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\title[Local-global principles for reductive groups over function fields]{Local-global principles for homogeneous spaces of reductive groups over global function fields}

\alttitle{Principe local-global pour les espaces homogènes de groupes réductifs sur les corps globaux de fonctions}


\subjclass{11E72, 11G35, 14G12, 14G25, 14M17, 20G30}

\keywords{Global function fields, Hasse principle, Weak and Strong approximation, Brauer--Manin obstruction, algebraic groups, homogeneous spaces}


\author[\initial{C.} \lastname{Demarche}]{\firstname{Cyril} \lastname{Demarche}}
\address{Sorbonne Universit\'e and\\
Universit\'e Paris Cit\'e, \\
CNRS, IMJ-PRG,\\
F-75005 Paris (France)}
\email{cyril.demarche@imj-prg.fr}


\author[\initial{D.} \lastname{Harari}]{\firstname{David} \lastname{Harari}}
\address{Universit\'e Paris-Saclay, CNRS,\\
Laboratoire de Math\'ematiques d'Orsay,\\
F-91405 Orsay (France)}
\email{david.harari@universite-paris-saclay.fr}




\begin{abstract}
Let $X$ be a homogeneous space of a reductive
group with reductive stabilizers, defined over a global field of positive characteristic. Using duality theorems for complexes of tori, we study cohomological obstructions to various arithmetic properties.
\end{abstract}

\begin{altabstract}
Soit $X$ un espace homogène d'un groupe réductif à stabilisateurs réductifs, défini sur un corps global de caractéristique positive. \`A l'aide de théorèmes de dualité pour les complexes de tores, on étudie les obstructions cohomologiques naturelles à différentes propriétés arithmétiques.
\end{altabstract}



\datereceived{2021-07-29}
\daterevised{2022-04-01}
\dateaccepted{2022-05-24}

\editors{X. Caruso and B. Edixhoven}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\dateposted{2022-11-14}
\begin{document}
\maketitle




\section{Introduction}\label{intro}

Let $K$ be a global field of characteristic $p \geq 0$ and let $\A_K$ denote the ring of ad\`eles of $K$. Let $G$ be a reductive group over $K$, and $X$ be a homogeneous space of $G$. We are interested in rational points on $X$, and more precisely, on various local-global principles associated to $X$: does $X$ satisfy the Hasse principle, i.e. does $X(\A_K) \neq \emptyset$ imply $X(K) \neq \emptyset$? If not, can we explain the failure using the so-called Brauer--Manin obstruction to the Hasse principle? Assuming that $X(K) \neq \emptyset$, can we estimate the size of $X(K)$ by studying the so-called weak and strong approximation on $X$ (with a Brauer--Manin obstruction if necessary), i.e. the closure of the set $X(K)$ in the topological space $X(\A_K^S)$, where $S$ is a (not necessarily finite) set of places of $K$ and $\A_K^S$ is the ring of $S$-adèles (with no components in $S$)?

The answer to those questions is known in the case where $K$ is a number field, provided that the geometric stabilizers of points in $X$ are connected (see~\cite{Borcrelle} and~\cite{BDcom}).

In the case of a global field of positive characteristic, the answer is known for semisimple simply connected groups (thanks to works by Harder, Kneser, Chernousov, Platonov, Prasad), but the general case is essentially open (see~\cite[Theorem~1.9]{Ros-Tam} for some related results). In this paper, we deal with these questions when both $G$ and the geometric stabilizers are smooth, connected and \emph{reductive}.

%\smallskip*


Several new ingredients are needed to obtain our results:
\begin{itemize}
\item To show that the Brauer--Manin obstruction to the Hasse principle is the only one (see Theorem~\ref{hptheo} for the precise result), one has to use Poitou--Tate duality for complexes of tori in positive characteristic (proven in~\cite{DHdualcompl}) and a (non-straightforward) compatibility result between Brauer--Manin and Poitou--Tate pairings.

\item The statement on weak approximation (Theorem~\ref{weakaptheo}) relies on some part of Poitou--Tate exact sequence (which is established in~\cite{DHdualcompl}) for a certain complex of tori, and on abelianization techniques (namely Lemma~\ref{compatweak}). Beforehand we define in section~\ref{sectcompat} a new abelianization map associated to a homogeneous space $X=G/H$ as above, and prove a rather intricate compatibility formula (Theorem~\ref{theo compatible strong approx}).

\item Theorem~\ref{mainstrong} presents the obstruction to strong approximation. As in the number field case (settled in~\cite{BDcom}), it is related to the Brauer--Manin pairing, but there are two important differences. The first one is that there is an additional term in the exact sequence describing the obstruction, which reflects the fact that the global reciprocity map of class field theory is not surjective in positive characteristic. The second difference is that a fibration method like in loc. cit. would probably not work here (see for instance Section~5 in loc. cit.). Therefore, one should again rely on abelianization techniques (in particular the compatibility formula of Theorem~\ref{theo compatible strong approx} in its full generality, and not only for elements of the algebraic Brauer group $\br_1 X$). An important role is also played by duality theorems for complex of tori, some of them extending results of~\cite{DHdualcompl}.
\end{itemize}

\subsection*{Notation and conventions}

In the whole article (except in Section~\ref{sectcompat}, where $k$ is an arbitrary field), we consider a finite field $k$ and a projective, smooth and irreducible $k$-curve $E$. We set $K = k(E)$, which is a global function field of characteristic $p$, and fix a separable closure $K^s$ of $K$. The absolute Galois group $\gal(K^s/K)$ of $K$ is denoted by $\Ga_K$. Denote by $\Omega_K$ the set of all places of $K$; for every $v \in \Omega_K$, we will identify the Brauer group $\br K_v$ of the completion $K_v$ to $\Q/\Z$ thanks to local class field theory. For every $K$-variety $X$, we set $\ov X=X \times_K K^s$. The (cohomological) Brauer group of $X$ is denoted $\br X$, and we set
\[
\br_1 X:= \ker \left[\br X \to \br \ov X\right].
\]
We still denote by $\br K$ the image of $\br K$ in $\br X$, even though the map $\br K \to \br X$ is not necessarily injective if $X$ has no rational point. Notation like $H^i(K,C)$ for a commutative $K$-group scheme $C$ (resp. a bounded complex of commutative $K$-group schemes) always denotes fppf cohomology (resp. fppf hypercohomology) of $C$. It coincides with étale (=Galois) cohomology when $C$ (resp. every group scheme occurring in $C$) is smooth. For every finite set of places $S$ of $K$, we set
\[
\Sha^i_S(K,C):=\ker \left[H^i(K,C) \to \prod_{v\,\not\,\in S} H^i\left(K_v,C\right)\right].
\]
\[
\Sha^i(K,C)=\Sha^i_{\emptyset} (K,C); \quad \Sha^i_{\omega}(K,C)=
\varinjlim_S \Sha^i_S(K,C),
\]
where the direct limit runs over all finite subsets $S$ of $\Omega_K$. The Pontryagin dual $A^D$ of a topological group $A$ is the group of continuous homomorphism from $A$ to $\Q/\Z$ (if topology is not specified, we assume that $A$ is discrete).

%\smallskip

Let $G$ be a reductive group (always meaning: smooth connected reductive) over $K$. Let $G^\ss$ denote the derived subgroup of $G$ and $G^\sc$ the simply connected cover of $G^\ss$, together with the obvious morphism $\rho : G^\sc \to G$. Set $G^{\tor}=G/G^{\ss}$ (it is the maximal toric quotient of $G$). Let $T^\sc \subset G^\sc$ and $T_G \subset G$ be maximal tori such that $\rho(T^\sc) \subset T_G$. Let $C_G$ be the complex $C_G := [T^\sc \xrightarrow{\rho} T_G]$, with $T_G$ in degree~$0$. Following Borovoi (cf.~\cite{BorAMS} in characteristic zero), we have a natural map of Galois (hyper)cohomology sets:
\[
\ab^1_G : H^1(K,G) \to H^1_{\ab}(K,G):=H^1\left(K,C_G\right) \,,
\]
which is functorial in $K$. There is an exact sequence of $K$-group schemes
\[
1 \to \mu_G \to G^{\sc} \to G^{\ss} \to 1,
\]
where $\mu_G$ is a finite $K$-group scheme of multiplicative type. It induces an exact triangle
\[
\mu_G[1] \to C_G \to G^{\tor} \to \mu_G[2].
\]
We denote by $Z_G$ the center of a reductive group $G$.


\section{Hasse principle for homogeneous spaces} \label{two}

We start with extending a well-known result on the abelianization maps to the positive characteristic case.

\begin{prop}\label{prop ab1 bij}
Let $F$ be either a global field with no real place, or a non-archimedean local field. Then there exists a natural exact sequence of groups
\[
G^\sc(F) \to G(F) \xrightarrow{\ab^0_G} H^0\left(F, C_G\right) \to 1
\]
and the map $\ab^1_G : H^1(F,G) \to H^1(F,C_G)$ is a bijection.
\end{prop}

\begin{proof}
If the characteristic of $K$ is zero, see~\cite[Section~5]{BorAMS}. We now assume that the characteristic of $F$ is positive.

By construction, one has a short exact sequence of groups and pointed sets
\[
G^\sc(F) \xrightarrow{\rho_*} G(F) \xrightarrow{\ab^0_G} H^0(F,C_G) \to H^1\left(F, G^\sc\right) \xrightarrow{\rho_*} H^1(F,G) \xrightarrow{\ab^1_G} H^1(F,C_G) \,.
\]

In the global case, following~\cite[Satz~A]{harder}, the set $H^1(F, G^\sc)$ is trivial, hence the map $\ab^1_G$ has trivial kernel and the first sequence in the statement is exact. A twisting argument implies that the map $\ab^1_G$ is even injective. For the local case, the injectivity (and exactness of the first sequence) follows from~\cite[Theorem~4.7$\MK$(ii)]{BT}.

The proof of the surjectivity is an adaptation of the proofs of~\cite[Theorems~5.4 and~5.7]{BorAMS}, except that the existence of anisotropic maximal tori over local fields of positive characteristic is provided by~\cite[Lemma~2.4.1]{DeB} (see also~\cite[Proposition~4.4]{RosBig}).
\end{proof}


We are interested in the Hasse principle for homogeneous spaces under $G$, with \emph{reductive} (recall that by definition this includes smoothness and connectedness) geometric stabilizers. Following Raynaud (see~\cite[Definition~VI.1.1 and Proposition~VI.1.2]{Ray}), if $G$ is a smooth group scheme over $K$, a homogeneous space of $G$ is a smooth $K$-scheme $X$ with an action of $G$, such that for any $x \in X(K^s)$ (such an $x$ exists since $X$ is smooth), the stabilizer of $x$ in $\ov G:=G \times_K K^s$ is a finite type (over $K^s$) subgroup scheme $H_x$ of $\ov G$ and $\ov X$ is isomorphic to the quotient $\ov G/H_x$.

\begin{defi}
Let $X$ be a homogeneous space of a reductive group $G$ with reductive $K^s$-stabilizer $\ov H=H_x$. Let $L_X$ be the $K$-kernel defined by $X$ and $\sigma_X \in H^2(K, L_X)$ be the Springer class, that is the class of the gerbe associated to $X$ (cf.~\cite[\S~5.2.]{FSS} or~\cite[\S~7.7]{BorDuke} in characteristic zero). By assumption, the $K^s$-stabilizers are reductive, hence following loc. cit., there exists a $K$-torus $T$, which is a $K$-form of $\bar{H}^\tor$, and a natural map of marked sets
\[
\ab^2_X : H^2\left(K,L_X\right) \to H^2(K,T) \,,
\]
which is functorial in $K$. On the other hand we have the class $\eta_X \in H^1(K,[T \to G^{\tor}])$ constructed by Borovoi in~\cite{borann} (where characteristic zero is assumed, but not used in the definition of $\eta_X$). There is an exact sequence in Galois (hyper)-cohomology
\begin{equation}\label{long}
\begin{split}
H^1(K,T) &\to H^1\left(K,G^{\tor}\right) \;\to H^1\left(K,\left[T \to G^{\tor}\right]\right)\\
&\to H^2(K,T) \quad\;\;\to H^2\left(K,G^{\tor}\right).
\end{split}
\end{equation}
\end{defi}

\begin{prop}\label{twoclass}
The image of $\eta_X$ in $H^2(K,T)$ is $\ab^2_X(\sigma_X)$.
\end{prop}

\begin{proof}
The method is similar to the one used in the proof of~\cite[Theorem~9.6]{BvH2}. We start with the case when $G$ itself is a torus. Then $T$ is a subtorus of $G$ and $[T \to G]$ is quasi-isomorphic to the quotient group $G/T$. Moreover, $X$ is a principal homogeneous space of $G/T$ with class $[X] \in H^1(K,G/T) \simeq H^1(K, [T \to G])$ corresponding to $\eta_X$. As $\sigma_X \in H^2(K,T)$ is (by definition) just the image of $[X]$ by the coboundary map $H^1(K,G/T) \to H^2(K,T)$, the result holds in this case.

%\smallskip

Assume now that $G$ is an arbitrary reductive group but satisfies the additional hypothesis:
\begin{equation}\label{eq;etoile}
\tag{$*$} \text{The canonical morphism $T \to G^{\tor}$ is injective}.
\end{equation}
%\smallskip


Then $Y:=X/G^{\ss}$ is a homogenous space of $G^{\tor}$ with stabilizer (defined over $K$ in this case) $T$. The Springer class $\sigma_Y \in H^2(K,T)$ is (by construction) just $\ab^2_X(\sigma_X)$, and it is also the image of $\eta_Y \in H^1(K,[T \to G^{\tor}])$ in $H^2(K,T)$ by the first case. But $\eta_Y=\eta_X$ by functoriality of the class $\eta_X$ (\cite[\S~1.6]{borann}), whence the result when~\eqref{eq;etoile} is satisfied.

%\smallskip

We now deal with the general case. By~\cite[Proposition~9.9]{BvH2} (whose proof is identical in characteristic $p$ thanks to the assumption that $G$ and $\ov H$ are reductive), there exists a homogeneous space $Z$ of a $K$-group $F=G \times P$, where $P$ is a quasi-trivial torus, such that: the homogeneous space $Z$ satisfies~\eqref{eq;etoile} and there is a $K$-morphism $\pi : Z \to X$, compatible with the respective actions of $F$, $G$ (via the projection~$F \to G$), which makes $Z$ an $X$-torsor under $P$. In particular the $G$-homogeneous space $Z$ still has geometric stabilizer $\ov H$, with $L_Z=L_X$ and $\sigma_Z=\sigma_X$. Since Proposition~\ref{twoclass} holds for $Z$ (because its satisfies~\eqref{eq;etoile}), it also holds for $X$ by functoriality of the class $\eta_X$ and commutativity of the diagrams
\begin{gather*}
\xymatrix{ H^2(K,L_Z) \ar[r]^{\ab^2_Z} \ar[d] & H^2(K, T) \ar[d]^= \\
H^2\left(K,L_X\right) \ar[r]^{\ab^2_X} & H^2(K,T) }
\\
{\xymatrix{ H^1\left(K,\left[T \to F^{\tor}\right]\right) \ar[r] \ar[d] & H^2(K, T) \ar[d]^= \\
H^1\left(K,\left[T \to G^{\tor}\right]\right) \ar[r] & H^2(K,T) }}
\\[-1.5em]\qedhere
\end{gather*}
\end{proof}

\begin{rema}
The previous proposition holds over an arbitrary field. Recall also that the existence of a $K$-point on $X$ implies that the class $\sigma_X$ is neutral, as well as the vanishing of $\eta_X$.
\end{rema}

\begin{theo}\label{hptheo}
Let $G$ be a reductive group over $K$. Then the Brauer-Manin obstruction to the Hasse principle is the only one for homogeneous spaces of $G$ with reductive geometric stabilizers. More precisely, such a torsor $X$ has a rational point if and only if $X$ has an adelic point orthogonal to the subgroup
\[
\Be(X) := \ker\left[(\br X/\br K) \to
\prod_{v\, \in\,\Omega_K} \left(\br X_{K_v}/\br K_v\right)\right]
\]
for the Brauer--Manin pairing:
\begin{equation}\label{bmp}
X (\A_K) \times (\br X/\br K)
\to \Q/\Z, \quad (P_v) \mapsto
\left\langle \alpha, (P_v) \right\rangle_{BM}:=
\sum_{v\,\in\,\Omega_K} \alpha(P_v).
\end{equation}
\end{theo}

Recall that by global class field theory, the sum $\sum_{v\,\in\,\Omega_K} \alpha(P_v)$ is zero for every element $\alpha \in {\Im} [\br K \to \br X]$, hence the Brauer--Manin pairing~\eqref{bmp} is well defined. Also, for $\alpha \in \Be(X)$, the element $\langle \alpha, (P_v) \rangle_{BM}$ is independent of the choice of $(P_v) \in X (\A_K)$, because the localisation $\alpha_v \in
\br X_{K_v}$ of $\alpha$ is a constant element (i.e. it comes from $\br K_v$) for every place $v$.

Before starting the proof of Theorem~\ref{hptheo}, we prove the following (well known) lemma, for which we didn't find an appropriate reference.

\begin{lemm}\label{lem ext red}
Let $k$ be a field and
\[
1 \to G_1 \to G_2 \to G_3 \to 1
\]
be an exact sequence of smooth connected linear algebraic groups over $k$.

Then $G_2$ is reductive if and only if $G_1$ and $G_3$ are reductive.
\end{lemm}

\begin{proof}
Without loss of generality, one can assume that $k$ is algebraically closed. We denote by $R_u(G)$ the unipotent radical of $G$.
\begin{itemize}
\item Assume $G_1$ and $G_3$ are reductive. The image of $R_u(G_2)$ inside $G_3$ is contained in $R_u(G_3)$, hence it is trivial. Therefore, $R_u(G_2) \subset G_1$, and $R_u(G_2)$ is unipotent connected and normal in $G_1$, hence $R_u(G_2)$ is trivial. So $G_2$ is reductive.
\item Assume that $G_2$ is reductive. Since $R_u(G_1)$ is a characteristic subgroup of $G_1$, it is normal in $G_2$, hence $R_u(G_1) \subset R_u(G_2)$, hence $R_u(G_1)$ is trivial, so $G_1$ is reductive. To conclude the proof, one uses the non-trivial classical fact that a quotient of a reductive group is reductive (see~\cite[Corollary~14.11]{Bor}).
\end{itemize}
\end{proof}


\begin{proof}[Proof of Theorem~\ref{hptheo}] Fix a point $x \in X(K^s)$ and let $\ov H = H_x$
be the stabilizer of $x$ in $\ov G$. Up to replacing $G$ by a flasque resolution (\cite[Proposition~3.1]{CTfl})
\[
1 \to S \to G' \to G \to 1,
\]
where $S$ is a flasque torus (which is central in $G'$) and $G'$ is a quasi-trivial group (that is: extension of a quasi-trivial torus by a semisimple simply connected group), we can assume that the group $G$ itself is quasitrivial. Indeed $X$ is also a homogeneous space of $G'$ such that the $K^s$-stabilizer of $x$ is reductive by Lemma~\ref{lem ext red} (the stabilizer is an extension of $H_x$ by $S_{K^s}$). In particular $\pic \ov G=0$ and the group of characters $\widehat {G^{\tor}}$ of $G^{\tor}$ is a permutation Galois module (\cite[Proposition~2.2]{CTfl}).


Since $H^1(K,G^{\tor})=0$ (the torus $G^{\tor}$ being quasi-trivial), we have by exact sequence~\eqref{long} that the canonical map $H^1(K,[T \to G^{\tor}]) \to H^2(K,T)$ is injective. In addition, the class $\eta_X$ (viewed as an element of $H^2(K,T)$) is just $\ab^2_X(\sigma_X)$ by Proposition~\ref{twoclass}.


Assume that $X(\A_K) \neq \emptyset$. Then the class $\sigma_X$ is neutral at every place $v$ of $K$, which implies that $\eta_X=\ab^2_X(\sigma_X) \in \Sha^2(K,T)$. The Brauer--Manin pairing defines a morphism $\beta_X : \Be(X) \to \Q / \Z$ (recall that if $\alpha \in \Be(X)$, then $\langle \alpha, (P_v) \rangle_{BM}
\in \Q/\Z$ is independent of $(P_v) \in X(\A_K)$).


By~\cite[\S~4]{BDH}, there is a complex $\upic(X)$ (up to a shift this is the complex $\upic \ov X$ of~\cite{BvH2}; we will recover this complex in Section~\ref{sectcompat}) such that $\br_1 X/\br K$ is isomorphic to $H^1(K,\upic(X))$ (this is valid over any field $K$ such that $H^3(K,\G)=0$). Moreover by~\cite[Theorem~5.8]{BvH2}, (whose proof is still valid in characteristic $p$ thanks to the additional assumption that $G$ and $H$ are reductive), the complex $\upic(X)$ is quasi-isomorphic to $[\widehat{G^{\tor}} \to \widehat T]$ (recall that $\pic \ov G=0$). Therefore, the exact sequence of complexes of Galois modules
\[
0 \to \widehat{T} \to \left[\widehat{G^\tor} \to \widehat{T}\right] \to \widehat{G^\tor}[1] \to 0
\]
induces an exact triangle in the derived category
\begin{equation}\label{triangle upic}
\widehat{G^\tor} \to \widehat{T} \xrightarrow{\lambda} \upic(X) \to \widehat{G^\tor}[1] \,.
\end{equation}
Since $\widehat{G^{\tor}}$ is a permutation Galois module, $H^1(F,\widehat{G^{\tor}})=0$ for any field extension $F/K$, and $\Sha^2(K, \widehat{G^\tor}) = 0$ by Shapiro's lemma and {\v C}ebotarev's Theorem. Hence the long exact sequence associated to the triangle~\eqref{triangle upic} induces an isomorphism of abelian groups
\[
\psi_X:=\lambda_* : \Sha^1\left(K,\widehat{T}\right) \xrightarrow{\sim} \Be(X).
\]

By~\cite[Theorem~5.2]{DHdualcompl}, there is a Poitou--Tate perfect duality of finite groups
\begin{equation}\label{duality sha2}
\langle, \rangle_{PT} : \,
\Sha^2(K,T) \times \Sha^1\left(K,\widehat{T}\right) \to \Q / \Z \,.
\end{equation}
This pairing and the class $\eta_X=\ab^2(\sigma_X)$ define a morphism $\alpha_X : \Sha^1(K,\widehat{T}) \to \Q/ \Z$.

\begin{lemm}\label{lemm2.7}
The following diagram
\[
\xymatrix{
\Sha^1\left(\widehat{T}\right) \ar[d]_{\psi_X} \ar[rd]^{\alpha_X} & \\
\Be(X) \ar[r]^{\beta_X} & \Q / \Z }
\]
is commutative (up to sign).
\end{lemm}

\begin{proof}
By~\cite[Theorem~9.6]{BvH2} (whose proof is identical in characteristic
$p$ thanks to our additional assumptions), the class $\eta_X \in H^1(K,[T \to G^{\tor}]) \hookrightarrow H^2(K,T)$ coincides (up to a sign) with the element of $\ext^2(\upic(X),\G)$ given by the map $w : \upic(X) \to \G[2]$ of exact sequence (1) in~\cite{HS-op-desc}. Therefore the class $\eta_X$ viewed in $\Sha^2(T) \subset H^2(K,T)=\ext^2(\widehat T,\G)$ is just $\partial(\lambda)$, where $\partial :
\Hom_{\Ga_K}(\widehat T,\upic(X)) \to H^2(K,T)$ is the map defined in exact sequence (2) of~\cite{HS-op-desc}. Now Theorem~3.5. of loc. cit. (whose proof in characteristic $p$ is identical) shows that for every $a \in
\Sha^1(\widehat T)$, we have
\[
\langle \eta_X,a \rangle_{PT}=\langle \partial(\lambda),a \rangle_{PT}=
\beta_X(\lambda_*(a)),
\]
or in other words:
\[
\alpha_X(a)=\beta_X(\psi_X(a)).
\]
This concludes the proof of the Lemma~\ref{lemm2.7}.
\end{proof}

We can now finish the proof of Theorem~\ref{hptheo}. Assuming that $X(\A_K)^{\Be(X)} \neq \emptyset$, we have $\beta_X = 0$, hence $\alpha_X = 0$ by the lemma. The exactness of the pairing~\eqref{duality sha2} and Proposition~\ref{twoclass} imply that $\ab^2_X(\sigma_X)=\eta_X=0$. Now, the analogue of~\cite{BorDuke}, Proposition~6.5 for global fields of positive characteristic (the proof of which is the same, except that~\cite[Lemma~5.7]{BorDuke} is replaced by Proposition~\ref{prop ab1 bij}) implies that the map $\ab^2_X$ has ``trivial kernel'', hence $\eta_X$ is neutral. As a consequence, there exists a $G$-equivariant morphism $P \to X$ defined over $K$, where $P$ is a $K$-torsor under $G$. Since $G$ is quasi-trivial, $H^1(K,G)$ is trivial (see~\cite[Satz~A]{harder}), hence $P(K) \neq \emptyset$, therefore $X(K) \neq \emptyset$.
\end{proof}

\section{Abelianization of homogeneous spaces and a compatibility formula} \label{sectcompat}

Let $H$ be a reductive subgroup of a reductive group $G$ over an arbitrary field $k$. Set $X=G/H$ and denote by $e \in X(k)$ the image of the neutral element of $G$ in $X$. Consider the complex of Galois modules (or of commutative smooth $k$-group schemes) defined
by
\[
C_X := \left[T_{H^\sc} \to T_H \oplus T_{G^\sc} \to T_G\right],
\]
with $T_G$ in degree~$0$.
In other words, we have $C_X := \Cone(C_H \to C_G)$. Denote by $\ab^0_X$ (see~\cite[\S~2.6]{DemEdinburgh}, where the characteristic zero assumption is not needed thanks to our additional assumptions that $G$ and $H$ are smooth and
reductive) the abelianization map
\[
\ab^0_X : X(k) \to H^0_{\ab}(k,X):=H^0\left(k,C_X\right).
\]
Following~\cite{DemJOA}, we set
\[
\br_1(X,G)=\ker \left[\br X \to \br \ov G\right],
\]
which is a subgroup of $\br X$ containing $\br_1 X$. We also consider the subgroup $\Br_{1,e}(X,G)$ of $\br_1(X,G)$ consisting of those elements $\alpha$ such that $\alpha(e)=0$. Quotienting by $\br k$ induces an isomorphism $\br_{1,e}(X,G) \simeq \br_1(X,G)/\br k$. We use a similar notation for $\br_{1,e} X \simeq \br_1 X/ \br k$.

%\smallskip

The goal of this section is to prove (see Theorem~\ref{theo compatible strong approx} below) that there is a natural isomorphism
\[
\phi_X : H^1\left(k, \widehat{C}_X\right) \xrightarrow{\sim} \Br_{1,e}(X,G) \,,
\]
and that the following diagram
\[
\xymatrix{ X(k) \ar@<-30pt>[d]_{\ab'_X} \times \Br_{1,e}(X,G) \ar[r]^(.65){\ev} & \Br(k) \ar[d]^= \\
H^0(k, C_X) \times H^1\left(k, \widehat{C}_X\right) \ar@<-30pt>[u]_{\phi_X} \ar[r]^(.7){\cup} & \Br(k) }
\]
is commutative ($\widehat{C}_X$ denotes the dual complex of $C_X$: it is the cone of the morphism of complexes
\[
\left[\widehat{T}_G \to \widehat{T}_{G^\sc}\right] \to \left[\widehat{T}_H \to \widehat{T}_{H^\sc}\right] \,
\]
where $\widehat{T}_H$ is in degree~$0$). Here $\ab'_X$ is an abelianization map, which we will define in section~\ref{subsec ab field} by changing a little the map $\ab^0_X$. In~\cite{Demeio}, Demeio proves such a compatibility in characteristic zero for $\ab^0_X$ with a long and impressive cocycle computation. It is quite likely that the maps $\ab^0_X$ and our new map $\ab'_X$ actually coincide, but we did not succeed in proving this. The modified map $\ab'_X$ seems more suitable to check the required compatibility.

\subsection{Abelianization map over a field} \label{subsec ab field}

From now on, let $k$ be a field, $X$ a smooth geometrically integral $k$-variety and $\pi : Y \to X$ a torsor under a reductive $k$-group $H$. We fix a point $y_0 \in Y(k)$ and let $x_0 := \pi(y_0) \in X(k)$. Following~\cite{DemJOA}, we define $Z$ to be $Y/Z_H$, $z_0$ to be the image of $y_0$ and $\upic'(\pi)$ to be the following complex of Galois modules:
\[
\upic'(\pi) := \left[\kbar(Z)^\times \to \divl\left(\bar{Z}\right) \to \pic'\left(\bar{Z}/\bar{X}\right)\right] \,,
\]
with $\kbar(Z)^\times$ in degree $-1$ (here $\pic'(\bar{Z}/\bar{X})$ is the relative Picard group of $\ov Z$ over $\bar{X}$). We also define
\[
\upic(\pi) := \left[\kbar(Z)^\times/\kbar^\times \to \divl\left(\bar{Z}\right) \to \pic'\left(\bar{Z}/\bar{X}\right)\right] \,,
\]
with the obvious natural exact sequence of complexes:
\[
0 \to \kbar^\times[1] \to \upic'(\pi) \to \upic(\pi) \to 0 \,.
\]

We will sometimes need the following pointed version of those complexes, which are canonically quasi-isomorphic to the previous ones:
\begin{align*}
\upic'(\pi)_0 :=& \left[\kbar(Z)_0^\times \to \divl\left(\bar{Z}\right)_0 \to \pic'\left(\bar{Z}/\bar{X}\right)\right]
\\
\intertext{and}
\upic(\pi)_0 :=& \left[\kbar(Z)_{0,1}^\times \to \divl\left(\bar{Z}\right)_0 \to \pic'\left(\bar{Z}/\bar{X}\right)\right] \,,
\end{align*}
where $\kbar(Z)_0^\times$ (resp. $\kbar(Z)_{0,1}^\times$) denotes the subgroup of rational functions defined at $z_0$ (resp. taking the value $1$ at $z_0$) and $\divl(\bar{Z})_0$ is the group of divisors $D$ such that $z_0$ is not contained in the support of $D$. We also have the classical complexes $\upic(Z)=[\kbar(Z)^*/\kbar^* \to \divl \ov Z]$, $\upic'(Z)$ etc., corresponding to the case when $\pi : Z\to Z$ is the identity map (we already encountered $\upic$ in the proof of Theorem~\ref{hptheo}).

The construction of $\upic'(\pi)$ is contravariant in $\pi$ (see for instance~\cite[Proposition~2.2]{DemJOA}), and for any $x \in X(k)$, the natural morphism 
\[
\kbar^\times[1] \to \upic'\left(\pi_x : Z_x \to \spec k\right)
\]
is a quasi-isomorphism, where $Z_x$ denotes the fiber of $Z$ at $x$.

Therefore, one gets a well-defined specialization map
\[
{\ab''}_\pi : X(k) \to \Hom_k\left(\upic'(\pi), \kbar^\times[1]\right) \,,
\]
where $\Hom_k$ denotes the set of morphisms in the derived category of bounded complexes of Galois modules.

In addition, when a point $y_0 \in Y(k)$ is given, one gets a natural splitting $\upic(\pi) \to \upic'(\pi)$, hence it defines the required map
\[
{\ab'}_\pi : X(k) \to \Hom_k\left(\upic(\pi), \kbar^\times[1]\right) \,.
\]
When $Y=X$ and $\pi = \id_X$, we denote ${\ab'}_\pi$ by ${\ab'}_X$.


We want to think about ${\ab'}_\pi$ as an abelianization map for the set of rational points of $X$, which is a replacement of $\ab^0_X$ when $X=G/H$.

We now study the dévissage of $\upic(\pi)$ in terms of $Y$ and $H$ :
\begin{lemm} \label{lem triangle exact}
There is a natural exact triangle
\[
\upic(\pi) \to \upic(Y) \to \upic(H) \to \upic(\pi)[1] \,.
\]
\end{lemm}

\begin{proof}
This is essentially the proof of~\cite[Corollary~3.3]{DemJOA}. Let $H'=H/Z_H$ be the quotient of $H$ by its center. Since $\upic(\pi)_0[1]$ is the cone of $\upic(Z)_0 \to \upic(H')_0$, the following commutative diagram of complexes
\[
\xymatrix{
\upic(Z)_0 \ar[r] \ar[d] & \upic(H')_0 \ar[d] \\
\upic(Y)_0 \ar[r]^{\varphi} & \upic(H)_0 }
\]
induces a canonical morphism of complexes
\[
\alpha : \upic(\pi)_0[1] \to \con(\varphi) \,,
\]
such that the following diagram of exact triangles is commutative:
\[
\xymatrix{
\upic(Z)_0 \ar[r] \ar[d] & \upic(H')_0 \ar[d] \ar[r] & \upic(\pi)_0[1] \ar[d]^\alpha \ar[r] & \upic(Z)_0[1] \ar[d] \\
\upic(Y)_0 \ar[r]^{\varphi} & \upic(H)_0 \ar[r] & \con(\varphi) \ar[r] & \upic(Y)_0[1] \,. }
\]

Let us now prove that $\alpha$ is a quasi-isomorphism. Since the complexes $\upic(\pi)_0[1]$ and $\con(\varphi)$ are concentrated in degrees $-2$ to $0$, we only compute the cohomology corresponding to those degrees: one has a commutative diagram of long exact sequences
{\footnotesize
\begin{equation}\label{diag qis}
\xymatrix{ 0 \ar[r] & H^{-2}\left(\upic(\pi)[1]\right) \ar[r] \ar[d]^{\alpha^{-2}} & U\left(\bar{Z}\right) \ar[r] \ar[d] & \widehat{H'} \ar[r] \ar[d] & H^{-1}\left(\upic(\pi)[1]\right) \ar@{-->}[r] \ar[d]^{\alpha^{-1}} & \\
0 \ar[r] & H^{-2}(\con(\varphi)) \ar[r] & U\left(\bar{Y}\right) \ar[r] & \widehat{H} \ar[r]^(.4){\beta} & H^{-1}(\con(\varphi)) \ar@{-->}[r] & \\
\ar@{-->}[r] & \pic\left(\bar{Z}\right) \ar[r] \ar[d] & \pic\left(\bar{H'}\right) \ar[r] \ar[d] & H^{0}\left(\upic(\pi)[1]\right) \ar[r] \ar[d]^{\alpha^0} & 0 \\
\ar@{-->}[r] & \pic\left(\bar{Y}\right) \ar[r] & \pic\left(\bar{H}\right) \ar[r] & H^{0}(\con(\varphi)) \ar[r] & 0 \,. }
\end{equation}}
The proof of~\cite[Corollary~3.3] {DemJOA} ensures that $\alpha^{-2}$ and $\alpha^0$ are isomorphisms.

Let us now prove that $\alpha^{-1}$ is an isomorphism (the proof of this fact in~\cite{DemJOA} is too sketchy): using the exact sequences $\widehat{Z_H} \to \pic(\bar{Z}) \to \pic(\bar{Y})$ and $\widehat{Z_H} \to \pic(\bar{H'}) \to \pic(\bar{H})$, diagram chasing in~\eqref{diag qis} proves that the result follows from the commutativity of the following diagram:
\[
\xymatrix{
\pic\left(\bar{X}\right) & H^{-1}\left(\upic(\pi)_0[1]\right) \ar[l]_(.6){\sim} \ar[d]^{\alpha^{-1}} \\
\widehat{H} \ar[r]^(.3){\beta} \ar[u]^\Delta & H^{-1}(\con(\varphi)) \,, }
\]
where $\Delta : \widehat{H} \to \pic(\bar{X})$ is the map defined by $\chi \mapsto \chi_* [\bar{Y}]$ and the isomorphism
\[
H^{-1}\left(\upic(\pi)_0[1]\right) \to \pic\left(\bar{X}\right)
\]
is constructed in~\cite{DemJOA}. To prove the required commutativity, given $\chi \in \widehat{H}$, functoriality of the various maps implies that it is sufficient to consider the case $H = \Gm$ (and $H'=1$, $Z=X$) and $\chi = \id : \Gm \to \Gm$. More precisely, let $D \in \divl(\bar{X})$ with support not containing $x_0$, let $\pi : Y_D \to \bar{X}$ be the associated $\Gm$-torsor. Let $U \subset \bar{X}$ be the complement of the support of $D$. By construction, the pullback $Y_{D,U} \to U$ of $Y_D \to \bar{X}$ admits a canonical section $s$ over $U$, inducing a map $f : Y_{D,U} \to \Gm$ which can be seen as an element $f \in \bar{k}(Y_D)^\times$. Let $y_0 := s(x_0) \in Y_{D,U}(k)$ and let $Y_0$ be the fiber of $\pi$ at $x_0$. Then the restriction of $f$ at $Y_0$ and the point $y_0 \in Y_0(k)$ induce a morphism $f_0 : \Gm \to \Gm$. Making explicit the maps $\alpha^{-1}$ and $\beta$, the required commutativity boils down to the natural equalities $\pi^*(D) = \divli(f)$ in $\divl(\bar{Y_D})$ and $f_0 = \id$.
\end{proof}

We now want to compare the map ${\ab'}_\pi$ defined earlier with the maps $\ab^0_H$ and $\ab^1_H$ defined by Borovoi in~\cite{BorAMS} for reductive $k$-groups $H$. We first prove the following

\begin{lemm} \label{lem compat ab ab'}
With the above notation, we have a commutative diagram (up to a sign) with exact rows
{\small
\[
\xymatrix{ H(k) \ar@{^{(}->}[r] \ar[d]^{\ab^0_H} & Y(k) \ar[d]^{{\ab'}_Y} \ar[r]^\pi & X(k) \ar[r] \ar[d]^{{\ab'}_\pi} & H^1(k,H) \ar[d]^{\ab^1_H} \\
H^0_{\ab}(k,H) \ar[r] & \Hom_k\left(\upic(Y), \bar{k}^\times[1]\right) \ar[r]^{\pi'} & \Hom_k\left(\upic(\pi), \bar{k}^\times[1]\right) \ar[r] & H^1_{\ab}(k,H) \,, }
\]}
where the second line comes from Lemma~\ref{lem triangle exact} and from the isomorphisms
\[
H^i_{\ab}(k,H) = \Hom_k\left(\upic(H), \bar{k}^\times[i+1]\right)
\]
constructed in~\cite[Theorem~4.8 and Corollary~4.9]{BvH}.

In addition, when $Y=G$ is a reductive $k$-group, then the map
\[
{\ab'}_G : G(k) \to \Hom_k\left(\upic(G), \kbar^\times[1]\right)=H^0_\ab(k,G)
\]
coincides with $\ab^0_G$.
\end{lemm}

\begin{proof}
The commutativity of the central square is a consequence of the functoriality of the map ${\ab'}_\pi$ with respect to morphisms of torsors.

Let us now prove the commutativity of the left hand side square : using the functoriality of the map ${\ab_\pi}$ and the fact that the morphism $H \to Y$, given by the action on $y_0 \in Y(k)$, is $H$-equivariant, it is sufficient to prove that ${\ab^0}_H = {\ab'}_H$, where ${\ab'}_H : H(k) \to \Hom_k(\upic(H), \bar{k}^\times[1])$ is defined as ${\ab'}_{\id_H}$. There exists a coflasque resolution (see~\cite[Proposition~4.1]{CTfl})
\[
1 \to P \to H_1 \to H \to 1 \,,
\]
where $P$ is a quasi-trivial $k$-torus and $H_1$ is an extension of a (coflasque) torus $T$ by a semisimple simply connected $k$-group $H^\ss$. Then functoriality of the map ${\ab'}_H$ and Hilbert 90 imply that it is enough to prove the required compatibility ${\ab'}_T = {\ab^0}_T$ for the $k$-torus $T$. By definition, the map
\[
{\ab'}_T : T(k) \to \Hom_k\left(\widehat{T}[1], \kbar^\times[1]\right)=\Hom_k\left(\widehat{T}, \kbar^\times\right)
\]
is given by $t \mapsto (\chi \mapsto \chi(t))$, and it clearly coincides with the map ${\ab^0}_T: T(k) \to T(k)$, composed with the natural identification $T(k) \xrightarrow{\sim} \Hom_k(\widehat{T}, \kbar^\times)$. It concludes the proof of the commutativity of the left hand side square. Note that this also proves the last statement in Lemma~\ref{lem compat ab ab'}.

Let us now prove the remaining commutativity, concerning the right hand side square. Let $x \in X(k)$ and consider the torsor $\pi_x : Y_x \to \spec k$ defined as the pullback of $\pi$ by $x$ (recall that $Y_x$ denotes the fiber of $Y$ at $x$). We have a commutative diagram
\[
\xymatrix{
\upic(\pi) \ar[d]^{\alpha} \ar[r]^{y} & \upic'(\pi) \ar[d]^{\alpha'} \ar[r]^{x^*} & \upic'(\pi_x) \ar[d]^{\alpha_x} & \kbar^\times[1] \ar[l] \ar[d]^= \\
\con(\varphi)[-1] \ar[r]^y & \con(\varphi')[-1] \ar[r]^{x^*} & \con(\varphi'_x)[-1] & \kbar^\times[1] \ar[l] \,, }
\]
where the vertical maps are quasi-isomorphisms (see the proof of Lemma~\ref{lem triangle exact}). We now prove that the diagram
\[
\xymatrix{
\con(\varphi)[-1] \ar[r]^y & \con(\varphi')[-1] \ar[r]^{x^*} & \con(\varphi'_x)[-1] & \kbar^\times[1] \ar[l] \\
\upic(H)[-1] \ar[u] \ar[r]^= & \upic(H)[-1] \ar[u] \ar[r]^= & \upic(H)[-1] \ar[u] & \upic(Y_x)[-1] \ar[u]^{\partial_{Y_x}} \ar[l]^{\varphi_x} }
\]
is commutative. The only non-obvious square is the right hand side one, but it comes from the functoriality of the cone in the category of complexes, together with the definition of $\partial_{Y_x}$.

The two previous commutative diagrams imply that the square
\[
\xymatrix{ X(k) \ar[r] \ar[d]^{{\ab'}_\pi} & H^1(k,H) \ar[d]^{{\ab'}^1_H} \\
\Hom_k\left(\upic(\pi), \bar{k}^\times[1]\right) \ar[r] & \Hom_k\left(\upic(H), \bar{k}^\times[2]\right) }
\]
commutes, where ${\ab'}^1_H$ maps the cohomology class of a torsor $W \to \spec k$ under $H$ to the rightmost morphism in the natural exact triangle
\[
\kbar^\times[1] \to \upic'(W) \to \upic(W) \xrightarrow{\partial_W} \kbar^\times[2]
\]
with the canonical identification $\upic(W) \xrightarrow{\sim} \upic(H)$.

We conclude the proof using~\cite[Theorem~5.5]{BvH}, which proves that ${\ab'}^1_H = - \ab^1_H$.
\end{proof}

We need to prove other properties of the map $\ab'_\pi$:

\begin{prop}\label{prop ab action}
Let $X=G/H$ be the quotient of a reductive group $G$ by a reductive subgroup $H$. Let $\pi : G \to X$ be the quotient map (pointed by $e \in G(k)$). Then for all $g \in G(k)$ and $x \in X(k)$,
\[
{\ab'}_\pi(g \cdot x) = \pi'\left(\ab^0_G(g)\right) + {\ab'}_\pi(x) \,.
\]
In particular, if $G$ is semi-simple and simply connected, we have ${\ab'}_\pi(g \cdot x) = {\ab'}_\pi(x) \, $.
\end{prop}

\begin{proof}
Consider $\id_G : G \to G$ as a torsor under the trivial group, and $\id \times \pi : G \times_k G \to G \times_k X$ as a natural torsor under $H$. Then we have natural morphisms of torsors:
\[
\xymatrix{ G \ar[d]^{\id_G} & G \times_k G \ar[r]^{p_2} \ar[l]_{p_1} \ar[d]^{\id_G \times \pi} & G \ar[d]^\pi \\
G & G \times_k X \ar[l]_{p_G} \ar[r]^{p_X} & X \,, }
\]
that induce, by functoriality of $\upic(\pi)$ and by Lemma~\ref{lem triangle exact}, a commutative diagram in the derived category, where the rows are exact triangles (see~\cite[Lemma~5.1]{BvH} for the third vertical map):
{\small
\[
\mkern-10mu\xymatrix{
\upic(H)[-1] \ar[r] \ar[d]^\sim & \upic(G) \oplus \upic(\pi) \ar[r] \ar[d]^{p_G^* + p_X^*} & \upic(G) \oplus \upic(G) \ar[d]^\sim \ar[r] & \upic(H) \ar[d]^\sim \\
\upic(1 \times H)[-1] \ar[r] & \upic\left(\id_G \times \pi\right) \ar[r] & \upic\left(G \times_k G\right) \ar[r] & \upic(1 \times H) \,. }
\]}
The five lemma implies that the morphism $p_G^* + p_X^* : \upic(G) \oplus \upic(\pi) \to \upic(\id_G \times \pi)$ is an isomorphism.

By functoriality of the construction of ${\ab'}_\pi$, the morphism of torsors
\[
\xymatrix{ G \times_k G \ar[r]^(.6){m_G} \ar[d]^{\id_G \times \pi} & G \ar[d]^\pi \\
G \times_k X \ar[r]^(.6){m} & X }
\]
induces a commutative diagram
\[
\xymatrix{ & \kbar^\times[1] & \kbar^\times[1] \oplus \kbar^\times[1] \ar[l]_{\times} \\
& \upic\left(\id_G \times \pi\right) \ar[u]_{(g,x)} & \upic(G) \oplus \upic(\pi) \ar@<0.8cm>[u]^g \ar@<-0.8cm>[u]^{x} \ar[l]_(.55){p_G^* + p_X^*} \\
\upic(\pi) \ar[ruu]^{g \cdot x} \ar[ru]^{m^*} \ar[rru]_{(\can, \id)} & && \,,}
\]
which concludes the proof. The non-trivial commutativity in this last diagram is that of the triangle at the bottom, which we explain now: recall that we are given a point $x_0 = \pi(e) \in X(k)$. Consider the following commutative diagram of morphisms of torsors:
\[
\xymatrix{ G \ar[r]^(.35){\left(\id_G,1\right)} \ar[d]^{\id_G} & G \times G \ar[d]^{\left(\id_G, \pi\right)} & G \ar[d]^{\pi} \ar[l]_(.3){\left(1, \id_G\right)} \\
G \ar[r]^(.4){\iota_{x_0}} & G \times X & X \ar[l]_(.4){\iota_1} \,, }
\]
where the bottom horizontal maps are defined by $\iota_{x_0}(g) = (g, x_0)$ and $\iota_1(x) = (1,x)$. If we denote by $\varpi_G$ (resp. $\varpi_X$) the projection from $\upic(G) \oplus \upic(\pi)$ to $\upic(G)$ (resp. $\upic(\pi)$), then we deduce from the previous diagram that $\varpi_G = \iota_{x_0}^* \circ (p_G^* + p_X^*)$ and $\varpi_X = \iota_1^* \circ (p_G^* + p_X^*)$. But $m \circ \iota_{x_0} = \pi : G \to X$ and $m \circ \iota_1 = \id_X : X \to X$, therefore we get that the required triangle commutes.

%\smallskip

Finally, if $G$ is assumed to be semi-simple and simply connected, then the complex $\upic (G)$ is quasi-isomorphic to $0$, hence the map $\pi'$ is trivial, which implies the required result.
\end{proof}

\begin{rema}
It is worth noting that the construction of the map $\ab_\pi$ depends on the choice of a $k$-point $y_0 \in Y(k)$. But one can prove, using the same kind of arguments as in the proof of Lemma~\ref{prop ab action}, that the map $\ab_\pi$ depends only on the image $x_0$ of $y_0$ in $X(k)$. More precisely, two points $y_0, y_0' \in Y(k)$ such that $\pi(y_0) = \pi(y_0')$ define the same map $\ab_\pi$, or equivalently, the construction of this map depends only on the choice of a point $x_0 \in \pi(Y(k))$.
\end{rema}

Lemma~\ref{lem compat ab ab'} and Proposition~\ref{prop ab action} imply the following proposition:

\begin{prop}\label{orbitesprop}
Let $F$ be a non-archimedean local field or a global field with no real place. Let $X = G/H$ be the quotient of a reductive group $G$ by a reductive subgroup $H$.

Then the map $\ab'_\pi : X(F) \to \Hom_k(\upic(\pi), \bar{F}^\times[1])$, associated to the torsor $\pi : G \to X$, is surjective, and for all $x, x' \in X(F)$, $\ab'_\pi(x) = \ab'_\pi(x')$ if and only if there exists $g \in G^\sc(F)$ such that $x' = g \cdot x$.
\end{prop}

\begin{proof}
Consider the following commutative diagram with exact rows (see Lem\-ma~\ref{lem compat ab ab'}):
{\small
\[
\mkern-25mu\xymatrix{ H(F) \ar[r] \ar[d]^{\ab^0_H} & G(F) \ar[r]^\pi \ar[d]^{\ab^0_G} & X(F) \ar[r] \ar[d]^{\ab'_\pi} & H^1(F,H) \ar[d]^{\ab^1_H} \ar[r] & H^1(F,G) \ar[d]^{\ab^1_G} \\
H^0_\ab(F,H) \ar[r] & H^0_\ab(F,G) \ar[r]^-{\pi'} & \Hom_F\left(\upic(\pi), \bar{F}^\times [1]\right) \ar[r] & H^1_\ab(F,H) \ar[r] & H^1_\ab(F,G) \,. }
\]}
Then diagram chasing proves that surjectivity of $\ab'_\pi$ is a consequence of surjectivity of $\ab^1_H$, injectivity of $\ab^1_G$, surjectivity of $\ab^0_G$ (those properties follows from Proposition~\ref{prop ab1 bij}), and of Proposition~\ref{prop ab action}.

Let us now prove the second part of the Proposition: let $x$, $x' \in X(F)$ such that $\ab'_\pi(x) = \ab'_\pi(x')$. Using the above diagram and the injectivity of $\ab^1_H$ (see Proposition~\ref{prop ab1 bij}), we get that $x$ and $x'$ have the same image in $H^1(F,H)$. Therefore, there exists $g \in G(F)$ such that $g \cdot x = x'$. Applying Lemma~\ref{prop ab action}, we get that $\pi'(\ab^0_G(g)) = 0$, hence by Proposition~\ref{prop ab1 bij}, $g$ lifts to $G^\sc(F)$, which concludes the proof.
\end{proof}

Let $X=G/H$, with $G$ and $H$ reductive over the field $k$. Let us now construct a canonical isomorphism $\phi_X : \widehat{C}_X \to \upic(\pi)$ in the derived category, inspired by~\cite[sections~4.1.2 and 4.1.3]{DemJOA}.

By construction, $\widehat{C}_X$ is the cone of the morphism of complexes
\[
\left[\widehat{T}_G \to \widehat{T}_{G^\sc}\right] \to \left[\widehat{T}_H \to \widehat{T}_{H^\sc}\right] \,
\]
where $\widehat{T}_H$ is in degree~$0$. Consider the following commutative diagram of complexes, where the vertical maps are either obvious or defined in~\cite[Section~4]{BvH}):
\begin{equation}\label{diag upic char}
\xymatrix{ \left[\widehat{T}_G \to \widehat{T}_{G^\sc}\right] \ar[r] \ar[d] & \left[\widehat{T}_H \to \widehat{T}_{H^\sc}\right] \ar[d] \\
\con\left(\upic(T_G)_0 \to \upic\left(T_{G^\sc}\right)_0\right) \ar[r] & \con\left(\upic(T_H)_0 \to \upic\left(T_{H^\sc}\right)_0\right) \\
\con\left(\upic(G)_0 \to \upic\left(G^\sc\right)_0\right) \ar[u] \ar[r] \ar[d] \ar[u] & \con\left(\upic(H)_0 \to \upic\left(H^\sc\right)_0\right) \ar[u] \ar[d] \\
\upic(G)_0 \ar[r]^\varphi & \upic(H)_0 \\
\upic(Z)_0 \ar[u] \ar[r] & \upic(H')_0 \ar[u] \,. }
\end{equation}
All the vertical maps, except the bottom ones, are quasi-isomorphisms. Hence this diagram induces a natural isomorphism $\phi'_X : \widehat{C}_X \to \con(\varphi)$, which we can compose with the quasi-isomorphism $\alpha : \upic(\pi)[1] \to \con(\varphi)$ induced by the two last lines of the above diagram (see the proof of Lemma~\ref{lem triangle exact}), to get a natural isomorphism $\phi_X : \widehat{C}_X \to \upic(\pi)[1]$.

By construction, this isomorphism fits into the following commutative diagram of exact triangles in the derived category:
\begin{equation}\label{diag devissage C_X}
\xymatrix{
\widehat{C}_H[-1] \ar[r] \ar[d]^{\phi_H} & \widehat{C}_X \ar[r] \ar[d]^{\phi_X} & \widehat{C}_G \ar[r] \ar[d]^{\phi_G} & \widehat{C}_H \ar[d]^{\phi_H} \\
\upic(H)[-1] \ar[r] & \upic(\pi) \ar[r] & \upic(G) \ar[r] & \upic(H) \,, }
\end{equation}
where the vertical maps are isomorphisms (morphisms $\phi_H$ and $\phi_G$ are defined in~\cite[Section~4]{BvH}, and also in the first four lines of diagram~\eqref{diag upic char}).


In addition, \cite[Lemma~3.1]{BDH} implies that the natural morphism $H^0(k, C_X) \to \Hom_k(\widehat{C}_X, \kbar^\times[1])$ is an isomorphism. These facts lead to the following:

\begin{defi}\label{def ab'}
We denote by ${\ab'}_X : X(k) \to H^0(k, C_X)$ the composition
\[
X(k) \xrightarrow{{\ab'}_\pi} \Hom_k\left(\upic(\pi), \kbar^\times[1]\right) \xrightarrow{\widehat{\phi_X}} \Hom_k\left(\widehat{C}_X, \kbar^\times[1]\right) \xrightarrow{\sim} H^0\left(k, C_X\right) \,,
\]
where $\pi : G \to X$ is the quotient morphism.
\end{defi}

In particular, Lemma~\ref{lem compat ab ab'}, Definition~\ref{def ab'} and diagram~\eqref{diag devissage C_X} imply that the following useful diagram is commutative (up to sign):
\begin{equation}\label{diag devissage ab}
\xymatrix{ H(k) \ar[r] \ar[d]^{\ab^0_H} & G(k) \ar[r]^\pi \ar[d]^{\ab^0_G} & X(k) \ar[r] \ar[d]^{{\ab'}_X} & H^1(k,H) \ar[r] \ar[d]^{\ab^1_H} & H^1(k,G) \ar[d]^{\ab^1_G} \\
H^0\left(k,C_H\right) \ar[r] & H^0\left(k,C_G\right) \ar[r] & H^0\left(k,C_X\right) \ar[r] & H^1\left(k,C_H\right) \ar[r] & H^1\left(k,C_G\right) \,, }
\end{equation}
where the unnamed maps are the natural ones.

\subsection{The compatibility result}

We can now prove the main compatibility result of this section, which can be seen as a generalization of~\cite[Theorem~4.14]{DemJOA} and~\cite[Theorem~6.2]{BDH}:

\begin{theo}\label{theo compatible strong approx}
Let $k$ be a field and $G$ be a reductive group. Let $H \subset G$ be a reductive $k$-subgroup and $X := G/H$. Let $\pi : G \to X$ be the quotient map (pointed by $e \in G(k)$ and its image $x_0:=\pi(e)$). The canonical isomorphism $\phi_X : \widehat{C}_X \to \upic(\pi)$ in the derived category induces an isomorphism
\[
\phi_X : H^1\left(k, \widehat{C}_X\right) \xrightarrow{\sim} \Br_{1,e}(X,G) \,,
\]
and the following diagram
\[
\xymatrix{ X(k) \ar@<-30pt>[d]_{{\ab'}_X} \times \Br_{1,e}(X,G) \ar[r]^(.65){\ev} & \Br(k) \ar[d]^= \\
H^0(k, C_X) \times H^1\left(k, \widehat{C}_X\right) \ar@<-30pt>[u]_{\phi_X} \ar[r]^(.7){\cup} & \Br(k) }
\]
is commutative, up to a universal sign (independent of all the data).
\end{theo}

\begin{proof}
Following~\cite{DemJOA}, there is a natural morphism $\upic'(\pi) \to \tau_{\leq 2} R \pi_*\linebreak \Gm[1]$ inducing an isomorphism $H^1(k, \upic(\pi)) \xrightarrow{\sim} \Br_{1,e}(X,G)$. Together with the isomorphism $\phi_X : \widehat{C}_X \to \upic(\pi)$, we get the required isomorphism. Let us now prove the commutativity.

For any $x \in X(k)$, one can define a natural splitting $\ab'(x)$ of $\kbar^\times \to \tau_{\leq 2} R \pi_* \Gm$ induced by $x$.

We can decompose the diagram above as the composition of the following diagrams:
\[
\xymatrix{ X(k) \ar@<-30pt>@/_10pc/[ddd]_{{\ab'}_\pi} \ar@<-30pt>[d]_{{\ab'}} \times \Br_{1,e}(X,G) \ar[r]^(.6){\ev} & \Br(k) \ar[d]^= \\
\Hom_k\left(\tau_{\leq\,2} R \pi_* \Gm [1], \kbar^\times[1]\right) \ar@<-30pt>[d] \times H^1\left(k, \tau_{\leq\,2} R \pi_* \Gm [1]\right) \ar@<-30pt>[u] \ar[r]^(.8)\cup & \Br(k) \ar[d]^= \\
\Hom_k\left(\upic'(\pi), \kbar^\times[1]\right) \ar@<-30pt>[d]_e \times H^1\left(k, \upic'(\pi)\right) \ar@<-30pt>[u] \ar[r]^(.75)\cup & \Br(k) \ar[d]^= \\
\Hom_k\left(\upic(\pi), \kbar^\times[1]\right) \ar@<-30pt>[d]_{\widehat{\phi_X}} \times H^1\left(k, \upic(\pi)\right) \ar@<-30pt>[u]_e \ar[r]^(.75)\cup & \Br(k) \ar[d]^= \\
\Hom_k\left(\widehat{C}_X, \kbar^\times[1]\right) \ar@<-30pt>[d]_\sim \times H^1\left(k, \widehat{C}_X\right) \ar@<-30pt>[u]_{\phi_X} \ar[r]^(.7){\cup} & \Br(k) \ar[d]^= \\
H^0(k, C_X) \times H^1\left(k, \widehat{C}_X\right) \ar@<-30pt>[u]_= \ar[r]^(.6){\cup} & \Br(k) \,. }
\]
By construction, and by functoriality of cup-products, this last diagram is commutative (up to sign).
\end{proof}

\begin{rema}
Let $X = G/H$ as in Definition~\ref{def ab'} and Theorem~\ref{theo compatible strong approx}. Up to replacing $G$ by a flasque resolution $G_1$, one can realise $X$ as the quotient of the quasi-trivial group $G_1$ by a reductive subgroup $H_1$. Since $\pic(\bar{G_1})=0$, one gets a natural isomorphism
\[
\widehat{C_X} \xrightarrow{\sim} \left[\widehat{G_1} \to \widehat{T_{H_1}} \to \widehat{T_{H_1^\sc}}\right],
\]
where $\widehat{G_1}$ is a permutation
Galois module. Assuming the group $G$ is quasi-trivial will be very useful in the next two sections.
\end{rema}

\subsection{The abelianization map over an arbitrary base}
In this section, we extend the definition of the map $\ab'_X$ for homogeneous spaces of reductive group schemes defined over an arbitrary base scheme $S$. It will be useful in the next sections in order to take integral points into account (case $S = \spec(\clO_v)$).

Let $S$ be an integral regular noetherian scheme and $H$ be a reductive group scheme over $S$. Let $\pi : Y \to X$ be a torsor under $H$. For any $S$-scheme $W$, let $p_W : W \to S$ denote the structure morphism.

Let $Z := Y / Z_H$ and $\varpi : Z \to X$ be the associated $H' := H/Z_H$-torsor. We define $\upic'(\pi)$ to be the following complex of étale sheaves over $S$:
\[
\underline{\upic'}(\pi) := \left[{p_Z}_* \underline{K}_{Z/X}^\times \to {p_Z}_* \underline{\divl}_{Z/X} \to {p_X}_* \underline{\pic}_{Z/X}\right] \,,
\]
where the sheaves $\underline{K}_{Z/X}^\times$ and $\underline{\divl}_{Z/X}$ are defined in~\cite[Appendix~A]{HS-op-desc}. By loc. cit., there is a natural exact sequence of étale sheaves over $X$
\[
0 \to (\Gm)_X \to \varpi_* \underline{K}_{Z/X}^\times \to \varpi_* \underline{\divl}_{Z/X} \to \underline{\pic}_{Z/X} \to 0 \,.
\]
Applying ${p_X}_*$, one gets a natural morphism $(\Gm)_S \to \underline{\upic'}(\pi)$, and we define $\underline{\upic}(\pi)$ to be the cone of this morphism, whence an exact triangle
\[
(\Gm)_S \to \underline{\upic'}(\pi) \to \underline{\upic}(\pi) \to (\Gm)_S[1] \,.
\]

Let $y_0 \in Y(S)$ be a section, and let $x_0 := \pi(y_0)$. Denote by $D(S)$ the derived category of bounded complexes of étale sheaves over $S$. Following the construction in section~\ref{subsec ab field}, we get a natural map, functorial in $S$ and $X$,
\[
\ab'_\pi : X(S) \to \Hom_{D(S)}\left(\underline{\upic}(\pi), (\Gm)_S\right) \,.
\]


Let now $G$ be a reductive group scheme over $S$ and $H \subset G$ a reductive subgroup scheme. Taking $Y=G$ and $X=G/H$, we can apply the previous constructions. Assume further that $H$ and $G$ admit compatible maximal tori $T_H \subset T_G$. The diagram~\eqref{diag upic char}, as a diagram of étale sheaves over $S$, still holds. All vertical morphisms, except the bottom ones, are quasi-isomorphisms of étale sheaves over $S$, since it is true over any separably closed field (see after diagram~\eqref{diag upic char}). Similarly, since the result holds over separably closed fields, diagram~\eqref{diag upic char} induces an isomorphism of complexes of étale sheaves $\phi_X : \widehat{C}_X \to \underline{\upic}(\pi)[1]$.

As a conclusion, composing the map $\ab'_\pi : X(S) \to \Hom_{D(S)}(\underline{\upic}(\pi), (\Gm)_S)$ with the isomorphism $\phi_X$, one gets the required map:
\[
\ab'_X : X(S) \to H^0(S, C_X) \,,
\]
that is functorial in $S$ and $(H,G)$, and that coincides with the definition of Section~\ref{subsec ab field} in the case $S$ is the spectrum of a field.

\section{Weak approximation} \label{four}

From now on, the setting is the following: $K=k(E)$ is again the function field of a projective, smooth and irreducible curve $E$ over a finite field $k$. We consider a reductive linear algebraic group $G$ over $K$, and $X$ a homogeneous space of $G$. We assume that $X(K) \neq \emptyset$. Let $e \in X(K)$ and let $H \subset G$ be the stabilizer of $e$ in $G$. Then $X = G/H$ (with $e$ identified to the image of the neutral element of $G$ in $X$) and we still suppose that $H$ is a reductive subgroup of $G$. Set $X(K_{\Omega})=\prod_{v\,\in\,\Omega_K} X(K_v)$.

%\smallskip

We are interested in the closure of $X(K)$ in $X(K_\Omega)$, for the product topology. We define $\Be_\omega(X)$ as the subgroup of $\br_{1,e} \, X \simeq \br_1 X/\br K$ consisting of those elements $\alpha$ such that their localization $\alpha_v \in \br_{1,e} \, X_{K_v}$ is zero for all but finitely many $v$. For each finite set of places $S$ of $K$, we set $K_S=\prod_{v\,\in\,S} K_v$
and we define
\[
\Be_S(X):=\ker \left[\br_{1,e} \, X \to \prod_{v\,\not\, \in\,S}
\br_{1,e} \, X_{K_v}\right].
\]
In particular $\Be(X)$ (cf. Section~\ref{two}) identifies to $\Be_{\emptyset}(X)$.

\begin{rema}
Assume that $X$ admits a regular compactification, i.e. there exists a regular proper $K$-variety $X^c$ and an open immersion $X \to X^c$. Then the group $\Be_\omega(X)$ is exactly the algebraic Brauer group of $X^c$ (see for example~\cite[Proposition~4.1]{BDH}).
\end{rema}

For all $\alpha \in \Be_\omega(X)$, $(P_v) \in X(K_{\Omega})$, the Brauer--Manin pairing:
\[
\langle \alpha,(P_v) \rangle_{BM}=\sum_{v\,\in\,\Omega_K} \alpha(P_v)
\]
is well-defined. For every subgroup $B$ of $\Be_\omega(X)$, we denote by $X(K_{\Omega})^B \subset X(K_{\Omega})$ the orthogonal of $B$ for the Brauer--Manin pairing.

\begin{theo}\label{weakaptheo}
Let $G$ be a reductive group over $K$ and $H$ a reductive subgroup. Then the Brauer--Manin obstruction to weak approximation on $X=G/H$ associated to $\Be_\omega(X)$ is the only one, i.e. $X(K)$ is dense in $X(K_\Omega)^{\Be_\omega(X)}$. More precisely, for any finite set $S$ of places of $K$, the Brauer--Manin pairing induces a surjective map $X(K_S) \to \left(\Be_S(X) / \Be(X) \right)^D$, whose kernel is exactly the closure of $X(K)$ inside $X(K_S)$.
\end{theo}



\begin{proof}
Up to replacing $G$ by a flasque resolution, one can assume that $G$ is quasi-trivial. Using an elementary instance of the fibration method, we see that $G$ satisfies weak approximation, since $G^\sc$ (by~\cite{prasad}) and the quasi-trivial (hence $K$-rational) torus $G^\tor$ do satisfy weak approximation. In addition, we have $H^1(K,G) = 1$ by~\cite{harder} and Hilbert's 90 (the latter shows that $H^1(K,G^{\tor})=0$ thanks to Shapiro's lemma).

%\smallskip

Let $C := [H^\tor \to G^\tor]$ and $C_X := \Cone(C_H \to C_G)$ (cf. Section~\ref{sectcompat}). We have the Cartier duals $\widehat C=[\widehat {G^\tor} \to \widehat {H^\tor}]$ and $\widehat {C_X}=\Cone(\widehat {C_G} \to \widehat {C_H})[-1]$.

%\smallskip

By construction, we have a natural commutative diagram of exact triangles of complexes:
\begin{equation}\label{diag hom spaces devissage}
\xymatrix{
\mu_H[1] \ar[d] & & \mu_H[2] \ar[r]^\sim \ar[d] & \mu_H[2] \ar[d] \\
C_H \ar[r] \ar[d] & C_G \ar[r] \ar[d]^{\qis} & C_X \ar[r] \ar[d] & C_H[1] \ar[d] \\
H^\tor \ar[r] \ar[d] & G^\tor \ar[r] & C \ar[r] \ar[d] & H^\tor[1] \ar[d] \\
\mu_H[2] & & \mu_H[3] \ar[r]^\sim & \mu_H[3] \,. }
\end{equation}

Since $G$ is assumed to be quasi-trivial, we have $G^{\ss}=G^{\sc}$, hence $\mu_G=0$ and $C_G$ is quasi-isomorphic to $G^{\tor}$, which is a quasi-trivial torus. Taking Cartier duals, we get an exact triangle:
\[
\widehat {C_X} \to \widehat {G^{\tor}}[1]
\to \widehat {C_H} \to \widehat {C_X} [1],
\]
whence (using $H^1(K,\widehat {G^{\tor}})=0$) an exact sequence:
\[
0 \to H^0\left(K,\widehat {C_H}\right) \to H^1\left(K,\widehat {C_X}\right)
\to H^2\left(K,\widehat {G^{\tor}}\right)
\]
and similarly replacing $K$ with a completion $K_v$. As $\widehat {G^{\tor}}$ is a permutation Galois module, we have $\Sha^2_S(K,\widehat {G^{\tor}})=0$ by Shapiro's lemma and \v Cebotarev's Theorem. Thus
\[
\Sha^0_S\left(K,\widehat{C_H}\right) \xrightarrow{\sim} \Sha^1_S\left(K, \widehat{C_X}\right).
\]
We also have an exact triangle
\[
\widehat {\mu_H} [-2] \to
\widehat C \to \widehat {C_X} \to \widehat {\mu_H} [-1],
\]
which yields an isomorphism
\[
\Sha^1_S(K, \widehat{C}) \to
\Sha^1_S(K, \widehat{C_X}).
\]
Summing up, we get isomorphisms
\[
\Sha^0_S\left(K,\widehat{C_H}\right) \xrightarrow{\sim} \Sha^1_S\left(K, \widehat{C_X}\right) \xleftarrow{\sim} \Sha^1_S\left(K, \widehat{C}\right) \,.
\]

%\smallskip

As recalled before (cf. proof of Th.~2.5.), the group $\br_{1,e} X \simeq
\br_1 X/ \br K$ is isomorphic to $H^1(K,\upic(X))=H^1(K,\widehat C)$ (and this is true over any field). Therefore $\Sha^1_S(K,\widehat C)$ (resp. $\Sha^1(K,\widehat C)$) identifies to $\Be_S(X)$ (resp. to $\Be(X)$), whence a Brauer--Manin pairing:
\[
X(K_S) \times \Sha^1_S\left(K,\widehat C\right) \to \Q/\Z, \left((P_v)_{v\,\in\,S}, \alpha\right)
\mapsto \sum_{v\,\in\,S} \alpha(P_v),
\]
which is trivial on $X(K_S) \times \Sha^1(K,\widehat C)$. It induces a map $\BM : X(K_S) \to (\Sha^1_S(K,\widehat{C}) /\linebreak \Sha^1(K, \widehat{C}))^D$. On the other hand, local duality for complex of tori (\cite[Theorem~3.1]{Demtores}) induces a map
\[
\theta_S : \prod_{v\,\in\,S} H^1_{\ab}(K_v,H)=\prod_{v\,\in\,S} H^1\left(K_v, C_H\right) \to
\left(\Sha^0_S\left(K, \widehat{C_H}\right)/\Sha^0\left(K, \widehat{C_H}\right)\right)^D.
\]
We also have a map in the derived category of Galois modules
\[
C_X \otimes^{\bfL} \widehat{C_X} \to \G[1],
\]
which induces (for each completion $K_v$) a cup-product pairing
\[
H^0\left(K_v,C_X\right) \times H^1\left(K_v,\widehat{C_X}\right) \to H^2\left(K_v,\G\right)=\Q/\Z.
\]
The latter is compatible (in an obvious sense) with local duality for the complexes $C_H$, $\widehat{C_H}$ via the maps $C_X \to C_H[1]$ and $\widehat{C_H} \to \widehat{C_X}[1]$. Finally, denote by $\partial_K : X(K) \to H^1(K,H)$ the coboundary map, which (by composing with the abelianization map) yields a morphism of pointed sets $\partial_K ^{\ab} : X(K) \to H^1_{\ab}(K,H)=H^1(K,C_H)$. Similarly we have (for every completion $K_v$) maps $\partial_v^{\ab} : X(K_v) \to H^1(K_v,C_H)$. By Lemma~\ref{lem compat ab ab'}, the map $\partial_v^{\ab}$ can also be obtained by composing the abelianization map $\ab'_{X_{K_v}} : X(K_v) \to H^0(K_v,C_X)$ (defined in Section~\ref{sectcompat}) with the natural map $H^0(K_v,C_X) \to H^1(K_v,C_H)$ (induced by diagram~\eqref{diag hom spaces devissage}).

\begin{lemm} \label{compatweak}
There is a commutative diagram with exact rows:
\small{
\begin{equation}\label{diag esp hom WA}
\xymatrix{ G(K) \ar[r] \ar[d] & X(K) \ar[r]^{\partial_K^{\ab}}
\ar[d] & H^1(K,C_H) \ar[r] \ar[d] & 1 \\
G(K_S) \ar[r] & X(K_S) \ar[d]^{BM} \ar[r]^{\partial_S^{\ab}=\prod_{v \in S} \partial_v^{\ab} } &
\prod_{v\,\in\,S} H^1\left(K_v,C_H\right) \ar[r] \ar[d]^{\theta_S} & 1 \\
& \left(\Sha^1_S\left(K,\widehat{C}\right) /
\Sha^1\left(K, \widehat{C}\right)\right)^D \ar[r]^{\sim} & \left(\Sha^0_S\left(K, \widehat{C_H}\right)/\Sha^0\left(K, \widehat{C_H}\right)\right)^D \ar[d] & \\
& & 0 & \,. }
\end{equation}}
\end{lemm}

\begin{proof}
The exactness of rows follows from the triviality of $H^1(K,G)$
and\linebreak $H^1(K_v, G)$ (recall that $G$ is quasi-trivial) combined with Proposition~\ref{prop ab1 bij}. The only non-trivial remaining point is the commutativity of the bottom square. By functoriality of the cup-product and Theorem~\ref{theo compatible strong approx}, there is a commutative diagram~:
\[
\xymatrix{ X(K_v) \ar[r]^(.4){{\ab'}_{X_{K_v}}} \ar[d] & H^0(K_v,C_X) \ar[r] \ar[d] & H^1(K_v,C_H) \ar[d] \\
\br_{1,e}(X_{K_v},G_{K_v})^D \ar[r]^(.6){\left(\Phi_{X_{K_v}}\right)^D} & H^1\left(K_v,\widehat{C_X}\right)^D \ar[r] & H^0\left(K_v,\widehat C_H\right)^D }
\]
where the left vertical map is given by the local evaluation pairing
\[
X(K_v) \times \br_{1,e}\left(X_{K_v},G_{K_v}\right) \to \Q/\Z, \, \left(P_v,\alpha_v\right)
\mapsto \alpha_v(P_v)
\]
and the right vertical map by local duality for the complex $C_H$. Now let $a \in \Sha^0_S(K,\widehat{C_H})$ with image $b \in \Sha^1_S(K,\widehat{C_X})$ and $c \in \Sha^1_S(K,\widehat{C})$. Set $\alpha=\Phi_X(b) \in \br_{1,e} X \subset \br_{1,e}(X,G)$. Let $v \in S$ and $P_v \in X(K_v)$. By the previous diagram, we have
\[
\alpha(P_v) =\left(\partial_v^{\ab}(P_v) \cup a_v\right) \in \br K_v=\Q/\Z,
\]
where $a_v \in H^0(K_v,\widehat{C_H})$ is the localization of $a$. Hence
\[
\big(BM((P_v))\big).c:=\sum_{v\,\in\,S} \alpha(P_v)=
\sum_{v\,\in\,S} \left(\partial_v^{\ab}\left(P_v\right) \cup a_v\right)=\left(\theta_S\left(\partial_S^{\ab}(P_v)\right)\right).a,
\]
which yields the required commutativity.
\end{proof}

\begin{rema}
Actually we used the difficult compatibility proven in Theorem~\ref{theo compatible strong approx} only for those elements of $H^1(K_v,\widehat C_X)$ coming from $H^1(K_v,\widehat C)$, so~\cite[Theorem~6.2]{BDH} (whose proof is much easier) would be sufficient at this stage. However, we will definitely need Theorem~\ref{theo compatible strong approx} in its full generality in Section~\ref{strongsect}.
\end{rema}

We resume the proof of Theorem~\ref{weakaptheo}. Let us now prove that the right hand side column of diagram~\eqref{diag esp hom WA} is exact. Consider the following commutative diagram:
\[
\xymatrix{ & \bigoplus\limits_{v\,\notin\,S} H^1\left(K_v,C_H\right) \ar[r]^-\sim \ar[d] & \left(\prod\limits_{v\,\notin\,S} H^0\left(K_v, \widehat{C_H}\right) \right)^D \ar[d] & \\
H^1\left(K, C_H\right) \ar[r] \ar[d]^= & \bigoplus\limits_{v\,\in\,\Omega} H^1\left(K_v,C_H\right) \ar[r] \ar[d]^p & \left(H^0\left(K, \widehat{C_H}\right) / \Sha^0\left(K, \widehat{C_H}\right)\right)^D \ar[r] \ar[d] & 0 \\
H^1\left(K, C_H\right) \ar[r] & \prod\limits_{v\,\in\,S} H^1\left(K_v,C_H\right) \ar[r] \ar[d] & \left(\Sha^0_S\left(K, \widehat{C_H}\right) / \Sha^0\left(K, \widehat{C_H}\right)\right)^D \ar[r] \ar[d] & 0 \\
& 0 & 0 & \,. }
\]
Using~\cite[Theorem~5.7]{DHdualcompl}, the second row is exact. And by construction, the columns are exact. Hence an easy diagram chase implies that the bottom row is exact. Therefore, the right hand side column in~\eqref{diag esp hom WA} is exact. In addition, we know that $G(K)$ is dense in $G(K_S)$. Therefore, an easy diagram chase in~\eqref{diag esp hom WA}, together with the comparison~\cite[Theorem~6.2]{BDH}, implies that the map
\[
X(K_S) \xrightarrow{BM} \left(\Sha^1_S(K,\widehat{C}) / \Sha^1(K, \widehat{C})\right)^D = \left(\Be_S(X) / \Be(X) \right)^D
\]
is surjective, and that the inverse image of $0$ is exactly the closure of $X(K)$, which concludes the proof.
\end{proof}

Theorem~\ref{weakaptheo} can be slightly refined when the homogeneous space $X$ is a reductive group:

\begin{coro} \label{groupcase}

Let $L$ be a reductive group over $K$. Let $C_L=[T^{\sc} \to T_L]$ be the complex of tori associated to $L$. Then there is an exact sequence of groups
\[
1 \to \overline{L(K)} \to L(K_\Omega) \to \Sha^1_\omega\left(K, \widehat{C_L}\right)^D \to \Sha^1(K,C_L) \to 1 \,.
\]
\end{coro}

\begin{proof}
Using a flasque resolution of $L$ (see for instance~\cite[Proposition~3.1]{CTfl}), one can view $L$ as a homogeneous space $L=G/H$, with $G$ quasi-trivial, and $H$ reductive. By Theorem~\ref{weakaptheo}, there
is an exact sequence of pointed sets
\[
1 \to \overline{L(K)} \to L(K_\Omega) \xrightarrow{BM} \Be_{\omega}(L)^D \simeq \Sha^1_{\omega}\left(K, \widehat{C_L}\right)^D.
\]
By Lemma~\ref{compatweak}, the Brauer--Manin map $L(K_\Omega) \to
\Sha^1_{\omega}(K, \widehat{C_L})^D$ is the composition of the abelianization map $\ab_L : L(K_\Omega) \to \prod_{v\,\in\,\Omega_K} H^0(K_v,C_L)$ with the map (which is induced by local duality) $\theta : \prod_{v\, \in\,\Omega_K} H^0(K_v,C_L) \to \Sha^1_{\omega}(K, \widehat{C_L})^D$. Therefore the Brauer--Manin map is a morphism of groups. Proposition~\ref{prop ab1 bij} implies that for every completion $K_v$, the map
\[
\ab_L ^v : L(K_v) \to H^0\left(K_v,C_L\right)
\]
is surjective.


%\smallskip

It is now sufficient to show that the sequence of abelian groups
\[
\prod_{v\,\in\,\Omega_K} H^0\left(K_v,C_L\right) \to \Sha^1_\omega\left(K, \widehat{C_L}\right)^D \to \Sha^1\left(K,C_L\right) \to 0
\]
is exact. This is done by observing that by~\cite[Theorem~3.1]{Demtores} and~\cite[Theorem~5.2]{DHdualcompl}, this sequence is the dual of the exact sequence of discrete abelian groups
\[
0 \to \Sha^1\left(K,\widehat{C_L}\right) \to \Sha^1_\omega\left(K, \widehat{C_L}\right) \to
\bigoplus_{v\,\in\,\Omega_K} H^1\left(K_v, \widehat{C_L}\right).\qedhere
\]
\end{proof}

\begin{rema}
The analogues of Theorems~\ref{hptheo} and~\ref{weakaptheo} were previously known in the context of number fields: they are proven by Borovoi in~\cite{Borcrelle} via more geometric techniques (namely fibration methods); the case of principal homogeneous spaces is due to Sansuc~\cite{San}. Another approach over an arbitrary global field is to use flasque resolutions, see~\cite[Theorem~3.9]{nguyen}. In the next section, we will see that the situation is slightly different for strong approximation.
\end{rema}


\section{Strong approximation} \label{strongsect}

Notation is as in the previous section. We set $A^\star :=\Hom(A,\Z)$ for every abelian group $A$. An abelian group is said to be \emph{in the class $\clE$} (cf.~\cite[Definition~3.10]{DHdualcompl}) if it is an extension of a finitely generated group by a profinite group.



Let $C=[T_1 \to T_2]$ be a short complex of $K$-tori with dual $\widehat C=[\widehat {T_2} \to \widehat{T_1}]$. For every $\chi \in H^{-1}(K,\widehat{C})$, denote by $\chi_v \in H^{-1}(K_v,\widehat{C})$ the localization of $\chi$ at the place $v$. The cup-product pairing
\[
H^0\left(K_v,C\right) \times H^{-1}\left(K_v,\widehat{C}\right) \to K_v^*,
\]
induces a pairing
\begin{equation}\label{pairing char c}
\begin{split}
H^0(\A_K, C) \times H^{-1}(K,\widehat{C}) & \to \Z \\
((g_v), \chi) & \mapsto \sum_v v(\chi_v.g_v) \cdot \left[k(v):k\right] \,.
\end{split}
\end{equation}
Since the degree of a principal divisor on the projective curve $E$ is zero, this pairing is trivial on the subgroup $H^0(K,C) \times H^{-1}(K,\widehat{C})$.

\begin{lemm} \label{wedgelem}
\leavevmode
\begin{enumerate}
\item \label{wedgelem1}
The kernel $P$ of the map $H^0(\A_K, C)/H^0(K,C) \to H^{-1}(K,\widehat{C})^\star$ induced by~\eqref{pairing char c} is profinite.
\item \label{wedgelem2}
The canonical morphism
\[
i : H^0\left(\A_K, C\right)_{\wedge}/H^0(K,C)_{\wedge} \to
\left(H^0\left(\A_K, C\right)/H^0(K,C)\right)_\wedge
\]
is an isomorphism.
\item \label{wedgelem3}
The map $H^0(\A_K,C) \to H^{-1}(K, \widehat{C})^\star$ is surjective and the diagram
\[
\xymatrix{ H^0(\A_K,C)/H^0(K,C) \ar[r] \ar[d] & \left(H^0\left(\A_K,C\right) / H^0(K,C) \right)_\wedge \ar[d] \\
H^{-1}\left(K, \widehat{C}\right)^\star \ar[r] & {H^{-1}\left(K, \widehat{C}\right)^\star}_\wedge }
\]
is cartesian.
\end{enumerate}
\end{lemm}

\begin{proof}
\begin{itemize}
\item[\eqref{wedgelem1}] Up to replacing $C$ by a quasi-isomorphic complex, one can assume that $T_1$ is quasi-trivial. This yields a commutative diagram with exact rows and surjective left vertical map:
\[
\xymatrix{ T_1(\A_K) \ar[r] \ar[d] & T_2(\A_K) \ar[r] \ar[d] & H^0(\A_K,C) \ar[r] \ar[d] & 0 \\
\widehat T_1(K)^{\star} \ar[r] & \widehat T_2(K)^{\star} \ar[r] & H^{-1}(K,\widehat C)^{\star} & \,. }
\]

The snake lemma now implies that the kernel of $H^0(\A_K,C)/H^0(K,C) \to H^{-1}(K, \widehat{C})^\star$ is a quotient of $\ker\left(T_2(\A_K)/T_2(K) \to \widehat{T_2}(K)^\star\right)$ by a closed subgroup, and this kernel is profinite by~\cite[Proposition~5.7.5]{RosBig}. Hence the group $\ker\left(H^0(\A_K,C)/H^0(K,C)
\to H^{-1}(K, \widehat{C})^\star \right)$ is profinite.

\item[\eqref{wedgelem2}] The exact sequence
\[
H^0(K,C) \to H^0\left(\A_K,C\right) \to H^0\left(\A_K,C\right)/H^0(K,C) \to 0
\]
and~\cite[Lemma~3.12$\MK$(a)]{DHdualcompl} show that $i$ is surjective. Since $H^{-1}(K, \widehat{C})^\star$ is a lattice, \eqref{wedgelem1} shows that $H^0(\A_K,C)/H^0(K,C)$ is in the class $\clE$. In particular, the canonical map $H^0(\A_K,C)/H^0(K,C) \to (H^0(\A_K,C)/H^0(K,C))_{\wedge}$ is injective, which shows that $P$ injects into $(H^0(\A_K,C)/H^0(K,C))_{\wedge}$ as well as in $H^0(\A_K, C)_{\wedge}/H^0(K,C)_{\wedge}$. The commutative diagram
\begin{equation}\label{diagwedge}
\xymatrix{ 0 \ar[r] & P \ar[r] \ar[d]^= & H^0\left(\A_K,C\right)/H^0(K,C) \ar[r] \ar[d] & H^{-1}\left(K,\widehat{C}\right)^\star \ar[d] \\
0 \ar[r] & P \ar[r] \ar[d]^= & H^0(\A_K,C)_{\wedge}/H^0(K,C)_{\wedge} \ar[r] \ar[d]^i & H^{-1}(K,\widehat{C})^\star _{\wedge} \ar[d] \\
0 \ar[r] & P \ar[r] & \left(H^0\left(\A_K,C\right)/H^0(K,C)\right)_{\wedge} \ar[r] & H^{-1}\left(K,\widehat{C}\right)^\star _{\wedge} }
\end{equation}
has exact first line by definition, and exact third line because it is obtained by completing the first line and $H^{-1}(K,\widehat{C})^\star$ is a lattice, so~\cite[Lemma~3.12$\MK$(b)]{DHdualcompl} applies. To prove the injectivity of $i$, it is sufficient (by diagram chasing) to show that the second line is exact as well. Let $\pi : H^0(\A_K,C) \to H^0(\A_K,C)/H^0(K,C)$ be the projection. The exact sequence
\[
0 \to \pi^{-1}(P) \to H^0\left(\A_K,C\right) \to H^{-1}\left(K,\widehat{C}\right)^\star
\]
induces (again by~\cite[Lemma~3.12$\MK$(b)]{DHdualcompl} an exact sequence
\[
\pi^{-1}(P)_{\wedge} \to H^0(\A_K,C)_{\wedge}
\to H^{-1}\left(K,\widehat{C}\right)^\star _{\wedge},
\]
whence an exact sequence
\[
\pi^{-1}(P)_{\wedge}/H^0(K,C)_{\wedge} \xrightarrow{j} H^0(\A_K,C)_{\wedge}/H^0(K,C)_{\wedge} \to
H^{-1}(K,\widehat{C})^\star _{\wedge}.
\]
But the exact sequence
\[
H^0(K,C) \to \pi^{-1}(P) \to P \to 0
\]
induces (as $P$ is profinite) a surjective map $u : \pi^{-1}(P)_{\wedge}/H^0(K,C)_{\wedge} \to P$ and $j$ factorizes through $u$, hence the second line of the diagram is exact, as required.

\item[\eqref{wedgelem3}] If $C = [0 \to \Gm]$, the required surjectivity is obviously true. If $C=[0 \to T]$, then one can find a resolution $0 \to R \to Q \to T \to 0$ (e.g. a flasque resolution) of $T$ by $K$-tori $R$ and $Q$, with $Q$ quasi-trivial. It induces an injective map of lattices $\widehat T(K) \to \widehat Q(K)$, hence a surjective map $\widehat{Q}(K)^\star \to \widehat{T}(K)^\star$; thus one can reduce to the case of a quasi-trivial torus. By Shapiro's Lemma, this case reduces to the known case of $\Gm$. Hence the map $H^0(\A_K,C) \to H^{-1}(K, \widehat{C})^\star$ is surjective as soon as $C = [0 \to T]$ for any torus $T$. Let $C = [T_1 \to T_2]$ be an arbitrary complex of tori. Then the map $\widehat{T_2}(K)^\star \to H^{-1}(K, \widehat{C})^\star$ is surjective (again by injectivity of $H^{-1}(K,\widehat C)
\to \widehat C(K)$), hence the surjectivity of $H^0(\A_K,C) \to H^{-1}(K, \widehat{C})^\star$ follows from that of $T_2(\A_K) \to \widehat{T_2}(K)^\star$. Since the diagram~\eqref{diagwedge} is commutative with exact rows, the second point follows.
\end{itemize}
\end{proof}

\begin{prop}\label{complexc}
\leavevmode
\begin{enumerate}
\item \label{complexc1}
There is an exact sequence
\begin{equation}\label{exact seq complex}
\begin{split}
H^0(K,C) &\to H^0\left(\A_K, C\right) \to \left(H^0\left(\A_K, C\right)/H^0(K,C)\right)_\wedge\\
 &\xrightarrow{\partial} {H^{-1}\left(K,\widehat{C}\right)^\star}_\wedge / H^{-1}\left(K,\widehat{C}\right)^\star \to 0 \,,
\end{split}
\end{equation}
where the morphism $\partial$ is given (after completion) by pairing~\eqref{pairing char c} for $C$.

\item \label{complexc2}
There is an exact sequence
\begin{align*}
H^0(K,C) \to H^0\left(\A_K,C\right) &\to \left(H^1\left(K,\widehat C\right)/\Sha^1\left(K,\widehat C\right)\right)^D \\
&\xrightarrow{\partial_C}
{H^{-1}\left(K,\widehat{C}\right)^\star}_\wedge / H^{-1}\left(K,\widehat{C}\right)^\star \to 0.
\end{align*}
\end{enumerate}
\end{prop}

Observe that for $C=\G$, we have $\widehat C=\Z[1]$ and~\eqref{complexc2} is just the classical exact sequence of global class field theory.

\begin{proof}
By~\cite[Theorems~5.7 and 5.10]{DHdualcompl}, there is a
commutative diagram with exact rows:
\begin{equation}\label{ptdiag}
\xymatrix{
H^0(K,C) \ar[r] \ar[d] & H^0\left(\A_K,C\right) \ar[r] \ar[d] & \left(H^1\left(K,\widehat C\right)/\Sha^1\left(K,\widehat C\right)\right)^D \ar[d] & \\
H^0(K,C)_{\wedge} \ar[r] & H^0\left(\A_K,C\right)_{\wedge} \ar[r] & \left(H^1\left(K,\widehat C\right)/\Sha^1\left(K,\widehat C\right)\right)^D \ar[r] & 0 \,,
}
\end{equation}
whence exactness of the sequence 
\[
H^0(K,C) \to H^0\left(\A_K, C\right) \to
\left(H^0\left(\A_K, C\right)/H^0(K,C)\right)_\wedge
\]
thanks to Lemma~\ref{wedgelem}$\MK$\eqref{wedgelem2}. The exactness of sequence~\eqref{exact seq complex} now follows from Lem\-ma~\ref{wedgelem}$\MK$\eqref{wedgelem3}, and part~\eqref{complexc2} of the proposition follows from its part~\eqref{complexc1} and diagram~\eqref{ptdiag}.
\end{proof}

We now consider a homogenous space $X=G/H$ with $G$ and $H$ reductive and we assume further that $G$ is quasi-trivial. As in the previous sections, we define $C=[H^{\tor} \to G^{\tor}]$ and $C_X=\Cone(C_H \to C_G)= [T_{H^{\sc}} \to T_H \to G^{\tor}]$. We denote by $\widehat C$ and $\widehat {C_X}$ their respective duals (cf. Section~\ref{four}). We have the analogue of the pairing~\eqref{pairing char c} with $C$ (resp. $\widehat C$) replaced by $C_X$ (resp. $\widehat{C_X}$), and again the pairing is trivial on $H^0(K,C_X) \times H^{-1}(K,\widehat{C_X})$.

%\smallskip

We observe that $H^{-1}(K,\widehat{C})=H^{-1}(K,\widehat{C_X})$ (and similarly over every completion $K_v$ of $K$) thanks to the exact triangle
\[
\widehat{\mu_H}[-2] \to \widehat C \to \widehat{C_X} \to \widehat{\mu_H}[-1].
\]
For every finite and non-empty set of places $S$ of $K$, we set
\[
\br_S \, X=\ker \left[\br_e X \to \prod_{v\,\in\,S} \br X_{K_v}\right]
\]
(not to be confused with the groups $\Be_S(X)$ of Section~\ref{four}) and $\br_S(X,G):=\br_S \, X \cap \br_{1,e}(X,G)$, $\br_{1,S} X:=\br_S \, X \cap \br_{1,e} X $. We will also use a smooth model $\clG$ (resp. $\HH$, $\XX=
\clG/\HH$) of $G$ (resp. $H$, $X$) over some non-empty Zariski open subset $U \neq E$ of the curve $E$. Shrinking $U$ if necessary, we can assume that $\HH$ and $\clG$ admit compatible maximal tori $T_{\HH} \subset T_{\clG}$. We have the corresponding complexes $C_{\HH}$, $\C=[\HH^{\tor} \to \clG^{\tor}]$, $C_{\XX}$ defined over $U$. For every bounded complex $\clF$ of flat commutative finite type group schemes over $U$, the compact support hypercohomology groups $H^i_c(U,\clF)$ are defined as in~\cite[\S~2]{DHdualcompl} and we set
\[
D^i(U,\clF):=\ker \left[H^i(U,\clF) \to\bigoplus_{v\,\not\,\in\,U}H^i\left(K_v,F\right)\right]=\Im \left[H^i_c(U,\clF) \to H^i(U,\clF)\right],
\]
where $F$ is the generic fibre of $\clF$ over $K$. For $v \in U$, we denote by $H^i_\nr(K_v,C_X)$ the image of $H^i(\clO_v,C_{\XX})$ in $H^i(K_v,C_X)$ and if $S$ is a finite set of places that does not meet $U$, we set
\[
\clP^i_S(U,C_X) := \prod_{v\,\notin\,U, v\,\notin\,S} H^i\left(K_v,C_X\right) \times \prod_{v\,\in\,U} H^i_\nr\left(K_v,C_X\right)
\]
(by convention, $\clP^i_\emptyset(U,C_X)$ is denoted by $\clP^i(U,C_X)$).

\begin{lemm} \label{surjab}
For every place $v \in U$, the abelianization map $\ab'_{{\XX}_v} : \XX(\clO_v) \to H^0(\clO_v,C_{\XX})$ is surjective.
\end{lemm}

\begin{proof}
Using the same method as in~\cite[Theorem~2.18]{DemEdinburgh}, it is sufficient (via the version of Lemma~\ref{lem compat ab ab'} over $\clO_v$) to show that $H^1_{\ab}(\clO_v,\HH)=0$ and $\ab^0 : \clG(\clO_v) \to H^0_{\ab}(\clO_v,\clG)$ is surjective. The nullity of $H^1_{\ab}(\clO_v,\HH)$ follows by devissage from the fact that for an $\clO_v$-torus $\T$, we have $H^1(\clO_v,\T)=H^2(\clO_v,\T)=0$ (see for example~\cite[proof of Theorem~2.10]{HSz-Crelle}). Finally $H^1(\clO_v, \clG^{\sc}) \simeq H^1(\F_v, \widetilde{G^{\sc}})$ is trivial by Lang's Theorem (here $\widetilde{G^{\sc}}$ is the reduction mod. $v$ of the reductive group scheme $\clG^{\sc}$; it is a connected linear group scheme over the residue field $\F_v$ of the curve $E$ at $v$). This implies that the abelianization map $\clG(\clO_v) \to H^0_{\ab}(\clO_v,\clG)$ is surjective because by definition of the abelianization map, there is an exact sequence
\[
\clG(\clO_v) \to H^0_{\ab}(\clO_v,\clG) \to H^1\left(\clO_v, \clG^{\sc}\right).
\]
\end{proof}

We need now to extend a few duality results of~\cite{DHdualcompl} to the three-term complex~$C_X$~:

\begin{prop}\label{dualcx}
\leavevmode
\begin{enumerate}
\item \label{dualcx1}
The group $H^1_c(U, C_{\XX})$ is in the class $\clE$ and $H^0(U,C_{\XX})$ is of finite type.

\item \label{dualcx2}
The groups $D^0(U,C_{\XX})$ and $D^1(U, \widehat C_{\XX})$ are finite.

\item \label{dualcx3}
The group $H^2_c(U,\widehat C_{\XX})$ is the dual of the discrete group $H^0(U,C_{\XX})$ and 
\[
H^1_c(U, C_{\XX})_\wedge\simeq H^1\left(U, \widehat{C}_{\XX}\right)^D.
\]
\end{enumerate}
\end{prop}

\begin{proof}
\begin{itemize}
\item[\eqref{dualcx1}] Consider the exact triangle
\begin{equation}\label{triangleC}
\mu_H[2] \to C_{\XX} \to \C \to \mu_H[3],
\end{equation}
which implies that $H^1_c(U, C_{\XX})$ is an extension of $H^1_c(U,\C)$ by the finite (cf. \cite[Theorem~1.1]{DHAMM}) group $H^3_c(U,\mu_H)$. Since $H^1_c(U,\C)$ is in $\clE$ (\cite[Proposition~3.13]{DHdualcompl}), so is $H^1_c(U, C_{\XX})$.

As $H^3(U,\mu_H)=0$ and $H^2(U,\mu_H)$ is finite (\cite[Theorem~1.1 and Corollary~4.9]{DHAMM}), we also get that $H^0(U,C_{\XX})$ is an extension of $H^0(U,\C)$ (which is of finite type by~\cite[Proposition~3.6.$\MK$(b)]{DHdualcompl}) by a finite group, hence it is also of finite type.

\item[\eqref{dualcx2}] The finiteness of $D^0(U,C_{\XX})$ follows from that of $D^0(U,\C)$ (see \cite[Lem\-ma~4.15]{DHdualcompl}) and that of $H^2(U, \mu_H)$ (\cite[Corollary~4.9]{DHAMM}) thanks to the commutative diagram with exact rows:
\[
\xymatrix{H^2(U,\mu_H) \ar[r] & H^0(U,C_{\XX}) \ar[r] \ar[d] & H^0(U,\C) \ar[d] \\
& \bigoplus\limits_{v\,\not\,\in\,U} H^0(K_v,C_X) \ar[r] & \bigoplus\limits_{v\,\not\,\in\,U} H^0(K_v,C) \,. }
\]

Similarly $D^1(U, \widehat C_{\XX})$ contains $D^1(U,\widehat{\C})$ as a finite index subgroup thanks to exact triangle
\[
\widehat {\mu_H} [-2] \to
\widehat {\C} \to \widehat {C_{\XX}} \to \widehat {\mu_H} [-1],
\]
and $D^1(U,\widehat {\C})$ is finite by~\cite[Theorem~5.2]{DHdualcompl}.

\item[\eqref{dualcx3}] We first show that Artin--Verdier duality induces an isomorphism $H^1_c(U, C_{\XX})_\wedge\linebreak \to H^1(U, \widehat{C}_{\XX})^D$. To prove this, we use a devissage given by the triangle~\eqref{triangleC}. By~\cite[Theorem~4.11$\MK$(b)]{DHdualcompl}, the discrete torsion group $H^i(U,\widehat{\C})$ is dual to the profinite group $H^{2-i}_c(U, {\C})_{\wedge}$ for $i=1,2$. There is a commutative diagram
\[
\xymatrix{H^0_c(U,\C)_{\wedge} \ar[r] \ar[d] & H^3_c(U,\mu_H) \ar[r] \ar[d] & H^1_c(U,C_{\XX})_{\wedge} \ar[r] \ar[d] & H^1_c(U,\C)_{\wedge} \ar[r] \ar[d] & 0 \\
H^2\left(U,\widehat{{\C}}\right)^D \ar[r] & H^0(U,\widehat \mu_H)^D \ar[r] & H^1\left(U,\widehat{{\C}}_{\XX}\right)^D \ar[r] & H^1\left(U,\widehat{\C}\right)^D \ar[r] & 0 \\
}
\]
The second line is exact as the dual of an exact sequence of discrete torsion groups. The first line is exact as well thanks to~\cite[Lemma~3.12]{DHdualcompl} because $H^3_c(U,\mu_H)$ is finite and $H^1_c(U,C_{\XX})$ belongs to the class $\clE$ (hence it injects into its completion). Now Artin--Mazur--Milne duality for $\mu_H$ (see~\cite[Theorem~1.1]{DHAMM}) and for $\C$ (see~\cite[Theorem~4.11$\MK$(b)]{DHdualcompl}) yield that the first, second, and fourth vertical maps are isomorphisms. The five lemma implies that the third one is also an isomorphism, as required.

The argument to prove the isomorphism $H^1_c(U, C_{\XX})_\wedge \simeq H^1(U, \widehat{C}_{\XX})^D$ is similar, using the commutative diagram with exact rows:
\[
\xymatrix{ 0 \ar[r] & H^2_c\left(U,\widehat{\C}\right) \ar[r] \ar[d] & H^2_c\left(U,\widehat{\C_{\XX}}\right)
\ar[r] \ar[d] & H^1_c\left(U,\widehat \mu_H\right) \ar[r] \ar[d] & H^3_c\left(U,\widehat {\C}\right) \ar[d] \\
0 \ar[r] & H^0(U,\C)^D \ar[r] & H^0\left(U,\C_{\XX}\right)^D \ar[r] & H^2(U,\mu_H)^D \ar[r] & H^{-1}\left(U,{\C}\right)^D \,. }
\]

Indeed the first and fourth vertical maps are isomorphisms by~\cite[Theorem~4.9$\MK$(a)]{DHdualcompl}, and so is the third by~\cite[Theorem~1.1]{DHAMM}.\qedhere
\end{itemize}
\end{proof}

\begin{prop}\label{ptsuit}\leavevmode
\begin{enumerate}
\item \label{ptsuit1}
There is an exact sequence
\[
H^0(K, C_X) \to H^0\left(\A_K, C_X\right) \to H^1\left(K, \widehat{C}_X\right)^D.
\]

%\smallskip

\item \label{ptsuit2}
There is an exact sequence
\[
H^1\left(K,\widehat C_X\right) \to H^1\left(\A_K, \widehat C_X\right) \to
H^0(K,C_X)^D.
\]
\item \label{ptsuit3}
There is a local duality isomorphism
\[
H^0(K_v, C_X)_\wedge \xrightarrow{\sim} H^1\left(K_v, \widehat{C}_X\right)^D,
\]
and the orthogonal of $H^1_{\nr}(K_v, \widehat{C}_X)$ in the local duality is the image of $H^0_{\nr}(K_v, C_X)$ in $H^0(K_v, C_X)_\wedge$.
\end{enumerate}
\end{prop}

\pagebreak
\begin{proof}
\begin{itemize}
\item[\eqref{ptsuit1}] We extend the proof of the exactness of the second line of Poitou--Tate exact sequence in~\cite[Theorem~5.7]{DHdualcompl}. Following the same method, it is sufficient to extend the first part of~\cite[Lemma~5.6~(a)] {DHdualcompl} to the complex $C_X$. Namely, it remains to show the exactness of
\[
H^0(U,C_{\XX}) \to \clP^0(U,C_X) \to
\left(H^1\left(K, \widehat{C}_X\right) / \Sha^1\left(K, \widehat{C}_X\right)\right)^D.
\]
The proof is exactly the same as for $C$, applying~\cite[Lemma~2.2]{DHdualcompl} to the complex $C_X$ and using the three following facts (proven in Proposition~\ref{dualcx}): for every non-empty Zariski open subset $V \subset U$, the group $H^1_c(V, C_{\XX})$ is in the class $\clE$, the group $D^0(V,C_{\XX})$ is finite, and Artin--Verdier duality induces an isomorphism $H^1_c(V, C_{\XX})_\wedge \simeq H^1(V, \widehat{C}_{\XX})^D$.

\item[\eqref{ptsuit2}] Similarly, it is sufficient to extend the second part of~\cite[Lemma~5.6$\MK$(b)]{DHdualcompl}, that is to show the exactness of
\[
H^1\left(U, \widehat C_{\XX}\right) \to \clP^1\left(U, \widehat C_X\right)
\to H^0(K,C_X)^D.
\]
Applying again~\cite[Lemma~2.2]{DHdualcompl} to $\widehat C_X$, one just has to check the following properties:
\begin{itemize}
\item for every non-empty Zariski open subset $V \subset U$,
we have
\[
H^2_c\left(V,\widehat C_{\XX}\right) \simeq \left(H^0(V,C_{\XX})_{\wedge}\right)^D =
\left(H^0(V,C_{\XX}\right)^D.
\]
This holds thanks to Proposition~\ref{dualcx} (the
finite type group $H^0(V,C_{\XX})$ has same dual as $H^0(V,C_{\XX})_{\wedge}$).

\item the group $D^1(U,\widehat C_{\XX})$ is finite, which is also proven in Proposition~\ref{dualcx}.
\end{itemize}

\item[\eqref{ptsuit3}] Using the exact triangle~\eqref{triangleC} and the vanishing of $H^3(K_v,\mu_H)$ (\cite[Propostion~III.6.4]{MilADT}), there is a commutative diagram with exact rows (the completed first line remains exact because $H^2(K_v,\mu_H)$ is finite by~\cite[Exemple~III.6.7]{MilADT}, and the second line is obtained by dualizing an exact sequence of discrete torsion groups):
\[
\xymatrix{ H^{-1}\left(K_v,C\right)_{\wedge } \ar[r] \ar[d] & H^2\left(K_v,\mu_H\right) \ar[r] \ar[d] & H^0\left(K_v,C_X\right)_{\wedge} \ar[r] \ar[d] & H^0\left(K_v,C\right)_{\wedge} \ar[r] \ar[d] & 0 \\
H^2\left(K_v,\widehat{C}\right)^D \ar[r] & H^0\left(K_v,\widehat \mu_H\right)^D \ar[r] & H^1\left(K_v, \widehat{C}_X\right)^D \ar[r] & H^1\left(K_v, \widehat{C}\right)^D \ar[r] & 0 \,. }
\]

Since the first, second, and fourth vertical map are isomorphisms by~\cite[Theorem~III.6.10]{MilADT} and~\cite[Theorem~3.1]{Demtores}, so is the third vertical map.

%\smallskip

It remains to show that the map 
\[
H^0\left(K_v,C_X\right)_{\wedge}/H^0_{\nr}\left(K_v,C_X\right)
\to H^1\left(\clO_v,\widehat{C}_{\XX}\right)^D
\]
is injective. There is a commutative diagram with exact lines:
\[
\xymatrix{ & & H^0\left(\clO_v,C_{\XX}\right) \ar[r] \ar[d] & H^0\left(\clO_v,\C\right) \ar[r] \ar[d] & 0 \\
H^{-1}(K_v,C)_{\wedge } \ar[r] \ar[d] & H^2\left(K_v,\mu_H\right) \ar[r] \ar[d] & H^0(K_v,C_X)_{\wedge} \ar[r] \ar[d] & H^0\left(K_v,C\right)_{\wedge} \ar[r] \ar[d] & 0 \\
H^2\left(\clO_v,\widehat{\C}\right)^D \ar[r] & H^0\left(\clO_v,\widehat \mu_H\right)^D \ar[r] & H^1\left(\clO_v, \widehat{C}_{\XX}\right)^D \ar[r] & H^1\left(\clO_v, \widehat{\C}\right)^D \ar[r] & 0 \,.}
\]

The fourth column is exact and the map $H^{-1}(K_v,C)_{\wedge } \to H^2(\clO_v,\widehat{\C})^D$ is surjective (\cite[Theorems~3.1. and 3.3]{Demtores}). Since $\widehat \mu_H$ is a finite group scheme, we have $H^0(\clO_v,\widehat \mu_H)=H^0(K_v,\widehat \mu_H)$, hence the map $H^2(K_v,\mu_H) \to H^0(\clO_v,\widehat \mu_H)^D$ is injective by~\cite[Theorem~III.6.10]{MilADT}. The required result follows by diagram chasing.\qedhere
\end{itemize}
\end{proof}
%\let\qed\relax

\begin{prop}\label{dualstep}
There is an exact sequence
\begin{equation}\label{short exact seq complex SA}
\begin{split}
H^0\left(K, C_X\right) &\to H^0\left(\A_K, C_X\right) \to \left(H^1\left(K, \widehat{C}_X\right) / \Sha^1\left(K, \widehat{C}_X\right)\right)^D \\
&\to {H^{-1}\left(K, \widehat{C}_X\right)^\star}_\wedge / H^{-1}\left(K, \widehat{C}_X\right)^\star \to 0 \,,
\end{split}
\end{equation}
where the last non-trivial map is defined via the natural map
\[
\left(H^1\left(K, \widehat{C}_X\right) / \Sha^1\left(K, \widehat{C}_X\right)\right)^D \to
\left(H^1\left(K, \widehat{C}\right) / \Sha^1\left(K, \widehat{C}\right)\right)^D
\]
and the map
\begin{multline*}
\left(H^1\left(K, \widehat{C}\right) / \Sha^1\left(K, \widehat{C}\right)\right)^D
\xrightarrow{\partial_C}{H^{-1}\left(K, \widehat{C}\right)^\star}_\wedge / H^{-1}\left(K, \widehat{C}\right)^\star\\
\simeq {H^{-1}\left(K, \widehat{C_X}\right)^\star}_\wedge / H^{-1}\left(K, \widehat{C_X}\right)^\star.
\end{multline*}
\end{prop}

\begin{proof}
We consider diagram~\eqref{diag hom spaces devissage}
comparing $C_X$ and $C$. Applying cohomology, we get a commutative diagram
{\footnotesize
\begin{equation}\label{devissage complex 3-terms}
\mkern-25mu\xymatrix{
H^2\left(\A_K, \mu_H\right) \ar[r] \ar[d] & \left(H^0\left(K, \widehat{\mu_H}\right)\right)^D \ar[r] \ar[d] & 0 & \\
H^0\left(\A_K, C_X\right) \ar[r] \ar[d] & \left(H^1\left(K, \widehat{C}_X\right) / \Sha^1\left(K, \widehat{C}_X\right)\right)^D \ar[r] \ar[d] & {H^{-1}\left(K, \widehat{C}_X\right)^\star}_\wedge / H^{-1}\left(K, \widehat{C}_X\right)^\star \ar[r] \ar[d]^{\simeq} & 0 \\
H^0\left(\A_K, C\right) \ar[r] \ar[d] & \left(H^1\left(K, \widehat{C}\right) / \Sha^1\left(K, \widehat{C}\right)\right)^D \ar[r] \ar[d] & {H^{-1}\left(K, \widehat{C}\right)^\star}_\wedge / H^{-1}\left(K, \widehat{C}\right)^\star \ar[r] & 0 \\
0 & 0 & }
\end{equation}}
Local duality for $\mu_H$ (cf.~\cite[Proposition~4.10$\MK$(b)]{Ces} for instance) and Proposition~\ref{complexc}$\MK$\eqref{complexc2} imply that the first row and the last one are exact. The exact triangles of~\eqref{diag hom spaces devissage} implies that the first column is exact.

The second column is exact, being the dual of the sequence of discrete groups
\[
0 \to H^1\left(K, \widehat{C}\right) / \Sha^1\left(K, \widehat{C}\right) \to H^1\left(K, \widehat{C}_X\right) / \Sha^1\left(K, \widehat{C}_X\right) \to H^0\left(K, \widehat{\mu_H}\right) \,,
\]
which is exact, thanks to the exact sequence (over $K$ and $K_v$)
\[
0 \to H^1\left(K, \widehat{C}\right) \to H^1\left(K, \widehat{C}_X\right) \to H^0\left(K, \widehat{\mu_H}\right).
\]

Now an easy diagram chase implies the exactness of the second line of~\eqref{devissage complex 3-terms}. The exactness of~\eqref{short exact seq complex SA} then follows from Proposition~\ref{ptsuit}$\MK$\eqref{ptsuit1}.
\end{proof}

\begin{prop}\label{flatcx}\leavevmode
\begin{enumerate}
\item \label{flatcx1}
The following sequence:
\begin{equation}\label{exact seq compl U 3}
H^0\left(K,C_X\right)_\wedge \to H^0\left(\A_K,C_X\right)_\wedge \to \left(H^1\left(K, \widehat{C}_X\right) / \Sha^1\left(K, \widehat{C}_X\right)\right)^D \to 0
\end{equation}
is exact.

\item \label{flatcx2}
Let $S$ be a finite set of places that does not meet $U$. Set
\[
H^1_S\left(K, \widehat{C}_X\right) =\ker \left[H^1\left(K, \widehat{C}_X\right)
\to \bigoplus_{v\,\in\,S} H^1\left(K_v, \widehat{C}_X\right)\right].
\]
Then the sequence
\begin{equation}\label{exact seq compl U S 3}
H^0(K,C_X)_\wedge \to H^0\left(\A_K ^S,C_X\right)_\wedge \to \left(H^1_S\left(K, \widehat{C}_X\right)/\Sha^1\left(K, \widehat{C}_X\right)\right)^D \to 0
\end{equation}
is exact.

\item \label{flatcx3}
There is an exact sequence
\[
H^0\left(U,C_{\XX}\right)_{\wedge} \to {\bfP}^0_S\left(U,C_{\XX})_{\wedge} \to
\left(H^1_S(K, \widehat{C}_X\right)/\Sha^1\left(K, \widehat{C}_X\right)\right)^D.
\]
\end{enumerate}
\end{prop}

\begin{proof}
\begin{itemize}
\item[\eqref{flatcx1}] Since $(H^1(K, \widehat{C}_X) /
\Sha^1(K, \widehat{C}_X))^D$ is profinite and 
\[
{H^{-1}\left(K, \widehat{C}_X\right)^\star}_\wedge / H^{-1}\left(K, \widehat{C}_X\right)^\star
\]
is uniquely divisible, the surjectivity of the map
\[
H^0\left(\A_K,C_X\right)_\wedge \to \left(H^1\left(K, \widehat{C}_X\right) / \Sha^1\left(K, \widehat{C}_X\right)\right)^D
\]
follows immediately from Proposition~\ref{dualstep} and~\cite[Lemma~3.12$\MK$(b)]{DHdualcompl}. Applying Proposition~\ref{ptsuit}$\MK$\eqref{ptsuit2} and taking into account that $H^1(K,\widehat C_X)$ is torsion, we get an exact sequence
\[
0 \to \Sha^1\left(K,\widehat C_X\right) \to
H^1\left(K,\widehat C_X\right) \to H^1\left(\A_K, \widehat C_X\right)_{\tors} \to
\left(H^0\left(K,C_X\right)_{\wedge}\right)^D.
\]
Indeed $(H^0(K,C_X)_{\wedge})^D=(H^0(K,C_X)^D)_{\tors}$ (cf.~\cite[Remark~5.9]{DHdualcompl}). Dualizing this exact sequence now yields the result, thanks to Proposition~\ref{ptsuit}$\MK$\eqref{ptsuit3}.

\item[\eqref{flatcx2}] Dualizing the exact sequence of discrete torsion groups
\[
0 \to H^1_S\left(K, \widehat{C}_X\right) \to H^1\left(K, \widehat{C}_X\right) \to \bigoplus_{v\,\in\,S} H^1\left(K_v, \widehat{C}_X\right) \,.
\]
and taking into account the local duality isomorphism $H^0(K_v, C_X)_\wedge \xrightarrow{\sim}\linebreak H^1(K_v, \widehat{C}_X)^D$, one gets a commutative diagram
\[
\xymatrix{ & \bigoplus\limits_{v\,\in\,S} H^0\left(K_v, C_X\right)_\wedge \ar[r]^= \ar[d]^i & \bigoplus\limits_{v\,\in\, S} H^0\left(K_v, C_X\right)_\wedge \ar[d] & \\
H^0\left(K,C_X\right)_\wedge \ar[r] \ar[d]^= & H^0\left(\A_K,C_X\right)_\wedge \ar[r] \ar[d]^p & \left(H^1\left(K, \widehat{C}_X\right)/\Sha^1\left(K, \widehat{C}_X\right)\right)^D \ar[d] \ar[r] & 0 \\
H^0\left(K,C_X\right)_\wedge \ar[r] & H^0\left(\A_K^S,C_X\right)_\wedge \ar[r] \ar[d] & \left(H^1_S\left(K, \widehat{C}_X\right)/\Sha^1\left(K, \widehat{C}_X\right)\right)^D \ar[d] & \\
& 0 & 0 \,, }
\]
where the second line (by~\eqref{flatcx1}) and the columns are exact (the left one because $H^0(K_v,C_X)$ is in the class $\clE$ by~\cite[Proposition~3.13]{DHdualcompl} and exact triangle~\eqref{triangleC}; so the sequence remains exact after completion thanks to loc. cit., Lemma~3.12). A simple diagram chase implies that the bottom line is exact, which proves~\eqref{flatcx2}.

\item[\eqref{flatcx3}] For $v \in U$ and $i \in \Z$, set
\[
H^i_r\left(K_v,\widehat C_X\right)=H^i\left(K_v,\widehat C_X\right)/H^i_{\nr}\left(K_v,\widehat C_X\right)
\]
(and similarly for $C_X$). Consider the commutative diagram:
{\small
\[
\xymatrix{ H^1_S\left(K,\widehat C_X\right) \ar[r] & \bigoplus\limits_{v\,\in\,U} H^1_r\left(K_v,\widehat C_X\right)
\oplus \bigoplus\limits_{v\,\not\,\in\,(S\,\cup\,U) } H^1\left(K_v,\widehat C_X\right) \ar[r] & H^0\left(U,C_{\XX}\right)^D \\
H^1_S\left(K,\widehat C_X\right) \ar[r] \ar[u]^= & \prod'\limits_{v\,\not\,\in\,S} H^1\left(K_v, \widehat C_X\right) \ar[r] \ar[u]^p & H^0\left(K,C_X\right)^D \ar[u] \\
& \prod_{v\,\in\,U} H^1_{\nr}\left(K_v,\widehat C_X\right) \ar[u] \ar[r]^= & \prod\limits_{v\,\in\,U} H^1_{\nr}\left(K_v,\widehat C_X\right) \ar[u] \,. }
\]}
The middle column is obviously exact. The right column is exact as well, because by Proposition~\ref{ptsuit}$\MK$\eqref{ptsuit3}, it is the dual of the sequence of discrete groups

\[
H^0(U,C_{\XX}) \to H^0(K,C_X) \to \bigoplus_{v\,\in\,U} H^0_r(K_v,C_X),
\]
which is exact by~\cite[Proposition~2.1]{DHdualcompl}. It follows immediately from Proposition~\ref{ptsuit}$\MK$\eqref{ptsuit2} that the second line of the diagram is exact. Since the map $p$ is obviously surjective, a diagram chase shows that the first line of the diagram is exact as well. Dualizing it (and observing that $H^0(U,C_{\XX})$ is an abelian group of finite type by Proposition~\ref{dualcx}$\MK$\eqref{dualcx1}) yields (thanks to Proposition~\ref{ptsuit}$\MK$\eqref{ptsuit3}) the exactness of
\[
H^0(U,C_{\XX})_{\wedge} \to {\bfP}^0_S(U,C_{\XX})_{\wedge} \to
H^1_S\left(K, \widehat{C}_X\right)^D \,.
\]
Since the canonical map ${\bfP}^0_S(U,C_{\XX})_{\wedge} \to H^1_S(K, \widehat{C}_X)^D$ factors through
\[
\left(H^1_S\left(K, \widehat{C}_X\right)/\Sha^1\left(K,\widehat{C}_X\right)\right)^D,
\]
the result is proven.\qedhere
\end{itemize}
\end{proof}


Recall that for a finite (possibly empty) set of places $S$ of $K$, we have the Brauer--Manin pairing
\[
\BM : X\left(\A_K ^S\right) \times \br X \to \Q/\Z, \quad
\left((P_v)_{v\,\not\,\in\,S}, \alpha\right) \mapsto \sum_{v\,\not\,\in\,S} \alpha(P_v).
\]
By global class field theory, elements of $X(K) \subset X(\A_K ^S)$ are orthogonal to $\Br_S(X)$ for this pairing; in particular when $S=\emptyset$, elements of $X(K)$ are orthogonal to $\Br_e X$ (hence to $\br X$). By continuity of the pairing, the same holds for the closure $\overline{{X(K)}}^S$ of $X(K)$ in $X(\A_K ^S)$ for the strong topology. The following theorem gives various converse statements.

\begin{theo}\label{mainstrong}
Let $X=G/H$ be a homogeneous space of a reductive group $G$ with $H$ reductive. Set $U(X):=K[X]^*/K^*$. Assume that $G^\sc$ satisfies strong approximation outside $S_0$ (finite set of places).
\begin{enumerate}
\item\label{theo5.8.1} There is a natural exact sequence of pointed topological spaces:
\begin{multline*}
1 \to \rho\left(\overline{G^\sc(K) \cdot G^\sc\left(K_{S_0}\right)}\right) \cdot X(K) \to X(\A_K)\\ \xrightarrow{BM} \left(\Br_{1,e}(X,G)/\Be(X)\right)^D \xrightarrow{\partial} {U(X)^\star}_\wedge / U(X)^\star \to 1 \,.
\end{multline*}
In particular, 
\[
X(\A_K)^{\Br X} = \rho\left(\overline{G^\sc(K) \cdot G^\sc\left(K_{S_0}\right)}\right) \cdot X(K)\quad \text{and} \quad X\left(\A_K^{S_0}\right)^{\Br X} \subset
\overline{X(K)}^{S_0}.
\]
\item\label{theo5.8.2} If $S$ is a non-empty finite set of places, there is a natural exact sequence of pointed topological spaces:
\[
1 \to \overline{\rho\left(G^\sc\left(K_{S_0}\right)\right) \cdot X(K)}^S \to X\left(\A_K^S\right) \xrightarrow{BM} \left(\Br_S(X,G)/\Be(X)\right)^D \to 1 \,.
\]
In particular, $X(\A_K^S)^{\Br_S X} =
\overline{\rho(G^\sc(K_{S_0})) \cdot X(K)}^S$.

If $S_0 \subset S$, we get an exact sequence
\[
1 \to \overline{X(K)}^S \to X\left(\A_K^S\right) \xrightarrow{BM} \left(\Br_S(X,G)/\Be(X)\right)^D \to 1 \,.
\]
\end{enumerate}
\end{theo}

\begin{proof}
As earlier, one can assume that the group $G$ is quasi-trivial, up to replacing $G$ by a flasque resolution
\[
1 \to S \to G' \to G \to 1.
\]

\pagebreak

Indeed $\pic \ov S=0$ (since $S$ is a torus), hence $\br \ov G \hookrightarrow
\br \ov G'$, which implies $\br_1(X,G)=\br_1(X,G')$ by~\cite[Proposition~6.10]{San}. Throughout the proof, we use Theorem~\ref{theo compatible strong approx} to translate results concerning complexes of tori and cup-products to results concerning homogeneous spaces and Brauer--Manin obstructions.
\begin{itemize}
\item[\eqref{theo5.8.1}] By~\cite[Theorem~4.14]{DemJOA}, the group $U(X)$ is isomorphic to $H^{-1}(K,\widehat{C_X})$. By Theorems~\ref{theo compatible strong approx} and~\ref{dualstep}, there is a commutative diagram with exact second row:
%\vspace{45pt}
{\scriptsize
\begin{center}
\begin{equation*}
\xymatrix@-1pc{ X(K) \ar[r] \ar[d]^{\ab'_K} & X(\A_K) \ar[d]^{\ab'_{\A_K}} \ar[r]^-\BM & \left(\Br_{1,e}(X,G)/\Be(X)\right)^D \ar[r] \ar[d]^{\Phi_X^D}_\sim & {U(X)^\star}_\wedge / U(X)^\star \ar[r] \ar[d]_\sim & 1 \\
H^0(K,C_X) \ar[r] & H^0\left(\A_K,C_X\right) \ar[r] & H^1\left(K,\widehat{C_X}\right)^D \ar[r] & {H^{-1}\left(K, \widehat{C_X}\right)^\star}_\wedge / H^{-1}\left(K, \widehat{C_X}\right)^\star \ar[r] & 1 }
\end{equation*}
\end{center}}

By Proposition~\ref{orbitesprop} and Lemma~\ref{surjab}, the maps $\ab ' _K$ and $\ab '_{\A_K}$ are surjective. By diagram chasing, the sequence that consists of the last three non-trivial terms on the first line is also exact. Furthermore, every element $x$ of $X(K) \subset X(\A_K)$ satisfies $\BM(x)=0$ by class field theory, and the same holds for an element of $X(\A_K)$ of the form $(g_v.x)$ with $(g_v) \in \rho(G^{\sc}(\A_K))$ thanks to the commutativity of the diagram and the property $\ab '(g_v.x)=\ab '(x)$ for every place $v$ of $K$ (Proposition~\ref{prop ab action}). It remains to show conversely that every $(P_v) \in X(\A_K)$ such that $\BM((P_v))=0$ comes from $\rho(\overline{G^\sc(K) \cdot G^\sc(K_{S_0})}) \cdot X(K)$ The diagram and the surjectivity of $\ab ' _K$ imply that there exists
$x \in X(K)$ such that
\[
\ab '_{\A_K}(x)=\ab '_{\A_K}(P_v)
\in H^0(K_v,C_X)
\]
for every place $v$.
By Proposition~\ref{orbitesprop}, there exists for each $v$ an element $g_v \in \rho(G^{\sc}(K_v))$ such that $P_v=g_v.x$, and we can assume that $g_v \in {\clG}^{\sc}(\clO_v)$ for $v \not \in S$, where $S \supset S_0$ is some finite set of places. Since $G^{\sc}$ satisfies strong approximation outside $S_0$, we finally obtain that $(P_v)$ belongs to $\rho\left(\overline{G^\sc(K) \cdot G^\sc(K_{S_0})}\right) \cdot X(K)$ as required. The two other assertions in 1. follow immediately.


\item[\eqref{theo5.8.2}] We follow the proof of~\cite[Theorem~6.1]{DemEdinburgh}.

Up to shrinking $U$, we can assume that it does not meet $(S_0 \cup S)$. We consider the following commutative diagram:


{\small
\begin{equation}\label{big diag}
\mkern-45mu\xymatrix@-1pc{ & & & \clP_S(U,C_H)_\wedge \ar[rr] \ar'[d][dd] & & H^1_S\left(K, \widehat{C}_X\right)^D \ar[dd] \\
& & \clP_S(U, H) \ar[ur] \ar[rr] \ar[dd] & & \left(\Br_S(H) \right)^D \ar[ru] \ar[dd] & \\
& H^0(U,C_G)_\wedge \ar'[d][dd] \ar'[r][rr] & & \clP^0_S(U,C_G)_\wedge \ar'[r][rr] \ar'[d][dd] & & H^1_S\left(K, \widehat{C}_G\right)^D \ar[dd] \\
G(U) \ar[ru] \ar[rr] \ar[dd] & & \clP^0_S(U,G) \ar[rr] \ar[dd] \ar[ru] & & \left(\Br_S(G) \right)^D \ar[ru] \ar[dd] & \\
& H^0(U, C_X)_\wedge \ar'[r][rr] \ar'[d][dd] & & \clP^0_S(U, C_X)_\wedge \ar'[r][rr] \ar'[d][dd] & & H^1_S\left(K, \widehat{C}_X\right)^D \\
X(U) \ar[ru] \ar[rr] \ar[dd] & & \clP^0_S(U,X) \ar[ru] \ar[rr] \ar[dd] & & \left(\Br_S(X,G)\right)^D \ar[ru] & \\
& H^1(U, C_H) \ar'[r][rr] & & \clP^0_S(U, C_H) & & \\
H^1(U,H) \ar[d] \ar[rr] \ar[ru]^\sim & & \clP^1_S(U,H) \ar[ru]^\sim & & & \\
1 & & & & & \,.}
\end{equation}}
The definition of the maps $H^0(U,C_X)_\wedge \to H^1(U,C_H)$ and $\clP^0_S(U,C_X)_\wedge \to \clP^1_S(U,C_H)$ is a consequence of~\cite[Lemma~6.5]{DemEdinburgh} (the proof works the same in the function field context). Using this lemma, we see that the columns in this diagram are exact. The surjectivity of $X(U) \to H^1(U,H)$ is a consequence of the vanishing of $H^1(U,G)$ for $U$ sufficiently small (Hilbert 90, together with Nisnevich Theorem as proven in~\cite[Theorem~5.1]{Gil-torseurs-affine}).

In addition, the fifth line of this diagram is exact by Proposition~\ref{flatcx}$\MK$\eqref{flatcx3}, and the maps $H^1(U,H) \to H^1(U, C_H)$ and $\clP^1_S(U,H) \to \clP^1_S(U,C_H)$ are bijections (see~\cite[Theorem~5.5 and Example~5.4$\MK$(iii)]{GA} and~\cite[Theorem~5.1]{Gil-torseurs-affine}).

Now, the proof of the second point of the Theorem is a diagram chase in diagram~\eqref{big diag}, following exactly the argument in~\cite[Theorem~6.1]{DemEdinburgh}. In particular, one uses the fact that $\clP^0_S(U,C_H)_\wedge$ and $\clP^0_S(U,C_H)$ (hence $\clP^0_S(H)$) have the same image inside $H^1_S(K, \widehat{C}_H)^D$ which is a consequence of the following proposition:

\begin{prop}\label{compactquot}
Let $\D=[\T_1 \to \T_2]$ be a complex (with $\T_2$ in degree~$0$) of flat, separated and finite type commutative group schemes over the affine curve $E-S$, such that the restriction of $\T_1$ and $\T_2$ to $U$ are tori. Let $D=[T_1 \to T_2]$ be the generic fibre of $\D$.
\begin{enumerate}\alphenumi
\item\label{compactquot1}
The group $H^0(U,\D) \backslash \clP^0_S(U,D) /
\prod_{v \notin S} H^0(\clO_v,\D)$ is finite.

\item\label{compactquot2}
The group $\clP^0_S(U,\D) / \bar{H^0(U,\D)}$ is compact.
\end{enumerate}
\end{prop}

\pagebreak
\begin{proof}
\begin{itemize}
\item[\eqref{compactquot1}] Let $S_1$ be the finite set of places of $K$ that are neither in $U$ nor in $S$. Recall (\cite[Corollary~2.3]{MilADT}) that the group $H^1(K_v,T)$ is finite for all places $v \in \Omega_K$. If $\T_1=0$, the result follows immediately from the finiteness of the class group of a torus, as proven in~\cite[Example~1.3.2]{Conrad-Compositio} (recall that $S \neq \emptyset$). In the general case, we have $H^1(\clO_v,\T_1)=0$ for $v \in U$, whence an exact sequence
\[
\clP^0_S(U,\T_2) \to \clP^0_S(U,D) \to
\prod_{v\,\in\,S_1} H^1(K_v,T_1),
\]
which shows in particular that the image of $\clP^0_S(U,\T_2)/\prod_{v\,\notin\,S} H^0(\clO_v,\T_2)$ is of finite index in $\clP^0_S(U,\D)/\prod_{v\,\notin\,S} H^0(\clO_v,\D)$. Thus one reduces to the already known case $\T_1=0$.

\item[\eqref{compactquot2}] follows from~\eqref{compactquot1} and the compactness of $\prod_{v\, \notin\,S} H^0(\clO_v,\D)$.\qedhere
\end{itemize}
\end{proof}

To finish the proof of the Theorem~\ref{mainstrong}, one deduces the surjectivity of the map
\[
X\left(\A_K^S\right) \to \left(\Br_S(X,G) / \Be(X)\right)^D
\]
from exact sequence~\eqref{exact seq compl U S 3} and from the surjectivity of $X(\A_K^S) \to H^0(\A_K^S,C_X)$ (which follows from Lemma~\ref{surjab} and
Proposition~\ref{orbitesprop}).\qedhere
\end{itemize}
\end{proof}

\begin{rema}\leavevmode

\begin{enumerate}
\item\label{rema5.10.1} In the special case when $X=G$ is a reductive group, we get exact sequences of topological groups:
\begin{multline*}
1 \to \rho\left(\overline{G^\sc(K) \cdot G^\sc(K_{S_0})}\right) \cdot G(K) \to G(\A_K)\\ \xrightarrow{BM} \left(\Br_{1,e} \, G/\Be(G)\right)^D \xrightarrow{\partial} {\widehat G(K)^\star}_\wedge / \widehat G(K)^\star \to 1 \,.
\end{multline*}
\[
1 \to \overline{\rho\left(G^\sc\left(K_{S_0}\right)\right) \cdot G(K)}^S \to G\left(\A_K^S\right) \xrightarrow{BM} \left(\Br_{1,S} \, G /\Be(G)\right)^D \to 1 \,.
\]
The fact that $BM$ is a group morphism in this situation follows from the same argument as in Corollary~\ref{groupcase}.
\item\label{rema5.10.2} In addition, using~\cite[Theorem~5.10]{DHdualcompl}, it is straightforward to continue the previous exact sequence as follows, for any non-empty finite set of places $S$:
\[
\xymatrix{ 1 \ar[r] & \overline{\rho\left(G^\sc\left(K_{S_0}\right)\right) \cdot G(K)}^S \ar[r] & G\left(\A_K^S\right) \ar[r]^{BM} & \left(\Br_{1,S} \, G\right)^D \ar[d] \\
& \left(\pic G\right)^D & \bigoplus\limits_v H^1\left(K_v,G\right) \ar[l] & H^1(K,G) \ar[l] \,, }
\]
where the pairing $H^1(K_v,G) \times \pic(G)$ can be defined via the natural isomorphism between $\pic(G)$ and the group $\ext^1_K(G, \Gm)$ of central extensions of $G$ by $\Gm$ (see for instance~\cite[Corollary~5.7 and Proposition~8.2]{CTfl}). One can even extend this exact sequence to a 9-term Poitou--Tate exact sequence involving non-abelian $H^2$, using~\cite[Theorem~5.10]{DHdualcompl}, following the methods of~\cite [Theorem~5.1]{DemAF}. See also~\cite[Theorem~5.16]{BorAMS} for a similar statement over a number field.
\end{enumerate}

\end{rema}



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