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Manin obstruction to strong approximation for homogeneous spaces

For a homogeneous space X (not necessarily principal) of a connected algebraic group G (not necessarily linear) over a number field k, we prove a theorem of strong approximation for the adelic points of X in the Brauer-Manin set. Namely, for an adelic point x of X orthogonal to a certain subgroup (which may contain transcendental elements) of the Brauer group Br(X) of X with respect to the Manin pairing, we prove a strong approximation property for x away from a finite set S of places of k. Our result extends a result of Harari for torsors of semiabelian varieties and a result of Colliot-Th\'el\`ene and Xu for homogeneous spaces of simply connected semisimple groups, and our proof uses those results.


Introduction
Let k be a number field. We denote by Ω the set of all places of k, by Ω ∞ the set of all infinite (archimedean) places of k, by Ω r the set of all real places of k, and by Ω f the set of all finite (nonarchimedean) places of k. For a finite set S ⊂ Ω, we set k S := v∈S k v , where k v denotes the completion of k at v. We write k ∞ for k Ω∞ . We denote by A the ring of adèles of k and by A S the ring of adèles without S-components. We have A = A S × k S . If S = Ω ∞ , we denote by A f := A Ω∞ the ring of finite adeles. If X is a k-variety, we have X(k S ) = v∈S X(k v ) and X(A) = X(A S ) × X(k S ). In particular X(A) = X(A f ) × X(k ∞ ).
Let X be a smooth geometrically integral k-variety over a field k of characteristic 0. Let Br(X) := H 2 et (X, G m ) denote the cohomological Brauer group of X. We set Br 1 (X) := ker[Br(X) → Br(X × k k)], where k is an algebraic closure of k.
Recall that when k is a number field, there exists a canonical pairing (Manin pairing) see [10], Section 3.1, or [40], Section 5.2. This pairing is additive in b ∈ Br(X) and continuous in x ∈ X(A). If x ∈ X(k) ⊂ X(A) or b comes from Br(k), then b, x = 0. For a subgroup B ⊂ Br(X) we denote by X(A) B the set of points of x ∈ X(A) orthogonal to B with respect to Manin pairing. We have One can ask whether any point x = (x v ) ∈ X(A) which is orthogonal to Br(X) can be approximated in a certain sense by k-rational points.
In this paper we consider the case when X is a homogeneous space of a connected algebraic k-group G (not necessarily linear) with connected geometric stabilizers. For such an X and x = (x v ) ∈ X(A) Br1(X) it was proved in [7], Appendix, Theorem A.1, that our X has a k-point and that x can be approximated by kpoints in the sense of weak approximation. We used a result of Harari [21] on the Manin obstruction to weak approximation for principal homogeneous spaces of semiabelian varieties. Here, using a result of recent Harari's paper [22] on the Manin obstruction to strong approximation for principal homogeneous spaces of semiabelian varieties together with a recent result of Colliot-Thélène and Xu [11] on strong approximation for homogeneous spaces of simply connected groups, we prove a theorem on strong approximation for our x.
For a connected k-group G we write G abvar for the biggest quotient of G which is an abelian variety, and we write G sc for "the simply connected semisimple part of G", see 1.1 below for details.
Theorem 0.1. Let G be a connected algebraic group (not necessarily linear) over a number field k. Let X := H\G be a right homogeneous space of G, where H is a connected k-subgroup of G. Assume that the Tate-Shafarevich group of the maximal abelian variety quotient G abvar of G is finite. Let S ⊃ Ω ∞ be a finite set of places of k containing all archimedean places. We assume that G sc (k) is dense in G sc (A S ). Let x = (x v ) ∈ X(A) be a point orthogonal to Br(X) with respect to the Manin pairing. Then for any open neighbourhood U S of the projection x S of x to X(A S ) there exists a rational point x 0 ∈ X(k) whose diagonal image in X(A S ) lies in U S . Moreover, we can ensure that for each archimedean place v, the points x 0 and x v lie in the same connected component of X(k v ).
Recall that G sc (k) is dense in G sc (A S ) if and only if for every k-simple factor G sc i of G sc the group G sc i (k S ) is noncompact (a theorem of Kneser and Platonov, cf. [36], Theorem 7.12).
In Theorem 0.1 we assume that our adelic point x is orthogonal to the whole Brauer group Br(X). Actually it is sufficient to require that x were orthogonal to a certain subgroup Br 1 (X, G) ⊂ Br(X). In general this subgroup Br 1 (X, G) contains transcendental elements (i.e. is not contained in Br 1 (X)). Note that Theorem 0.1 with Br 1 (X) instead of Br(X) would be false, see Counter-example 1.6 below. However this theorem still holds with Br 1 (X) instead of Br(X), if S contains at least one nonarchimedean place: Theorem 0.2. Let G be a connected algebraic group (not necessarily linear) over a number field k. Let X := H\G be a right homogeneous space of G, where H is a connected k-subgroup of G. Assume that the Tate-Shafarevich group of the maximal abelian variety quotient G abvar of G is finite. Let S ⊃ Ω ∞ be a finite set of places of k containing all archimedean places and at least one nonarchimedean place. We assume that G sc (k) is dense in G sc (A S ). Let x = (x v ) ∈ X(A) be a point orthogonal to Br 1 (X) with respect to the Manin pairing. Then for any open neighbourhood U S of the projection x S of x to X(A S ) there exists a rational point x 0 ∈ X(k) whose diagonal image in X(A S ) lies in U S . Moreover, we can ensure that for each archimedean place v, the points x 0 and x v lie in the same connected component of X(k v ).
Remark 0.3. Let X be a right homogeneous space of a connected group k-group G over a number field k such that the stabilizers of the geometric points are connected. By [7], Theorem A.1, if there exists x ∈ X(A) which is orthogonal to Br 1 (X), then the variety X must have a k-point, hence X = H\G, where H is a connected subgroup of G. Therefore we could reformulate Theorems 0.1, 0.2, 1.4 and 1.7 for a general homogeneous space X of G (without assuming that X is of the form X = H\G).
Our proof is somewhat similar to that of Theorem A.1 of [7]. We use the reductions and constructions of Subsections 3.1 and 3.3 of [7] in order to reduce the assertion to the case when X is a k-torsor under a semiabelian variety (treated by Harari [22]) and to the case when X is a homogeneous space of a simply connected semisimple group with connected geometric stabilizers (the Hasse principle was proved in [4]; for strong approximation, see Theorem 6.1 below, which is actually due to Colliot-Thélène and Xu [11]).
1 Main results 1.1. Let G be a connected algebraic group (not necessarily linear) over a field k of characteristic 0. Then G fits into a canonical short exact sequence where G lin is a connected linear k-group and G abvar is an abelian variety over k. We use the following notation: G u is the unipotent radical of G lin ; G red := G lin /G u , it is a reductive k-group; G ss is the commutator subgroup of G red , it is a semisimple k-group; G sc is the universal covering of G ss , it is a simply connected semisimple k-group; G tor := G red /G ss , it is a k-torus; G ssu := Ker(G lin → G tor ), it is an extension of G ss by G u . G sab := (G/G u )/G ss , it is a semiabelian variety over k, it fits into a short exact sequence 1 → G tor → G sab → G abvar → 1.
We define the group G scu as the fibre product G scu := G sc × G red G lin , it fits into an exact sequence 1 → G u → G scu → G sc → 1.
We have a canonical homomorphism G scu → G lin → G.

1.2.
Let X be a smooth geometrically integral k-variety. We write X for X × k k, where k is a fixed algebraic closure of k. Recall that Br(X) is the cohomological Brauer group of X and that Br 1 (X) = ker[Br(X) → Br(X)]. We set Br a (X) := coker[Br(k) → Br 1 (X)]. If x 0 ∈ X(k) is a k-point of X, we set Br x0 (X) := ker[x * 0 : Br(X) → Br(k)] and Br 1,x0 (X) := ker[x * 0 : Br 1 (X) → Br(k)]. We have a canonical isomorphism Br 1,x0 (X) ∼ → Br a (X). Let X = H\G be a homogeneous space of a connected k-group G. Let x 0 ∈ X(k). Consider the map π x0 : G → X, g → x 0 .g, it induces a homomorphism π * x0 : Br(X) → Br(G). Consider the commutative diagram Let Br 1 (X, G) denote the kernel of any of the two equal composed homomorphisms Br(X) → Br(G). In other words, Br 1 (X, G) is the subgroup of elements b ∈ Br(X) such that π * x0 (b) ∈ Br 1 (G). We show in Lemma 4.4 below, that Br 1 (X, G) does not depend on x 0 . Let It is easy to see that the structure map X → Spec (k) induces an embedding Br(k) ֒→ Br 1 (X, G) (because X has a k-point) and that Br 1 (X, G) = Br 1,x0 (X, G)+ Br(k). It follows that an adelic point x ∈ X(A) is orthogonal to Br 1 (X, G) with respect to Manin pairing if and only if it is orthogonal to Br 1,x0 (X, G).
1.3. Let X be a smooth geometrically integral k-variety over a number field k. We denote by X(A) • the set X(A f ) × v∈Ω∞ π 0 (X(k v )), where π 0 (X(k v )) is the set of connected components of X(k v ). The set X(A) • has a natural topology, which we call the adelic topology. We have a canonical continuous map X(A) → X(A) • and a canonical embedding X(k) ֒→ X(A) • . The pairing (1) of the Introduction induces a pairing For a subgroup B ⊂ Br(X) we denote by (X(A) • ) B the set of points of x ∈ X(A) • orthogonal to B with respect to Manin pairing. Let X be a homogeneous space of a connected k-group G. Then G(A) acts on X(A) and on X(A) • .
Main Theorem 1.4. Let G be a connected algebraic group (not necessarily linear) over a number field k. Let X := H\G be a right homogeneous space of G, where H is a connected k-subgroup of G. Assume that the Tate-Shafarevich group of the maximal abelian variety quotient G abvar of G is finite. Let S ⊃ Ω ∞ be a finite set of places of k containing all archimedean places. We assume that G sc (k) is dense in G sc (A S ). Set S f := S ∩ Ω f = S Ω ∞ . Then the set (X(A) • ) Br1(X,G) coincides with the closure of the set X(k).G scu (k S f ) in X(A) • for the adelic topology.
We are interested also in the algebraic Manin obstruction, that is, the obstruction coming from the subgroup Br 1 (X) ⊂ Br(X). We can easily see that in general the algebraic Manin obstruction is not the only obstruction to strong approximation.
First we notice that in general Br 1 (X) Br 1 (X, G). Indeed, assume that G is semisimple and simply connected and that H is connected semisimple but not simply connected. In particular, H fits into an exact sequence where µ H is finite and abelian. Consider X := H\G. Then we know by [38], Proposition 6.9(iv), that the groups Pic(G) and Br 1,e (G) are trivial. By [11] Proposition 2.10(ii) we have a canonical isomorphism Pic(H) ∼ = Br x0 (X), where x 0 ∈ X(k) is the image of e ∈ G(k). Since H is semisimple, by the exact sequence in [38], Lemma 6.9(i), we know that the map Pic(H) → Pic(H) is injective, therefore any non-trivial element of Br x0 (X) is a transcendental element of Br(X), i.e. is not killed in Br(X). We see that Br 1,x0 (X) = 0, hence Br a (X) = 0. In addition, Proposition 2.6(iii) in [11] (or the corollary in the introduction of [18]) implies that Br(G) = 0, hence Br 1,x0 (X, G) = Br x0 (X). Therefore in this case Br 1 (X)/Br(k) = Br a (X) = 0, while Br 1 (X, An explicit example is given by H = SO n ⊂ SL n = G for n ≥ 3. In this case µ H = µ 2 and µ H (k) = Z/2Z = 0, hence Br 1 (X, G) ⊂ Br 1 (X). We take k to be a totally imaginary number field, e.g. k = Q(i). Take S = Ω ∞ , then A S = A f . We show that in this case the algebraic Manin obstruction is not the only obstruction to strong approximation away from S.
We may and shall identify X := H\G with the variety of symmetric n × nmatrices T with determinant 1. Then an element of X(A) can be written as denote the Hasse invariant of the quadratic form defined by T v (see for instance [33], p.167, before Example 6:14). Note that for v ∈ Ω ∞ we have because for all v ∈ Ω ∞ we have ε v (T 0 ) = 1, and because by [33], Theorem 72:1, the last product equals 1. Since ε f is a continuous function, for any Fix v 0 ∈ Ω f . Let U v0 ∈ X(k v0 ) be a symmetric matrix with determinant 1 with ε v0 (U v0 ) = −1 (there exists such a symmetric matrix, see [33], Theorem 63: 22).
It is well known that for our G = SL n and S = Ω ∞ , the group G(k) is dense in G(A S ) = G(A f ). We have Br 1 (X) = Br(k), hence all the points of X(A) • = X(A f ) are orthogonal to Br 1 (X), in particular our point U f . However, U f does not lie in the closure of Of course, our U f is not orthogonal to Br 1 (X, G) (otherwise it would lie in the closure of X(k) by Main Theorem 1.4). Indeed, consider the map induced by the Manin pairing (2) from 1.3. It can be shown that this map coincides with the map ε f : The above counter-example shows that Main Theorem 1.4 does not hold with Br 1 (X) instead of Br 1 (X, G). Nevertheless, we can prove a similar result about the algebraic Manin obstruction, assuming that S contains at least one nonarchimedean place. Theorem 1.7. Let G be a connected algebraic group (not necessarily linear) over a number field k. Let X := H\G be a right homogeneous space of G, where H is a connected k-subgroup of G. Assume that the Tate-Shafarevich group of the maximal abelian variety quotient G abvar of G is finite. Let x ∈ X(A) be an adelic point orthogonal to Br 1 X with respect to the Manin pairing. Let S ⊃ Ω ∞ be a finite set of places of k containing all archimedean places and at least one nonarchimedean place v 0 . We assume that  Let π : Y → X be a (left) torsor under a connected linear k-group H. We define Br 1 (X, Y ) to be the following group: We denote by the evaluation map.

2.2.
Before recalling the result of Sansuc, we give a few more definitions and notations. Let A be an abelian category and F : Var/k → A be a contravariant functor from the category of k-varieties to A . If X and Y are k-varieties, the projections p X , p Y : X × k Y → X, Y induce a morphism in A (see [38], Section 6.b): where π X , π Y are the projections F (X) ⊕ F (Y ) → F (X), F (Y ) and the group law in the right-hand side is the law in Hom( (see [38], (6.4.1)).
Lemma 2.3. Let F : Var/k → A be a contravariant functor. Let X, Y be two k-varieties, m : X × k Y → Y be a k-morphism. Assume that: • F (Spec(k)) = 0.
, and the third assumption of the lemma implies that F (x Y ) • F (m) = id. Therefore, we deduce that But by (3) we have so (4) and (5) give exactly We shall apply those constructions and this lemma to the functors F = Pic(.) and F = Br a (.) and to the morphism m : H × Y → Y defined by an action of an algebraic group H on a variety Y . In this context, those functors satisfy the assumptions of Lemma 2.3 by [38], Lemma 6.6.
We now recall Sansuc's result.
Here m : H × Y → Y denotes the left action of H on Y , p Y : H × Y → Y the natural projection and ∆ ′ Y /X : Pic(H) → Br(X) is a map defined in the proof of Proposition 6.10 in [38]. For the definition of the map ϕ 1 , see (4) or [38], (6.4.1).
Proof. For the part of the exact sequence up to Br(X) π * − −− → Br(Y ) see [38], Proposition 6.10. Concerning the last map Br(Y ) → Br(H × Y ), it implicitly appears in Sansuc's paper in the last term of the exact sequence (the second sequence of the four short exact sequences on page 45). By definitioň Sansuc noticed that the maps pr 1 and pr 2 correspond under this isomorphism to the maps m and p Y , respectively (see [38], the formulas for the faces of the simplicial system on page 44 before Lemma 6.12). Thus we see thať A computation using theČech spectral sequence (6.12.0) in [38] shows that the map Br(X) → Br(Y ) defined by the composition is the pullback morphism π * : Br(X) → Br(Y ). This concludes the proof of the exactness of (6).
Such an exact sequence will be very useful in the following, but we need another exact sequence: we need a version of this exact sequence with the map ∆ Y /X : Pic(H) → Br(X), defined in [11] before Proposition 2.3. We recall here the definition of ∆ Y /X due to Colliot-Thélène and Xu.

2.5.
Definition of ∆ Y /X . We use the above notation. Since H is connected, we have a canonical isomorphism c H : Ext c k (H, G m ) ∼ = Pic(H) (see [8], Corollary 5.7), where Ext c k (H, G m ) is the abelian group of isomorphism classes of central extensions of k-algebraic groups of H by G m . Given such an extension corresponding to an element p ∈ Pic(H), we get a coboundary map inétale cohomology see [19], IV.4.2.2. This coboundary map fits in the natural exact sequence of pointed sets (see [19], Remark IV.4.2.10 ) The element ∆ Y /X (p) is defined to be the image of the class [Y ] ∈ H 1 (X, H) of the torsor π : Y → X by the map ∂ H1 . This construction defines a map which is functorial in X and H (this map was denoted by δ tors (Y ) in [11]). We can compare the map ∆ Y /X : Pic(H) → Br(X) with another useful map α Y /X : H 1 (k, H) → Br 1 (X) defined by the formula where p X : X → Spec(k) is the structure morphism and [Y ] ∈ H 1 (X, H).
Lemma 2.6. The following diagram is commutative : Proof. Define Z to be the quotient of Y by the action of H ssu . Then Z → X is a torsor under H tor . By functoriality, and using the isomorphism H tor ∼ = H, it is sufficient to prove the commutativity of the following diagram : Consider the groups Ext n k (H tor , G m ) in the abelian category of fppf-sheaves over Spec(k). By [29], Lemmas A.3.1 and A.3.2, we know that the diagram is commutative, where η ′ H tor : H 1 (k, H tor ) → Ext 1 k (H tor , G m ) is the edge map from the local to global Ext's spectral sequence H p (k, Ext q k (H tor , G m k )) =⇒ Ext p+q k (H tor , G m ) (see [1], V.6.1). By [35], Proposition 17.5, there exists a canonical map Ext 1 k (H tor , G m ) → Ext c k (H tor , G m ). Composing this map with c H tor : Ext c k (H tor , G m ) → Pic(H tor ) (used in the construction of ∆ Z/X ), we get a map c ′ H tor : It is now sufficient to prove that the diagram tors from the category of fppf-sheaves over Spec(k) to the category of Gal(k/k)modules induces a morphism of spectral sequences from the local to global Ext's spectral sequence to Leray's spectral sequence. This morphism implies that the induced diagram between edge maps is commutative. We need to prove that the map τ 0 induced by τ coincides with the map c ′ be an exact sequence of fppf-sheaves on Spec(k) such that I is injective. Then the long exact sequences associated to the functors H 0 (H tor , .) and Hom k (H tor , .) give rise to the following commutative diagram where the vertical maps are the coboundary maps and the horizontal ones are induced by τ . With this diagram, it is clear that the image by τ 0 of a given group extension is the same as the image of this extension by the map c ′ H tor , which concludes the proof.
The goal of the following theorem is to give an equivalent of Sansuc's exact sequence (6) with the map ∆ ′ Y /X replaced by the map ∆ Y /X . It will be very important in the following. Here is the map of [11], see 2.5 above, the homomorphisms ϕ 1 and ϕ 2 are defined in [38] (6.4.1) (or see (4)), the homomorphisms ι X and ι Y are the inclusion maps, and the injective homomorphism ν is given as the composite of the following natural injective maps: In particular, if Y (k) = ∅ and y ∈ Y (k), x = π(y), then the maps ϕ i are induced by the map i y : H → Y defined by h → h.y, and we have an exact sequence Remark 2.9. Recall that the exact sequences (8) and (9) can be extended to the left by (see [38], Prop. 6.10).
Proof. It is a direct application of Theorem 2.8, using Pic(T ) = 0.
Remark 2.11. This corollary compares the algebraic Brauer groups of X and Y . Concerning the transcendental part of those groups, Theorem 2.8 can be used to study the injectivity of the map Br(X) → Br(Y ). We cannot describe easily the image of this map in general. However, Harari and Skorobogatov studied this map in particular cases (universal torsors for instance): see [23], Theorems 1.6 and 1.7.
Corollary 2.12 (cf. [38], Corollary 6.11). Let be an exact sequence of connected algebraic groups over a field k of characteristic 0. Assume that G ′ is linear. Then there is a commutative diagram with exact rows .g (where the product denotes the group law in G), ι ′′ and ι are the inclusion homomorphisms, and the injective homomorphism ν is defined as in Theorem 2.8.
If the homomorphism Pic(G) → Pic(G ′ ) is surjective (e.g. when G ′ is a k-torus, or when G ′ ss is simply connected, or when all the three groups G ′ , G and G ′′ are linear), then Br 1,e (G ′′ , G) = Br 1,e (G ′′ ), and we have a commutative diagram with exact rows Proof of the corollary. The short exact sequence of algebraic groups defines a structure of (left) G ′′ -torsor under G ′ on G (G ′ acts on G by left translations). Now from the diagram with exact rows (8) we obtain diagram (10), which differs from diagram (11) by the middle term in the bottom row. From diagram (8) we obtain an exact sequence If the homomorphism i * : Pic(G) → Pic(G ′ ) is surjective, then the homomorphism j * : Br(G ′′ ) → Br(G) is injective, hence Br 1,e (G ′′ , G) = Br 1,e (G ′′ ), and we obtain diagram (11) from diagram (10). If G ′ is a k-torus or if G ′ ss is simply connected, then Pic(G ′ ) = 0, and the homomorphism Pic(G) → Pic(G ′ ) is clearly surjective. If all the three groups G ′ , G and G ′′ are linear, then again the homomorphism Pic(G) → Pic(G ′ ) is surjective, see [38], proof of Corollary 6.11, p. 44.
For the proof of Theorem 2.8 we need a crucial lemma.
. Remark 2.14. In [24], Harari and Skorobogatov studied this question of composition of torsors. Their results (see Theorem 2.2 and Proposition 2.5 in [24]) deal with torsors under multiplicative groups and not only under G m as here, but they require additional assumptions concerning the type of the torsor and on invertible functions on the varieties. Those additional assumptions are not satisfied in our context.
In this setting, Lemma 2.3 implies that Formula (13) means that the push-forward of the torsor In particular, we get the following commutative diagram: where m ′ is defined to be the composite of the two upper horizontal maps. The situation is very similar to that in the proof of Theorem 5.6 in [8]: the map m ′ fits into commutative diagram (14) and for all t 1 , t 2 ∈ G m and all We want to use m ′ to define a group action of H 1 on Z. Formula (15) implies that the morphism m ′ (e, .) : z → m ′ (e, z) is an automorphism of the Y -torsor Z, and we can define a map m ′′ : H 1 × Z → Z to be the composition Then we get a commutative diagram where the map m ′′ still satisfies formula (15) and now, for all z ∈ Z, we have We wish to prove that m ′′ is a left group action of H 1 on Z.
By Rosenlicht's lemma (see [37], Theorem 3, see also [38], Lemma 6.5), the map ϕ has to be trivial, i.e. ϕ z (h 1 , h 2 ) = 1 for all z, h 1 , h 2 . Therefore we have Formulas (17) and (18) show that m ′′ is a left group action of hence the action m ′′ extends the action of G m on Z. From diagram (16) with m ′′ instead of m ′ we see that the action m ′′ induces the action m of H on Y .
Consider the following commutative diagram (see (16)): where , y), y), and the unnamed morphisms are the natural ones.
Therefore the map φ Z is an isomorphism of k-varieties, which proves that the action m ′′ makes Z → X into a torsor under H 1 . Let us prove the uniqueness of the class of the extension H 1 . If H 2 is a central extension of H that satisfies the conditions of the lemma, then the analogues of diagram (16) and formula (15) with Comparing with (13), we see that  (8) is a complex. Let p ∈ Pic(Y ) and let us prove that Let p ∈ Pic(H), and let us prove that π * (∆ Y /X (p)) = 0. The element p corresponds to the class of an extension 1 → G m → H 1 → H → 1, and we have Let us prove that the top row of diagram (8) is exact. For the exactness at the term Br(Y ), see Proposition 2.4. So it remains to prove the exactness of the top row at Pic(H) and at Br(X). Let us now prove the exactness of the top row of (8) at Br(X). Assume first that the k-variety X is quasi-projective. Let A ∈ Br(X) such that π * A = 0 ∈ Br(Y ). By a theorem of Gabber also proven by de Jong (see [12]), we know that there exists a positive integer n and an X-torsor Z → X under PGL n such that −A is the image of the class of Z in H 1 (X, PGL n ) by the coboundary we see that the assumption π * A = 0 implies that the torsor W We have the following picture: Since W → X is a torsor under the connected linear group H × PGL n , we can apply Lemma 2.13 to get a central extension and a structure of X-torsor under L on V → X, compatible with the action of H × PGL n on W . In particular, the natural injections of H and PGL n into H × PGL n define two central extensions obtained by pulling back the extension (20): Since PGL n acts trivially on Y , the action of L PGL n (as a subgroup of L) on V defines a commutative diagram , w), and the vertical maps are the natural ones. We see easily that φ V is an isomorphism, hence V → Y is a torsor under L PGL n . This action of L PGL n extends the action of G m on V above W , and is compatible with the action of PGL n on W above Y via the extension (22) and the map V → W . Therefore, the unicity result in Lemma 2.13 implies that exact sequence (22) is equivalent to the usual extension 1 → G m → GL n → PGL n → 1, and in particular that Consider the direct product of the exact sequences (21) and (22) Define the morphism µ : L H × L PGL n → L by µ(l, l ′ ) := l.l ′ , where the product is taken inside the group L. By definition of L H and L PGL n , we see that the image of the commutator morphism c : L H × L PGL n → L defined by c(l, l ′ ) := l.l ′ .l −1 .l ′ −1 is contained in the central subgroup G m of L. Therefore, since c(e, l ′ ) = c(l, e) = 1, Rosenlicht's lemma implies that c(l, l ′ ) = 1 for all (l, l ′ ) ∈ L H × L PGL n . Hence the morphism µ is a group homomorphism, and the following diagram is commutative with exact rows where m : G m × G m → G m is the group law. Therefore we get a commutative diagram of coboundary maps In particular, this diagram implies that  (8) is exact under the assumption that X is quasi-projective.
Let us now deduce the general case: X is not supposed to be quasi-projective anymore. By Nagata's theorem (see [32]) we know that there exists a proper kvariety Z and an open immersion of k-varieties X → Z. By Chow's lemma (see [15], II.5.6.1 and II.5.6.2, or [39], Chapter VI, §2.1), there exists a projective kvariety Z ′ and a projective, surjective birational morphism Z ′ → Z. Moreover, using Hironaka's resolution of singularities (see [27], see also [3] and [17]), there exists a smooth projective k-variety Z and a birational morphism Z → Z ′ . Define X ′ to be the fibred product X ′ := Z × Z X. Then X ′ is a open subvariety of Z, hence X ′ is a smooth quasi-projective k-variety, and the natural map X ′ → X is a birational morphism. Define Y ′ to be the product Y × X X ′ . By the quasi-projective case, we know that in the commutative diagram the first row is exact. Since the map X ′ → X is a birational morphism, we have a commutative diagram where the bottom horizontal arrow is an isomorphism. Since both X and X ′ are smooth, by [20], II, Corollary 1.8, the vertical arrows are injective. It follows that the homomorphism Br(X) → Br(X ′ ) is injective. Now a diagram chase in diagram (23) proves the exactness of the second row in that diagram. This completes the proof of the exactness of the top row of diagram (8).
This formula implies immediately that the rightmost square of diagram (8) commutes, hence this diagram is commutative.
Since the top row of (8) is exact, and the diagram (8) is commutative, it is easy to conclude that the bottom row of (8) is exact.
2.17. Proof of Theorem 2.8: Exact sequence (9). We consider the following diagram: In this diagram the second row is the second row of diagram (8), hence it is exact.
In the first row, Br 1,x (X, Y ) is a subgroup of Br 1 (X, Y ) and Br 1,y (Y ) is a subgroup of Br 1 (Y ). We easily check that all the arrows in the first row are well-defined.
This diagram is commutative: for the first and the last squares it is a consequence of Lemma 6.4 of [38], and for the two central squares it is clear. We know that the second row of this diagram is exact. Since all the vertical arrows are injective, we see that the first row is a complex. An easy diagram chase shows that the first row is also exact, i.e. sequence (9) is exact.

Compatibility
In this section we use Theorem 2.8 to get a compatibility result between the evaluation map and the action of a linear group. We begin with a lemma: Lemma 3.1. Let k be a field of characteristic 0 and X, X ′ be two smooth k- is well-defined and is an isomorphism.
Proof. Consider the three exact sequences associated to the torsors Y → X, Y ′ → X ′ and Y × Y ′ → X × X ′ (see Theorem 2.8). We get the following commutative diagram with exact rows: The two first and the two last vertical arrows are isomorphisms by [38], Lemma 6.6, hence the five lemma implies that the central vertical arrow is an isomorphism.
Corollary 3.2. If in Lemma 3.1 we also have Y (k) = ∅, y ∈ Y (k) and x = π(y), then the homomorphism is an isomorphism.

Lemma 3.3 (Compatibility).
Let k be a field of characteristic 0. Let π : Y → X be a (left) torsor under a connected linear k-group H. Let G be a connected algebraic k-group, acting on the right on Y and X such that π is G-equivariant. Assume that Y is k-rational and Y (k) = ∅ and let y 0 ∈ Y (k), x 0 = π(y 0 ). Then for any Proof. We consider two torsors: π : Y H − → X and π ′ = id : G → G. Let m X : X × G → X and m Y : Y × G → Y denote the actions of G. Since m Y (y, e) = y, we see that if b ∈ Br y (X, Y ), then m * X b ∈ Br (y,e) (X × G, Y × G). By functoriality of the evaluation map, we have a commutative diagram where the first pairing is the evaluation map on X × G and the second one is the sum of the evaluation maps on X and on G, i.e. ev ′ ((B, C), (x, g)) := B(x) + C(g) ∈ Br(k) for B ∈ Br 1,x0 (X, Y ), C ∈ Br 1,e (G) .
where π : G → X is the quotient map and x 0 = π(e).
Corollary 3.6. Let k, G, H and X be as in Corollary 3.5. Let ϕ : G ′ → G be a homomorphism of k-groups, where G ′ is a simply connected k-group. Let b ∈ Br 1,x0 (X, G), x ∈ X(A), g ′ ∈ G ′ (A). Then b, x.ϕ(g ′ ) = b, x .

Some lemmas
For an abelian group A we write A D := Hom(A, Q/Z). Proof. We have Br a (P ) = H 2 (k, P ), see [38], Lemma 6.9(ii). By [38], (8.11.2), the map λ : P (A) → Br a (P ) D = H 2 (k, P ) D is given by the canonical pairing Consider the map µ from the Tate-Poitou exact sequence see [25], Theorem 5.6 or [14], Theorem 6.3. By (P (A) • ) ∧ we mean the completion of P (A) • for the topology of open subgroups of finite index. Then the map µ is induced by λ. Since P is a quasi-trivial torus, we have H 1 (k, P ) = 0, and we see from (28) that the map µ is surjective. But by [22], Lemma 4, im µ = im λ. Thus λ is surjective. Note that the geometric quotient X/N exists in the category of k-varieties by [5], Lemma 3.1.

Proof.
If v is a nonarchimedean place of k, we denote by O v the ring of integers of k v , and by κ v the residue field of Since the morphism π is smooth, the map X(k v ) → Y (k v ) is open for any place v of k.
Let S be a finite set of places of k containing all the archimedean places. Write O S for the ring of elements of k that are integral outside S. Taking S sufficiently large, we can assume that G and N extend to smooth group schemes G and N over Spec(O S ), and that X and Y extend to homogeneous spaces X of G and Y of G /N over Spec(O S ) such that Y = X /N . In particular, the reduction , the morphisms being given by π and by y v : Spec (O v ) → Y . It is an O v -scheme. Then its reduction X yv is a homogeneous space of the connected κ v -group N v over the finite field κ v . By Lang's theorem ( [31], Theorem 2) X yv has a κ v -point. By Hensel's lemma X yv has an O v -point. This means that y v ∈ π(X (O v )). Thus 4.3. Let X = H\G be a homogeneous space of a connected k-group G. Let x 1 ∈ X(k). Consider the map π x1 : G → X, g → x 1 .g, it induces a homomorphism π * x1 : Br(X) → Br(G). Let Br 1 (X, G) x1 denote the subgroup of elements b ∈ Br(X) such that π * x1 (b) ∈ Br 1 (G). The following lemma shows that Br 1 (X, G) x1 does not depend on x 1 , so we may write Br 1 (X, G) instead of Br 1 (X, G) x1 . Note that Br 1 (X, G) = Br 1,x1 (X, G) + Br(k). Proof. We have a commutative diagram We see that it suffices to prove that the kernel of π * x1 does not depend on x 1 . Now if x 2 ∈ X(k) is another k-point, then x 2 = x 1 .g for some g ∈ G(k), hence where l g denotes the left translation on G by g. Thus Since l g is an isomorphism of the underlying variety of G, we see that l * g : Br(G) → Br(G) is an isomorphism, hence ker π * x2 = ker π * x1 , which proves the lemma.
Lemma 4.5. Let X be a right homogeneous space of a unipotent k-group U over a field k of characteristic 0. Then X(k) is non-empty and is one orbit of U (k).
Corollary 4.6. Let X be a right homogeneous space of a unipotent R-group U . Then X(R) is non-empty and connected.
Proof. Since U (R) is connected and X(R) = x.U (R), we conclude that X(R) is connected.
Lemma 4.7. Let G be a unipotent k-group over a number field k. Let X be a right homogeneous space of G. Let S ⊂ Ω be any non-empty finite set of places. Then X(k) is is non-empty and dense in X(A S ).
Proof. By [5], Lemma 3.2(i), X(k) is non-empty. Let x 0 ∈ X(k), and let H ⊂ G denote the stabilizer of x 0 in G. We have X = H\G. Set g = Lie(G). Since g is a vector space and S = ∅, by the classical strong approximation theorem g is dense in g ⊗ k A S . Since char(k) = 0, we have the exponential map g → G, which is an isomorphism of k-varieties. We see that Since H is unipotent, we have H 1 (k v , H) = 0 for any v ∈ Ω, and therefore x 0 G(k v ) = X(k v ) for any v. It follows that x 0 G(A S ) = X(A S ) (we use Lang's theorem and Hensel's lemma). Thus x 0 G(k) is dense in X(A S ), and X(k) is dense in X(A S ).

Brauer group
We are grateful to A.N. Skorobogatov, E. Shustin, and T. Ekedahl for helping us to prove Theorem 5.1 below.
where Br(X) n denotes the group of elements of order dividing n in Br(X). Since every element of Br(X) is torsion (because Br(X) embeds in Br(k(X)), cf. [20], II, Corollary 1.8), it is enough to prove the following Theorem 5.2. Proof in characteristic 0. By the Lefschetz principle, we may assume that k = C. Let g ∈ G(C). We must prove that g acts trivially on the Betti cohomology H i B (X, A). Since G is connected, the group G(C) is connected, hence we can connect g with the unit element e ∈ G(C) by a path. We see that the automorphism of X g * : X → X, x → x.g is homotopic to the identity automorphism e * : X → X, x → x.
It follows that g * acts on H i B (X, A) as e * , i.e. trivially.
To prove Theorem 5.2 in any characteristic, we need two lemmas. Proof. Consider the commutative diagram here X × k Y is the fibre product of X and Y with respect to the structure morphisms s X and s Y . Clearly R i (s X ) * A is the constant sheaf on Spec (k) with stalk H i (X, A). By [13], Th. finitude, Theorem 1.9(ii), the sheaf R i (p Y ) * A on Y is the pullback of the constant sheaf R i (s X ) * A on Spec (k) along the morphism s Y : Y → Spec (k), which concludes the proof.
Let y ∈ Y (k). It defines a canonical morphism f y : X → X × k Y such that Lemma 5.4. Let X, Y be two smooth algebraic varieties over an algebraically closed field k (of any characteristic). Let A be a finite abelian group of order invertible in k. For a closed point y ∈ Y consider the map f y : does not depend on y.
Proof. Let η ∈ H i (X × Y, A). Then η defines a global section λ(η) of the sheaf A). It is easy to see that Since Y is irreducible, it is connected, hence the global section λ(η) of the constant sheaf R i (p Y ) * A on Y is constant, and therefore λ(η)(y) does not depend on y.
Thus f * y (η) does not depend on y.
Proof of Theorem 5.2 in any characteristic. Consider the map . For a k-point g ∈ G(k) consider the map f g : X → X × G defined by x → (x, g), as above. Since x.g = m(x, g) = m(f g (x)), we have By Lemma 5.4 f * g η does not depend on g. Thus g * ξ does not depend on g. This means that G(k) acts on H i (X, A) trivially.

Homogeneous spaces of simply connected groups
In the proof of the main theorem we shall need a result about strong approximation in homogeneous spaces of semisimple simply connected groups with connected stabilizers. If X = H\G is such a homogeneous space, since G is semisimple and simply connected, the group Br(G) is trivial (see [18], corollary in the Introduction), hence Br 1,x0 (X, G) = Br x0 (X). Theorem 6.1 (Colliot-Thélène and Xu). Let G be a semisimple simply connected k-group over a number field k, and let H ⊂ G be a connected subgroup. Set X := H\G. Let S be a non-empty finite set of places of k such that G(k) is dense in G(A S ). Then the set of points x ∈ X(A) such that b, x = 0 for all b ∈ Br 1,x0 (X, G) = Br x0 (X) coincides with the closure of the set X(k).G(k S ) in X(A) for the adelic topology.
Proof. This is very close to a result of Colliot-Thélène and Xu, see [11], Theorem 3.7(b). Since their result is not stated in these terms in [11], we give here a proof of Theorem 6.1, following their argument.
We prove the nontrivial inclusion of the theorem. Let x ∈ X(A) be orthogonal to Br x0 (X). Then by [11], Theorem 3.3, there exists x 1 ∈ X(k) and g ∈ G(A) such that x = x 1 .g. Let U X ⊂ X(A) be an open neighbourhood of x. Clearly there exists an open neighbourhood U G ⊂ G(A) of g such that for any g ′ ∈ U G we have x 1 .g ′ ∈ U X . By assumption G(k) is dense in G(A S ), hence G(k).G(k S ) is dense in G(A). It follows that there exist g 0 ∈ G(k) and g S ∈ G(k S ) such that g 0 g S ∈ U G , then x 1 .g 0 .g S ∈ U X . Set x 2 = x 1 .g 0 , then x 2 ∈ X(k) and x 2 .g S ∈ U X . Thus x lies in the closure of the set X(k).G(k S ) in X(A) for the adelic topology. This proves the nontrivial inclusion.
We prove the trivial inclusion. Let x 1 ∈ X(k), g S ∈ G(k S ) and b ∈ Br 1,x0 (X, G). Clearly we have b, x 1 = 0. By Corollary 3.6 we have b, x 1 .g S = b, x 1 = 0. By Lemma 6.2 below we have b, x = 0 for any x in the closure of the set X(k).G(k S ). Proof. Arguing as in [38], Proof of Lemma 6.2, we can reduce our lemma to the local case. In other words, it is enough to prove that for any completion k v of k the function is locally constant in x. This follows from non-published results from the thesis of Antoine Ducros, cf. [16], Part II, Propositions (0.31) and (0.33).
Since those results of Ducros are not published, we give another proof of this fact. We are grateful to J.-L. Colliot-Thélène and to the referee for this proof. Let x ∈ X(k v ). The problem is local, so we can replace X by an affine open subset containing x. From now on, X is assumed to be affine over k v .
To show that the map φ b is locally constant around x, we may replace b by b − b(x) ∈ Br(X). Now we have b(x) = 0 and we want to prove that φ b is zero in a topological neighbourhood of x.
Since X is a smooth affine k v -variety, by a result by Hoobler (see [28], Corollary 1) there exists a class η ∈ H 1 (X, PGL n ) such that b is the image of η by the usual coboundary map. The class η is represented by an X-torsor f : Y → X under PGL n . For It is clear that b(x ′ ) = ∆(η(x ′ )). From the exact sequence we see that b(x ′ ) = 0 if and only if η(x ′ ) = 1. On the other hand, clearly η(x ′ ) is the class of the k v -torsor x ′ = f (y) for some y ∈ Y (k v ). Hence the set of points x ′ ∈ X(k v ) such that b(x ′ ) = 0 is exactly the subset f (Y (k v )) of X(k v ). We now conclude by the implicit function theorem: since f : Y → X is a smooth morphism of k v -schemes, the image f (Y (k v )) is an open subset of X(k v ). Therefore, φ b is zero on the open neighbourhood f (Y (k v )) of x, which concludes the proof.

Proof of the main theorem
Throughout this section we consider X = H\G satisfying the assumptions of Theorem 1.4. Let x ∈ X(A) be an adelic point, we write . Let S be a finite set of places of k containing all archimedean places, and we set and where is a connected topological group, see [34], Theorem 5.2.3). Then U X and U ′ X are open neighbourhoods of x in X(A). We say that U X is the special neighbourhood of x defined by U f X . For the sake of the argument it will be convenient to introduce Property (P S ) of a pair (X, G): (P S ) For any point x ∈ X(A) orthogonal to Br 1 (X, G), and for any open neighbourhood U f X of x f , the set X(k).G scu (k S )∩U X (or equivalently the set X(k)∩U ′ X , or equivalently the set X(k).G scu (k S f )∩U X ) is non-empty, where U X is the special neighbourhood of x defined by U f X . The nontrivial part of Theorem 1.4 precisely says that property (P S ) holds for any X, G and S as in the theorem.
We start proving Theorem 1.4. The structure of the proof is somewhat similar to that of Theorem A.1 of [7].

First reduction.
We reduce Theorem 1.4 to the case G u = 1. Let X and G be as in the theorem. We represent G as an extension where G lin is a connected linear algebraic k-group and G abvar is an abelian variety over k. We use the notation of 1.1. Set G ′ := G/G u , Y := X/G u . We have a canonical epimorphism ϕ : G → G ′ and a canonical smooth ϕ-equivariant morphism ψ : X → Y . We have G ′ lin = G lin /G u , hence G ′ u = 1. We have G ′ abvar = G abvar , hence X(G ′ abvar ) is finite.
Assume that the pair (Y, G ′ ) has Property (P S ). We prove that the pair (X, G) has this property. Let x ∈ X(A) be a point orthogonal to Br 1 (X, G). Set y := ψ(x) ∈ Y (A). By functoriality, y is orthogonal to Br 1 (Y, G ′ ). Let U f X be as in (P S ), and let U X , U ′ X be the special neighbourhoods of x defined by U f X . Note Since the pair (Y, G ′ ) has Property (P S ), there exists a k-point y 0 ∈ Y (k) ∩ U ′ Y . Let X y0 denote the fibre of X over y 0 . It is a homogeneous space of the unipotent group G u . By Lemma 4.7, X y0 (k) = ∅ and X y0 has the strong approximation property away from Ω ∞ : the set X y0 (k) is dense in X y0 (A f ). Consider the set V f := X y0 (A f ) ∩ U ′ f X . By Corollary 4.6, for any v ∈ Ω ∞ the set X y0 (k v ) is connected, and by Lemma 4.5, Thus the pair (X, G) has Property (P S ). We see that in the proof of Theorem 1.4 we may assume that G u = 1.

Second reduction.
By [7], Proposition 3.1 we may regard X as a homogeneous space of another connected group G ′ such that G ′ u = {1}, G ′ ss is semisimple simply connected, and the stabilizers of the geometric points of X in G ′ are linear and connected. We have G ′ sc = G sc , hence G ′ scu = G scu (because G scu = G sc and G ′ scu = G ′ sc ). It follows from the construction in the proof of Proposition 3.1 of [7] that there is a surjective homomorphism G abvar → G ′ abvar . Since by assumption X(G abvar ) is finite, we obtain from [7], Lemma A.3 that X(G ′ abvar ) is finite.
Let us prove that if a point x ∈ X(A) is orthogonal to Br 1 (X, G), then it is orthogonal to Br 1 (X, G ′ ). More precisely, we prove that Br 1 (X, G ′ ) is a subgroup of Br 1 (X, G).
By construction (see [7], proof of Proposition 3.1), there is an exact sequence of connected algebraic groups where G 1 is the quotient of G by the central subgroup Z(G)∩H and S is a k-torus. Consider the following natural commutative diagram where the maps π, π ′ and π 1 are the natural quotient maps. From this diagram, we deduce the following one, where the second line is exact (see the top row of diagram (11)): Br(X) π * 1 π * 5 5 π ′ * 6 6  I  I  I  I  I  I  I  I Therefore, the injectivity of the map q * : Br(G 1 ) → Br(G ′ ) implies that the natural inclusion Br 1,x0 (X, G 1 ) ⊂ Br 1,x0 (X, G ′ ) is an equality. And by functoriality Br 1,x0 (X, G 1 ) is a subgroup of Br 1,x0 (X, G). Thus Br 1,x0 (X, G ′ ) = Br 1,x0 (X, G 1 ) is a subgroup of Br 1,x0 (X, G). It follows that if a point x ∈ X(A) is orthogonal to Br 1 (X, G), then it is orthogonal to Br 1 (X, G ′ ).
Thus if Theorem 1.4 holds for the pair (X, G ′ ), then it holds for (X, G). We see that we may assume in the proof of Theorem 1.4 that G lin is reductive, G ss is simply connected, and the stabilizers of the geometric points of X in G are linear and connected. Now in order to prove Theorem 1.4 it is enough to prove the following Theorem 7.3. Theorem 7.3. Let k be a number field, G a connected k-group, and X := H\G a homogeneous space of G with connected stabilizer H. Assume: H is linear, (iii) G ss is simply connected, (iv) X(G abvar ) is finite. Let S ⊃ Ω ∞ be a finite set of places of k containing all archimedean places. We assume that G sc (k) is dense in G sc (A S ). Then the pair (X, G) has Property (P S ).
The homogeneous space X defines a natural homomorphism H tor → G sab . We first prove a crucial special case of Theorem 7.3. Construction 7.5. Set Y := X/G ss . Then Y is a homogeneous space of the semiabelian variety G sab , hence it is a torsor of some semiabelian variety G ′ . We have G ′ abvar = G abvar , hence X(G ′ abvar ) is finite. We have a canonical smooth morphism ψ : X → Y .
To prove Proposition 7.4 we need a number of lemmas and propositions. The following proposition is crucial for our proof of Proposition 7.4 by dévissage.
Construction 7.7. Consider the map π x0 : G → X, g → x 0 .g. This map identifies X (resp. X y0 ) with a quotient of G (resp. G ss ) by a connected subgroup H ′ (resp. H ′ ssu ), so we have We define a k-variety Z by where the first two rows and the last column are exact sequences of connected algebraic groups, and the other maps are the natural maps between the different homogeneous spaces.
The following two lemmas are versions of exact sequence (9) of Theorem 2.8.
Lemma 7.8. The following sequence is exact: Proof. We use the functoriality of exact sequence (9) of Theorem 2.8 to get the following commutative diagram with exact columns. Here the second column is the exact sequence (9) for Z = H ′ ssu \G and the third column is exact sequence (9) applied to X y0 = H ′ ssu \G ss .
We have Pic(G ss ) = 0 and Br 1,e (G ss ) = 0 by [38], Lemma 6.9(iv), because G ss is simply connected. From the bottom row of diagram (11) of Corollary 2.12 we get an exact sequence where l : G ss → G is the canonical embedding. But Br 1,e (G ss ) = 0, therefore the homomorphism p * : Br 1,e (G sab ) → Br 1,e (G) is surjective. The composition r • f being the trivial morphism, the second row of the diagram is a complex. A diagram chase in diagram (29) proves the exactness of the sequence of the lemma.
Lemma 7.9. The sequence is exact.
Proof. We consider the following diagram, in which the middle column and the last row are the exact sequences coming from exact sequence (9) of Theorem 2.8, and the last column is the exact sequence coming from the exact bottom row of diagram (11) of Corollary 2.12: We prove that the diagram is commutative. In this diagram the two first columns define a commutative diagram by functoriality, and the second row is a complex. By construction we have g • v = π Z • h, hence in the diagram we have v * • g * = h * •π * Z . Let us prove the commutativity of the square in the top right-hand corner, i.e. let us prove that g * •∆ G/Z = ∆ H ′ /H tor . We observe that the following diagram of torsors under H ′ ssu is cartesian: i.e. π −1 Z (H tor ) = H ′ . In other words, the H tor -torsor H ′ is the pullback of the Z-torsor G by the morphism H tor g − → Z. Therefore, if is a central extension representing an element p ∈ Pic(H ′ ssu ) via the isomorphism Ext c k (H ′ ssu , G m ) ∼ = Pic(H ′ ssu ), we get a commutative diagram: in Br(H tor ). Therefore the top right-hand square in diagram (30) is commutative.
Returning to diagram (30), we see that its commutativity and the exactness of the last two columns and of the last row imply, via an easy diagram chase, that the second row of (30) is also exact, hence the sequence of the lemma is exact.
For an alternative proof of Lemma 7.9 we need the following generalization of Proposition 2.8: Proposition 7.10. Let k be a field of characteristic zero, and be an exact sequence of connected linear algebraic groups over k. Let π : Z → X and π ′ : Y → Z be two morphisms of algebraic varieties such that the composite Y → X is an X-torsor under G, such that the restriction to H 1 of the action of G on Y defines the structure of a Z-torsor under H 1 on Y , and such that Z → X is a torsor under H 2 via the induced action. Then we have a natural exact sequence If in addition z ∈ Z(k), we have an exact sequence where x ∈ X(k) is the image of z.
Proof. As in the end of the proof of Theorem 2.8, we construct the exact sequence (31) from the top row of diagram (8) applied to the torsor Z → X: Define a map ϕ ′ 2 : Br(Z, Y ) → Br 1,e (H 2 ) to be the composition where the morphism p * Z + p * H2 is an isomorphism by Lemma 3.1, and π H2 is the projection onto the first factor. Note that Br 1,e (H 2 , G) = Br 1,e (H 2 ) as a consequence of the injectivity of the homomorphism Br(H 2 ) → Br(G), which comes from the exactness of the top row of diagram (11) of Corollary 2.12 and the fact that Pic(G) → Pic(H 1 ) is onto (see [38], Remark 6.11.3).
Consider the diagram This diagram is commutative. Since i H is a canonical embedding, we see that ker ϕ ′ 2 = ker(m * − p * Z ). We prove that sequence (31) is exact. We use the exactness of (33). We know from Theorem 2.8 that the map ∆ Z/X : Pic(H 2 ) → Br(X) lands in Br 1 (X, Z). Since Br 1 (X, Y ) ⊃ Br 1 (X, Z), the map ∆ Z/X : Pic(H 2 ) → Br 1 (X, Y ) is defined. We obtain a commutative diagram with an exact long horizontal line and exact vertical lines: Now we see immediately that sequence (31) is exact at Pic(H 2 ) and Br 1 (X, Y ). Since ker(m * − p * Z ) = ker ϕ ′ 2 in diagram (34), it follows from the exactness of (33) at Br(Z) that sequence (31) is exact at Br 1 (Z, Y ). Thus (31) is exact, which completes the proof. The composition π x0 : G → X is naturally a torsor under H ′ . Therefore, we can apply Proposition 7.10 to get the exact sequence which concludes the proof.
7.12. Proof of Proposition 7.6. We have a commutative diagram 0 Br 1,x0 (X, G) q * 9 9 P P P P P P P P P P P P where the last row is exact by assumption, and the two slanted sequences are also exact by Lemmas 7.8 and 7.9. A diagram chase shows that the homomorphism is surjective, which completes the proof of Proposition 7.6.
For the proof of Proposition 7.4 we need three lemmas.
Here r 1 and r 2 are surjective by Proposition 7.6, while µ 1 and µ 2 are injective by Lemma 7.14 below. Clearly we can define the dotted arrow (in a unique way) such that the diagram with this new arrow will be also commutative. The top square of this new diagram is the desired diagram (36).
Lemma 7.14. Let X := H\G, where G is a simply connected semisimple k-group over a field k of characteristic 0, and H ⊂ G is a connected k-subgroup such that H tor = 1. Then Br(X)/Br(k) is finite and the canonical homomorphism Proof. Let x 0 denote the image of e ∈ G(k) in X(k). We have a canonical isomorphism Br(X)/Br(k) ∼ = Br x0 (X).
We have H = H ssu , hence Pic(H) ∼ = Pic(H ss ). In the commutative diagram the right vertical arrow is clearly injective, hence so is the left one.
Proof. It is an immediate consequence of Lemma 6.2, using the finiteness of M .
Now let x 1 ∈ X(k) be any other point. Set X 1 := x 1 .G ss . It follows from Lemma 7.13 that M ⊂ Br 1 (X, G)/Br(k) surjects onto Br(X 1 )/Br(k). such that the corresponding special neighbourhood U of x in X(A) is orthogonal to M . We may assume that U f X ⊂ U f , then U X ⊂ U , hence U X is orthogonal to M . Let Y and ψ : X → Y be as in Construction 7.5 (i.e. Y := X/G ss ). Set y := ψ(x) ∈ Y (A). Since x is orthogonal to Br 1,x0 (X, G), we see by functoriality that y is orthogonal to Br 1,y0 (Y, G sab ). Clearly there is a semiabelian variety G ′ such that Y is a (trivial) principal homogeneous space of G ′ . We have a morphism of pairs (Y, G sab ) → (Y, G ′ ), hence a homomorphism Br 1 (Y, G ′ ) → Br 1 (Y, G sab ). But Br 1 (Y, G ′ ) = Br 1 (Y ), hence y is orthogonal to the group Br 1 (Y ). As in the First reduction, see 7.1, we define U f Y := ψ(U f X ) and we construct the corresponding special open neighbourhood U Y of y. By [7], Lemma A.2, for any v ∈ Ω ∞ we have ψ(U X,v ) = U Y,v . We see that ψ(U X ) is an open subset of Y (A) of the is the connected component of y v . Then U ∞ is the connected component of x ∞ in X(k ∞ ). Now since Y is a torsor of the semiabelian variety G ′ with finite Tate-Shafarevich group, by [22], Theorem 4, there exists a k-point y 1 ∈ Y (k) ∩ ψ(U X ).
Let X y1 denote the fibre of X over y 1 . Consider the set V := X y1 (A) ∩ U X , it is open in X y1 (A). Since y 1 ∈ ψ(U X ), the set V is non-empty: there exists a point x ′ = (x ′ v ) ∈ V . In particular, X y1 (k v ) = ∅ for any v ∈ Ω r . The variety X y1 is a homogeneous space of G ss with geometric stabilizer H ∩ G ss = H ssu . The group G ss is semisimple simply connected by (iii has points in all real completions of k is enough to ensure that X y1 has a k-point. Note that Br(G ss ) = 0 by [18], hence Br 1 (X y1 , G ss ) = Br(X y1 ). Since U X is orthogonal to M , we see that V ⊂ U X is orthogonal to M . Since M surjects onto Br(X y1 )/Br(k), see 7.16, we see that V is orthogonal to Br(X y1 ). By Theorem 6.1 (due to Colliot-Thélène and Xu) there is a point of the form x 1 .g S in V , where x 1 ∈ X y1 (k) and g S ∈ G ss (k S ). It follows that the set V .G ss (k S ) contains a k-point of X y1 . Clearly V .G ss (k S ) ⊂ U X .G ss (k S ). Thus U X .G ss (k S ) contains a k-point of X, which shows that the pair (X, G) has Property (P S ). This completes the proof of Proposition 7.4.
Let us resume the proof of Theorem 7.3. We need a construction.
Construction 7.18. We follow an idea of a construction in the proof of [7], Theorem 3.5. By Lemma 3 in [9] there exists a coflasque resolution of the torus H tor , i.e. an exact sequence of k-tori 0 → H tor → P → Q → 0 where P is a quasi-trivial torus and Q is a coflasque torus. Recall that a torus is coflasque if for any field extension K/k we have H 1 (K, Q) = 0, where Q is the character group of Q. Consider the k-group F := G × P . The group H maps diagonally into F , and we can consider the quotient homogeneous space W := H\F of F . There is a natural morphism t : W → X. We have F abvar = G abvar , hence X(F abvar ) is finite. We have a canonical homomorphism H tor → F sab , and this homomorphism is clearly injective. Let us prove the following fact, which is necessary to apply Proposition 7.4 to the homogeneous space W of F . where π 1 (H) Γv denotes the groups of coinvariants of Γ v in π 1 (H), and ( ) tors denotes the torsion subgroup. Here we set λ ′ v := λ v . For v ∈ Ω ∞ we defined in [6], Proposition 4.2, a canonical isomorphism π 1 (H)).
Here we define a homomorphism λ ′ v as the composition For any v ∈ Ω we define the Kottwitz map β v as the composition This map β v is functorial in H. Note that for v ∈ Ω f the maps ab 1 : H 1 (k v , H) → H 1 ab (k v , H) and β v are bijections. Thus for v ∈ Ω f we have a canonical and functorial in H bijection β v : H 1 (k v , H) ∼ → (π 1 (H) Γv ) tors . For any v ∈ Ω we define a map µ v as the composition where cor v is the obvious corestriction map. We write v H 1 (k v , H) for the set of families (ξ v ) v∈Ω such that ξ v = 1 for almost all v. We define a map w ∈ S be a nonarchimedean place, then the map β w is bijective. It follows that the map µ w : H 1 (k w , H) → M Γ is surjective (because µ w = cor w • β w ). Since w is nonarchimedean, we have H 1 (k w , G) = 1 (because G is simply connected), hence ker[H 1 (k w , H) → H 1 (k w , G)] = H 1 (k w , H). It follows that the map µ w : ker[H 1 (k w , H) → H 1 (k w , G)] → M Γ is surjective. Now it is clear that the map µ S = v∈S µ v is surjective.
We prove the proposition. We must prove that the localization map is surjective. In the language of Galois cohomology, we must prove that the localization map Theorem 8.4. Let G be a simply connected k-group over a number field k, and let H ⊂ G be a connected geometrically character-free subgroup (i.e. H tor = 1). Set X := H\G. Let S be a finite set of places of k containing at least one nonarchimedean place v 0 . Assume that G ss (k) is dense in G ss (A S ). Then X has strong approximation away from S in the following sense. Let x = (x v ) ∈ X(A) and let U S ⊂ X(A S ) be any open neighbourhood of the A S -part x S of x. Then there exists a k-point x 0 ∈ X(k) ∩ U S . Moreover, one can ensure that for v ∈ Ω ∞ ∩ S the points x 0 and x v lie in the same connected component of X(k v ). More precisely, there exists y v0 ∈ X(k v0 ) such that the point x ′ ∈ X(A) defined by x ′ v0 := y v0 and x ′ v := x v for v = v 0 belongs to the closure of the set X(k).G(k S ) in X(A) for the adelic topology.