k$ of
virtual cohomological dimension $2$.  Then Conjecture \ref{conjIII} holds for $G$.  
\end{theorem}



%%\subsection{Link with Bogomolov's conjecture}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



\begin{proof} Let $O$ be the valuation ring of $K$ and denote by
$O_{nr}/O$ the  extension of henselian local rings  
 associated to $\kappa_s/\kappa$.
The fraction field $K_{nr}$ of $O_ {nr}$ is called  the unramified closure of $K$
and is of Kato dimension $\leq 1$ (REF).
By the extension of Steinberg's theorem \cite[th. 11.12]{St} by 
Borel-Springer \cite[\S 8.6]{BoS}, we know that
$H^1(K_{nr},G)=1$ (see also \cite[\S 4.2, th. 4']{Se2}. Hence 
$$
H^1\bigr(\Gal(\kappa_s/\kappa), G(K_{nr}) \bigl) \simlgr H^1(K,G).
$$ 
Apply Bruhat-Tits decomposition {\bf COMPLETE}.
\end{proof}
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