Manin obstruction to strong approximation for homogeneous spaces

8 Mikhail Borovoi and Cyril Demarcheand if y ∈ Y (k), x = π(y), define Br 1,x (X, Y ) <strong>to</strong> beBr 1,x (X, Y ) := ker[x ∗ : Br 1 (X, Y ) → Br(k)] = {b ∈ Br(X) : π ∗ (b) ∈ Br 1,y (Y )} .We denote bythe evaluation map.〈, 〉: Br(X) × X(k) → Br(k): (b, x) ↦→ b(x)2.2. Before recalling the result of Sansuc, we give a few more definitions andnotations. Let A be an abelian category and F : Var/k → A be a contravariantfunc<strong>to</strong>r from the category of k-varieties <strong>to</strong> A . If X and Y are k-varieties, theprojections p X , p Y : X × k Y → X, Y induce a morphism in A (see [38], Section6.b):F (p X ) + F (p Y ): F (X) ⊕ F (Y ) → F (X × k Y )such thatF (p X ) + F (p Y ) = F (p X ) ◦ π X + F (p Y ) ◦ π Y , (3)where π X , π Y are the projections F (X) ⊕ F (Y ) → F (X), F (Y ) and the group lawin the right-hand side is the law in Hom(F (X) ⊕ F (Y ), F (X × k Y )).Let m: X × k Y → Y be a morphism of k-varieties. Assume that the morphismF (p X ) + F (p Y ) is an isomorphism. We define a mapϕ : F (Y ) → F (X × k Y ) → F (X) ⊕ F (Y ) → F (X)by the formula(see [38], (6.4.1)).ϕ := π X ◦ (F (p X ) + F (p Y )) −1 ◦ F (m) (4)Lemma 2.3. Let F : Var/k → A be a contravariant func<strong>to</strong>r. Let X, Y be twok-varieties, m : X × k Y → Y be a k-morphism. Assume that:• F (Spec(k)) = 0.• F (p X ) + F (p Y ): F (X) ⊕ F (Y ) → F (X × k Y ) is an isomorphism.• There exists x ∈ X(k) such that the morphism m(x, .) : Y → Y is the identityof Y .Then F (m) = F (p X ) ◦ ϕ + F (p Y ): F (Y ) → F (X × k Y ).Proof. Consider the morphism x Y : Y → X × k Y defined by x. Then F (x Y ) ◦F (p X ) = 0, since the morphism p X ◦x Y : Y → X fac<strong>to</strong>rs through x : Spec(k) → Xand F (Spec(k)) = 0. Since p Y ◦ x Y = id Y , we have F (x Y ) ◦ F (p Y ) = id, and the

10 Mikhail Borovoi and Cyril Demarchewhere pr i : Y × X Y → Y denote the two projections. Since Y → X is a <strong>to</strong>rsor, wehave a canonical isomorphism H × Y → Y × X Y defined by (h, y) ↦→ (m(h, y), y).Sansuc noticed that the maps pr 1 and pr 2 correspond under this isomorphism<strong>to</strong> the maps m and p Y , respectively (see [38], the formulas for the faces of thesimplicial system on page 44 before Lemma 6.12). Thus we see thatȞ 0 (Y/X, Br ′ ) = ker[Br(Y ) m∗ −p ∗ Y−−−−−→ Br(H × Y )] .A computation using the Čech spectral sequence (6.12.0) in [38] shows that themap Br(X) → Br(Y ) defined by the compositionH 2 (X, G m ) p −→ Ȟ0 (Y/X, Br ′ ) ⊂ H 2 (Y, G m )is the pullback morphism π ∗ : Br(X) → Br(Y ). This concludes the proof of theexactness of (6).Such an exact sequence will be very useful in the following, but we need anotherexact 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 herethe definition of ∆ Y/X due <strong>to</strong> Colliot-Thélène and Xu.2.5. Definition of ∆ Y/X . We use the above notation. Since H is connected, wehave a canonical isomorphism c H : Ext c k(H, G m ) ∼ = Pic(H) (see [8], Corollary5.7), where Ext c k(H, G m ) is the abelian group of isomorphism classes of centralextensions of k-algebraic groups of H by G m . Given such an extension1 → G m → H 1 → H → 1corresponding <strong>to</strong> an element p ∈ Pic(H), we get a coboundary map in étale cohomology∂ H1 : H 1 (X, H) → H 2 (X, G m ),see [19], IV.4.2.2. This coboundary map fits in the natural exact sequence ofpointed sets (see [19], Remark IV.4.2.10 )H 1 (X, H 1 ) → H 1 (X, H) ∂ H−−→1H 2 (X, G m ) = Br(X). (7)The element ∆ Y/X (p) is defined <strong>to</strong> be the image of the class [Y ] ∈ H 1 (X, H) ofthe <strong>to</strong>rsor π : Y → X by the map ∂ H1 . This construction defines a map∆ Y/X : Pic(H) → Br(X), p ↦→ ∂ H1 ([Y ]),which is func<strong>to</strong>rial in X and H (this map was denoted by δ <strong>to</strong>rs (Y ) in [11]).We can compare the map ∆ Y/X : Pic(H) → Br(X) with another useful mapα Y/X : H 1 (k, Ĥ) → Br 1(X) defined by the formulaα Y/X (z) := p ∗ X(z) ∪ [Y ] ∈ H 2 (X, G m ) ,

<strong>Manin</strong> <strong>obstruction</strong> <strong>to</strong> strong approximation for homogeneous spaces 11where p X : X → Spec(k) is the structure morphism and [Y ] ∈ H 1 (X, H).Recall that we have a canonical map η H : H 1 (k, Ĥ) → Pic(H) coming fromLeray’s spectral sequence (see for instance [38], Lemma 6.9).Lemma 2.6. The following diagram is commutative :H 1 (k, Ĥ) α Y /XBr 1 (X)η HPic(X) ∆ Y /X Br(X) .Proof. Define Z <strong>to</strong> be the quotient of Y by the action of H ssu . Then Z → X isa <strong>to</strong>rsor under H <strong>to</strong>r . By func<strong>to</strong>riality, and using the isomorphism Ĥ<strong>to</strong>r ∼ = Ĥ, it issufficient <strong>to</strong> prove the commutativity of the following diagram :H 1 (k, Ĥ<strong>to</strong>r ) α Z/XBr 1 (X)η H <strong>to</strong>rPic(X)∆ Z/X Br(X) .Consider the groups Ext n k(H <strong>to</strong>r , G m ) in the abelian category of fppf-sheavesover Spec(k). By [29], Lemmas A.3.1 and A.3.2, we know that the diagramH 1 (k, Ĥ<strong>to</strong>r ) α Z/XBr 1 (X)η ′ H <strong>to</strong>rExt 1 k(H <strong>to</strong>r , G m ) ∆ Z/X Br(X)is commutative, where η ′ H <strong>to</strong>r: H 1 (k, Ĥ<strong>to</strong>r ) → Ext 1 k(H <strong>to</strong>r , G m ) is the edge mapfrom the local <strong>to</strong> global Ext’s spectral sequence H p (k, Ext q k (H<strong>to</strong>r , G m k)) =⇒Ext p+qk(H <strong>to</strong>r , G m ) (see [1], V.6.1).By [35], Proposition 17.5, there exists a canonical map Ext 1 k(H <strong>to</strong>r , G m ) →Ext c k(H <strong>to</strong>r , G m ). Composing this map with c H <strong>to</strong>r : Ext c k(H <strong>to</strong>r , G m ) → Pic(H <strong>to</strong>r )(used in the construction of ∆ Z/X ), we get a map c ′ H: Ext 1 k(H <strong>to</strong>r , G <strong>to</strong>rm ) →Pic(H <strong>to</strong>r ).It is now sufficient <strong>to</strong> prove that the diagramH 1 η ′(k, Ĥ<strong>to</strong>r H)<strong>to</strong>r Ext 1 k(H<strong>to</strong>r , G m )η H <strong>to</strong>rc ′ H <strong>to</strong>rPic(H <strong>to</strong>r )

<strong>Manin</strong> <strong>obstruction</strong> <strong>to</strong> strong approximation for homogeneous spaces 13Theorem 2.8. Let k be a field of characteristic zero. Let H a connected lineark-group, X a smooth k-variety and π : Y H −→ X a (left) <strong>to</strong>rsor under H. Then wehave a commutative diagram with exact rows, func<strong>to</strong>rial in (X, Y, π, H):Pic(Y )ϕ 1 Pic(H) ∆ Y /X Br(X)π ∗ Br(Y ) m∗ −p ∗ Y Br(H × Y )(8)ι Xι YνPic(Y )ϕ 1 Pic(H) ∆ Y /X Br 1 (X, Y )π ∗Br 1 (Y )ϕ 2Br a (H) .Here m : H × Y → Y denotes the left action of H on Y , the homomorphism∆ Y/X : Pic(H) → Br(X) is the map of [11], see 2.5 above, the homomorphismsϕ 1 and ϕ 2 are defined in [38] (6.4.1) (or see (4)), the homomorphisms ι X andι Y are the inclusion maps, and the injective homomorphism ν is given as thecomposite of the following natural injective maps:Br a (H) → Br a (H) ⊕ Br 1 (Y )∼ =−−→ Br 1 (H × Y ) ↩→ Br(H × Y ).In particular, if Y (k) ≠ ∅ and y ∈ Y (k), x = π(y), then the maps ϕ i are inducedby the map i y : H → Y defined by h ↦→ h.y, and we have an exact sequencePic(Y ) i∗ y−→ Pic(H) ∆ Y /X−−−−→ Br 1,x (X, Y ) −→ π∗Br 1,y (Y ) i∗ y−→ Br 1,e (H). (9)Remark 2.9. Recall that the exact sequences (8) and (9) can be extended <strong>to</strong> theleft by0 → k[X] ∗ /k ∗ → k[Y ] ∗ /k ∗ → Ĥ(k) → Pic(X) → Pic(Y )(see [38], Prop. 6.10).Corollary 2.10. Let k be a field of characteristic zero. Let T be a k-<strong>to</strong>rus, X asmooth k-variety and π : Y T −→ X a (left) <strong>to</strong>rsor under T . Then we have an exactsequence :Pic(Y ) −→ ϕ1Pic(T ) ∆ Y /X−−−−→ Br 1 (X) −→ π∗Br 1 (Y ) −→ ϕ2Br a (T ) .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 <strong>to</strong>study the injectivity of the map Br(X) → Br(Y ). We cannot describe easily theimage of this map in general. However, Harari and Skoroboga<strong>to</strong>v studied this mapin particular cases (universal <strong>to</strong>rsors for instance): see [23], Theorems 1.6 and 1.7.Corollary 2.12 (cf. [38], Corollary 6.11). Let1 → G ′ i−−→ Gj−−→ G ′′ → 1

14 Mikhail Borovoi and Cyril Demarchebe an exact sequence of connected algebraic groups over a field k of characteristic0. Assume that G ′ is linear. Then there is a commutative diagram with exact rowsPic(G)i ∗ Pic(G ′ ) ∆ G/G ′′ Br(G ′′ )ι ′′j ∗ Br(G) m∗ −p ∗ G Br(G ′ × G)iPic(G)∗ Pic(G ′ ) ∆ G/G ′′ Br 1,e (G ′′ , G) Br 1,e (G) Br 1,e (G ′ ) .(10)Here p G : G ′ × G → G is the projection map, the map m : G ′ × G → G is definedby m(g ′ , g) := i(g ′ ).g (where the product denotes the group law in G), ι ′′ and ι arethe inclusion homomorphisms, and the injective homomorphism ν is defined as inTheorem 2.8.If the homomorphism Pic(G) → Pic(G ′ ) is surjective (e.g. when G ′ is a k-<strong>to</strong>rus,or when G ′ ss is simply connected, or when all the three groups G ′ , G and G ′′ arelinear), then Br 1,e (G ′′ , G) = Br 1,e (G ′′ ), and we have a commutative diagram withexact rowsj ∗ιi ∗νPic(G)i ∗ Pic(G ′ ) ∆ G/G ′′ Br(G ′′ )j ∗ Br(G) m∗ −p ∗ G Br(G ′ × G)(11)ι ′′ινPic(G)i ∗ Pic(G ′ ) ∆ G/G ′′ Br 1,e (G ′′ )j ∗Br 1,e (G)i ∗Br 1,e (G ′ ) .Proof of the corollary. The short exact sequence of algebraic groups defines astructure of (left) G ′′ -<strong>to</strong>rsor under G ′ on G (G ′ acts on G by left translations).Now from the diagram with exact rows (8) we obtain diagram (10), which differsfrom diagram (11) by the middle term in the bot<strong>to</strong>m row.From diagram (8) we obtain an exact sequencePic(G) −→ i∗Pic(G ′ ) ∆ G/G−−−−→ ′′Br(G ′′ ) −→ j∗Br(G) . (12)If the homomorphism i ∗ : Pic(G) → Pic(G ′ ) is surjective, then the homomorphismj ∗ : Br(G ′′ ) → Br(G) is injective, hence Br 1,e (G ′′ , G) = Br 1,e (G ′′ ), and we obtaindiagram (11) from diagram (10).If G ′ is a k-<strong>to</strong>rus or if G ′ ss is simply connected, then Pic(G ′ ) = 0, and thehomomorphism Pic(G) → Pic(G ′ ) is clearly surjective. If all the three groupsG ′ , G and G ′′ are linear, then again the homomorphism Pic(G) → Pic(G ′ ) issurjective, see [38], proof of Corollary 6.11, p. 44.For the proof of Theorem 2.8 we need a crucial lemma.Lemma 2.13. Let k be a field of characteristic zero. Let H a connected lineark-group, X a smooth k-variety and π : Y H −→ X a (left) <strong>to</strong>rsor under H. Let

<strong>Manin</strong> <strong>obstruction</strong> <strong>to</strong> strong approximation for homogeneous spaces 15τ : Z −−→ GmY be a <strong>to</strong>rsor under G m . Then there exists a central extension ofalgebraic k-groups1 → G m → H 1 → H → 1and a left action H 1 × Z → Z, extending the action of G m on Z and compatiblewith the action of H on Y . This action makes Z → X in<strong>to</strong> a <strong>to</strong>rsor under H 1 .Moreover, the class of such an extension H 1 in the group Ext c k(H, G m ) is uniquelydetermined, namely [H 1 ] = ϕ 1 ([Z]) ∈ Ext c k(H, G m ) = Pic(H).Remark 2.14. In [24], Harari and Skoroboga<strong>to</strong>v studied this question of compositionof <strong>to</strong>rsors. Their results (see Theorem 2.2 and Proposition 2.5 in [24]) dealwith <strong>to</strong>rsors under multiplicative groups and not only under G m as here, but theyrequire additional assumptions concerning the type of the <strong>to</strong>rsor and on invertiblefunctions on the varieties. Those additional assumptions are not satisfied in ourcontext.Proof. Let p H : H × Y → H and p Y : H × Y → Y denote the two projections. Let1 → G m → H 1 → H → 1be a central extension such that its class in Ext c k(H, G m ) = Pic(H) is exactlyϕ 1 ([Z]).In this setting, Lemma 2.3 implies thatm ∗ [Z] = p ∗ H[H 1 ] + p ∗ Y [Z] . (13)Formula (13) means that the push-forward of the <strong>to</strong>rsor H 1 ×Z −−−−−→ Gm×GmH ×Yby the group law homomorphism G m ×G m → G m is isomorphic (as a H ×Y -<strong>to</strong>rsorunder G m ) <strong>to</strong> the pullback m ∗ Z of the <strong>to</strong>rsor Z → Y by the map m : H × Y → Y .In particular, we get the following commutative diagram:m ′ G mH 1 × Z m ∗ Z Z(14)H × YG m×G mwhere m ′ is defined <strong>to</strong> be the composite of the two upper horizontal maps. Thesituation is very similar <strong>to</strong> that in the proof of Theorem 5.6 in [8]: the map m ′fits in<strong>to</strong> commutative diagram (14) and for all t 1 , t 2 ∈ G m and all h 1 ∈ H 1 , z ∈ Z,we havem ′ (t 1 .h 1 , t 2 .z) = t 1 t 2 .m ′ (h 1 , z) . (15)We want <strong>to</strong> use m ′ <strong>to</strong> define a group action of H 1 on Z. Formula (15) implies thatthe morphism m ′ (e, .) : z ↦→ m ′ (e, z) is an au<strong>to</strong>morphism of the Y -<strong>to</strong>rsor Z, andwe can define a map m ′′ : H 1 × Z → Z <strong>to</strong> be the compositionG mm YG mm ′′ := m ′ (e, .) −1 ◦ m ′ : H 1 × Z → Z.

16 Mikhail Borovoi and Cyril DemarcheThen we get a commutative diagramH 1 × Z m′′ Z(16)H × Ym Ywhere the map m ′′ still satisfies formula (15) and now, for all z ∈ Z, we havem ′′ (e, z) = z. (17)We wish <strong>to</strong> prove that m ′′ is a left group action of H 1 on Z.Since m: H × Y → Y is a left action, we haveτ(m ′′ (h 1 h 2 , z)) = τ(m ′′ (h 1 , m ′′ (h 2 , z))) for h 1 , h 2 ∈ H 1 , z ∈ Z,where τ : Z → Y is the canonical map. Since τ : Z → Y is a <strong>to</strong>rsor under G m ,there is a canonical mapZ × Y Z → G m ,(z 1 , z 2 ) ↦→ z 1 z −12 .We obtain a morphism of k-varietiessuch thatϕ: H 1 × H 1 × Z → G m (h 1 , h 2 , z) ↦→ ϕ z (h 1 , h 2 )m ′′ (h 1 h 2 , z) = ϕ z (h 1 , h 2 ).m ′′ (h 1 , m ′′ (h 2 , z)) for h 1 , h 2 ∈ H 1 , z ∈ Z.Then (17) implies thatϕ z (h, e) = 1 and ϕ z (e, h) = 1.By Rosenlicht’s lemma (see [37], Theorem 3, see also [38], Lemma 6.5), the mapϕ has <strong>to</strong> be trivial, i.e. ϕ z (h 1 , h 2 ) = 1 for all z, h 1 , h 2 . Therefore we havem ′′ (h 1 h 2 , z) = m ′′ (h 1 , m ′′ (h 2 , z)) . (18)Formulas (17) and (18) show that m ′′ is a left group action of H 1 on Z. Sincem ′′ satisfies (15), we have for t ∈ G m , z ∈ Zm ′′ (te, z) = t.m ′′ (e, z) = t.z,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 .

<strong>Manin</strong> <strong>obstruction</strong> <strong>to</strong> strong approximation for homogeneous spaces 17Consider the following commutative diagram (see (16)):H 1 × k Z φ Z Z × X Z(19)H × k Yφ Y Y × X Y ,where φ Z (h 1 , z) := (m ′′ (h 1 , z), z), φ Y (h, y) := (m(h, y), y), and the unnamed morphismsare the natural ones. Since Y → X is a <strong>to</strong>rsor under H, the morphism φ Yis an isomorphism. The group G m × G m acts on H 1 × k Z via (t 1 , t 2 ).(h 1 , z) :=(t 1 h 1 , t 2 .z), making H 1 × k Z → H × k Y in<strong>to</strong> a <strong>to</strong>rsor under G m × G m . We definean action of G m × G m on Z × X Z by(t 1 , t 2 ).(z 1 , z 2 ) := ((t 1 t 2 ).z 1 , t 2 .z 2 ),then Z × X Z → Y × X Y is a <strong>to</strong>rsor under G m × G m . By formula (15) the map φ Zin (19) is a morphism of <strong>to</strong>rsors under G m × G m compatible with the isomorphismφ Y of k-varieties. Therefore the map φ Z is an isomorphism of k-varieties, whichproves that the action m ′′ makes Z → X in<strong>to</strong> a <strong>to</strong>rsor under H 1 .Let us prove the uniqueness of the class of the extension H 1 . If H 2 is a centralextension of H that satisfies the conditions of the lemma, then the analogues ofdiagram (16) and formula (15) with H 2 instead of H 1 define an isomorphism ofH × Y -<strong>to</strong>rsors under G m between the push-forward of H 2 × Z by the morphismG m × G m → G m and the <strong>to</strong>rsor m ∗ Z. Therefore, we getm ∗ [Z] = p ∗ H[H 2 ] + p ∗ Y [Z].Comparing with (13), we see that p ∗ H [H 2] = p ∗ H [H 1]. Since p ∗ H + p∗ Y : Pic(H) ⊕Pic(Y ) → Pic(H ×Y ) is an isomorphism, we see that p ∗ H : Pic(H) → Pic(H ×Y ) isan embedding, hence [H 2 ] = [H 1 ], which completes the proof of Lemma 2.13.2.15. Proof of Theorem 2.8: Top row of the diagram. First we prove that the<strong>to</strong>p row in diagram (8) is a complex. Let p ∈ Pic(Y ) and let us prove that∆ Y/X (ϕ 1 (p)) = 0. Let Z → Y be a <strong>to</strong>rsor under G m such that [Z] = p ∈ Pic(Y ).Let p ′ := ϕ 1 (p) ∈ Pic(H), and 1 → G m → H 1 → H → 1 be a central extensionof H by G m corresponding <strong>to</strong> p ′ via the isomorphism Ext c k(H, G m ) ∼ = Pic(H).Then ∆ Y/X (p ′ ) is equal (by definition) <strong>to</strong> ∂ H1 ([Y ]) ∈ H 2 (X, G m ), where ∂ H1 :H 1 (X, H) → H 2 (X, G m ) is the coboundary map coming from the extension H 1 ,and [Y ] is the class of the <strong>to</strong>rsor Y → X in H 1 (X, H).Lemma 2.13 implies that the class [Y ] ∈ H 1 (X, H) is in the image of the mapH 1 (X, H 1 ) → H 1 (X, H). From exact sequence (7) we see that the class ∂ H1 ([Y ])is trivial in H 2 (X, G m ), i.e. ∆ Y/X (p ′ ) = ∂ H1 ([Y ]) = 0 ∈ H 2 (X, G m ), hencePic(Y ) −→ j∗Pic(H) ∆ Y /X−−−−→ Br(X) is a complex.

18 Mikhail Borovoi and Cyril DemarcheLet p ∈ Pic(H), and let us prove that π ∗ (∆ Y/X (p)) = 0. The element pcorresponds <strong>to</strong> the class of an extension 1 → G m → H 1 → H → 1, and we have∆ Y/X (p) = ∂ H1 ([Y ]). We have a commutative diagramH 1 (X, H)∂ H1H 2 (X, G m )π ∗H 1 (Y, H)π ∗∂ H1 H 2 (Y, G m )so that π ∗ (∆ Y/X (p)) = π ∗ (∂ H1 ([Y ])) = ∂ H1 (π ∗ [Y ]). The <strong>to</strong>rsor π ∗ [Y ] is trivial inthe set H 1 (Y, H), hence π ∗ (∆ Y/X (p)) = 0. Consequently the <strong>to</strong>p row of diagram(8) is a complex.Let us prove that the <strong>to</strong>p row of diagram (8) is exact. For the exactness at theterm Br(Y ), see Proposition 2.4. So it remains <strong>to</strong> prove the exactness of the <strong>to</strong>prow at Pic(H) and at Br(X).Let p ∈ Pic(H) be such that ∆ Y/X (p) = 0. Such a p corresponds <strong>to</strong> the classqof an extension 1 → G m → H 1 −→ H → 1 such that ∂H1 ([Y ]) = 0. From exactsequence (7) we see that there exists an X-<strong>to</strong>rsor Z −−→ H1X under H 1 , with anH 1 -equivariant map Z → Y , making Z → Y in<strong>to</strong> a <strong>to</strong>rsor under G m . Then by theuniqueness part of Lemma 2.13 we know that the class of Z → Y in Pic(Y ) maps<strong>to</strong> the class of H 1 in Pic(H), i.e. ϕ 1 ([Z]) = p, which proves the exactness of the<strong>to</strong>p row of diagram (8) at Pic(H).Let us now prove the exactness of the <strong>to</strong>p row of (8) at Br(X).Assume first that the k-variety X is quasi-projective. Let A ∈ Br(X) such thatπ ∗ A = 0 ∈ Br(Y ). By a theorem of Gabber also proven by de Jong (see [12]), weknow that there exists a positive integer n and an X-<strong>to</strong>rsor Z → X under PGL nsuch that −A is the image of the class of Z in H 1 (X, PGL n ) by the coboundarymap H 1 (X, PGL n ) ∂GLn−−−→ H 2 (X, G m ). Let W denote the product Y × X Z. Fromthe commutative diagram with exact rowsH 1 (X, GL n ) H 1 (X, PGL n ) Br(X)H 1 (Y, GL n ) H 1 (Y, PGL n ) Br(Y )π ∗we see that the assumption π ∗ A = 0 implies that the <strong>to</strong>rsor W −−−−→ PGLnY isdominated by some Y -<strong>to</strong>rsor under GL n , i.e. there exists a <strong>to</strong>rsor V −−−→ GLnYand a morphism of Y -<strong>to</strong>rsors V → W compatible with the quotient morphism

<strong>Manin</strong> <strong>obstruction</strong> <strong>to</strong> strong approximation for homogeneous spaces 19GL n → PGL n . We have the following picture:YHPGL nGL nWHG mVX Z .PGL nSince W → X is a <strong>to</strong>rsor under the connected linear group H × PGL n , we canapply Lemma 2.13 <strong>to</strong> get a central extension1 → G m → L → H × PGL n → 1 (20)and a structure of X-<strong>to</strong>rsor under L on V → X, compatible with the actionof H × PGL n on W . In particular, the natural injections of H and PGL n in<strong>to</strong>H × PGL n define two central extensions obtained by pulling back the extension(20):1 → G m → L H → H → 1 (21)1 → G m → L PGLn → PGL n → 1 . (22)Since PGL n acts trivially on Y , the action of L PGLndefines a commutative diagram(as a subgroup of L) on VL PGLn × Vφ VV × Y VPGL n × W φ W W × Y Wwhere φ V (l, v) := (l.v, v), φ W (p, w) := (p.w, w), and the vertical maps are thenatural ones. We see easily that φ V is an isomorphism, hence V → Y is a <strong>to</strong>rsorunder L PGLn . This action of L PGLn extends the action of G m on V above W , andis compatible with the action of PGL n on W above Y via the extension (22) andthe map V → W . Therefore, the unicity result in Lemma 2.13 implies that exactsequence (22) is equivalent <strong>to</strong> the usual extension 1 → G m → GL n → PGL n → 1,and in particular that ∂ LPGLn ([Z]) = ∂ GLn ([Z]) ∈ H 2 (X, G m ).Consider the direct product of the exact sequences (21) and (22)1 → G m × G m → L H × L PGLn → H × PGL n → 1 .Define the morphism µ : L H × L PGLn → L by µ(l, l ′ ) := l.l ′ , where the product istaken inside the group L. By definition of L H and L PGLn , we see that the image ofthe commuta<strong>to</strong>r morphism c : L H ×L PGLn → L defined by c(l, l ′ ) := l.l ′ .l −1 .l ′ −1 is

20 Mikhail Borovoi and Cyril Demarchecontained 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 PGLn . Hence themorphism µ is a group homomorphism, and the following diagram is commutativewith exact rows1 G m × G m L H × L PGLnH × PGL n 1mµ1 G m L H × PGL n 1 ,where m: G m × G m → G m is the group law. Therefore we get a commutativediagram of coboundary mapsH 1 (X, H) × H 1 (X, PGL n ) ∂ L H ×∂ LPGLn H 2 (X, G m ) × H 2 (X, G m )∼ =∼ =H 1 (X, H × PGL n )H 1 (X, H × PGL n )∂ LH ×L PGLn H 2 (X, G m × G m )m∂ L H 2 (X, G m ) .In particular, this diagram implies that∂ L ([W ]) = ∂ LH ([Y ]) + ∂ LPGLn ([Z])in H 2 (X, G m ). Since W −−−−−−→ H×PGLnX is dominated by the X-<strong>to</strong>rsor Vknow that ∂ L ([W ]) = 0, therefore the above formula implies thatL −→ X, we∂ LH ([Y ]) = −∂ LPGLn ([Z]) = A ∈ Br(X) ,i.e. A = ∆ Y/X ([L H ]), with [L H ] ∈ Ext c k(H, G m ) ∼ = Pic(H). Hence the first row ofdiagram (8) is exact under the assumption that X is quasi-projective.Let us now deduce the general case: X is not supposed <strong>to</strong> be quasi-projectiveanymore. By Nagata’s theorem (see [32]) we know that there exists a proper k-variety 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 k-variety Z ′ and a projective, surjective birational morphism Z ′ → Z. Moreover,using Hironaka’s resolution of singularities (see [27], see also [3] and [17]), thereexists a smooth projective k-variety ˜Z and a birational morphism ˜Z → Z ′ . DefineX ′ <strong>to</strong> 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 ′ → Xis a birational morphism. Define Y ′ <strong>to</strong> be the product Y × X X ′ . By the quasi-

24 Mikhail Borovoi and Cyril DemarcheIn the case x = x 0 = π(y 0 ) we obtain(π ∗ b)(y 0 .g) = b(π(y 0 ).g) = b(π(y 0 )) + π G (m ∗ b)(g) = π G (m ∗ b)(g), (27)since b(π(y 0 )) = b(x 0 ) = 0. Consequently, (26) and (27) give the expected formula,that is:b(x.g) = b(x) + (π ∗ b)(y 0 .g) .Corollary 3.4. Let X := H\G be a right homogeneous space of a connected k-group G over a field k of characteristic 0, where H ⊂ G is a connected lineark-subgroup. Then for all b ∈ Br 1,x0 (X, G), x ∈ X(k), g ∈ G(k) we haveb(x.g) = b(x) + π ∗ b(g),where π : G → X is the quotient map and x 0 = π(e).Proof. We take Y = G, y 0 = e ∈ G(k), and x 0 = π(y 0 ) = π(e) in Lemma 3.3, thenBr 1,x0 (X, Y ) = Br 1,x0 (X, G) and π ∗ b(y 0 .g) = π ∗ b(g).Corollary 3.5. Let k be a number field. Let X := H\G be a homogeneous spaceof a connected k-group G, where H ⊂ G is a connected linear k-subgroup. Let 〈, 〉denote the <strong>Manin</strong> pairingBr(X) × X(A) → Q/Z.Let b ∈ Br 1,x0 (X, G), x ∈ X(A), g ∈ G(A). Then〈b, x.g〉 = 〈b, x〉 + 〈π ∗ (b), 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 bea 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). ThenProof. By Corollary 3.5 we haveBy func<strong>to</strong>riality we have〈b, x.ϕ(g ′ )〉 = 〈b, x〉.〈b, x.ϕ(g ′ )〉 = 〈b, x〉 + 〈π ∗ (b), ϕ(g ′ )〉.〈π ∗ (b), ϕ(g ′ )〉 = 〈ϕ ∗ π ∗ (b), g ′ 〉.Since b ∈ Br 1,x0 (X, G), we have π ∗ b ∈ Br 1,e (G) and ϕ ∗ π ∗ b ∈ Br 1,e (G ′ ) = 0. Thusϕ ∗ π ∗ b = 0, hence 〈π ∗ (b), ϕ(g ′ )〉 = 0, and the corollary follows.

<strong>Manin</strong> <strong>obstruction</strong> <strong>to</strong> strong approximation for homogeneous spaces 254 Some lemmasFor an abelian group A we write A D := Hom(A, Q/Z).Lemma 4.1. Let P be a quasi-trivial k-<strong>to</strong>rus over a number field k. Then thecanonical map λ: P (A) → Br a (P ) D induced by the <strong>Manin</strong> pairing is surjective.Proof. We have Br a (P ) = H 2 (k, ̂P ), see [38], Lemma 6.9(ii). By [38], (8.11.2), themapλ: P (A) → Br a (P ) D = H 2 (k, ̂P ) Dis given by the canonical pairingP (A) × H 2 (k, ̂P ) → Q/Z.Consider the map µ from the Tate-Poi<strong>to</strong>u exact sequence(P (A) • ) ∧ µ−−→ H 2 (k, ̂P ) D → H 1 (k, P ), (28)see [25], Theorem 5.6 or [14], Theorem 6.3. By (P (A) • ) ∧ we mean the completionof P (A) • for the <strong>to</strong>pology of open subgroups of finite index. Then the map µ isinduced by λ. Since P is a quasi-trivial <strong>to</strong>rus, we have H 1 (k, P ) = 0, and we seefrom (28) that the map µ is surjective. But by [22], Lemma 4, im µ = im λ. Thusλ is surjective.Lemma 4.2. Let X be a right homogeneous space (not necessarily principal) ofa connected k-group G over a number field k. Let N ⊂ G be a connected normalk-subgroup. Set Y := X/N, and let π : X → Y be the canonical map. Then theinduced map X(A) → Y (A) is open.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 ofk v , and by κ v the residue field of O v . For an O v -scheme Z v we set ˜Z v := Z v × Ov κ v .Since the morphism π is smooth, the map X(k v ) → Y (k v ) is open for any placev of k.Let S be a finite set of places of k containing all the archimedean places. WriteO S for the ring of elements of k that are integral outside S. Taking S sufficientlylarge, we can assume that G and N extend <strong>to</strong> smooth group schemes G and Nover Spec(O S ), and that X and Y extend <strong>to</strong> homogeneous spaces X of G andY of G /N over Spec(O S ) such that Y = X /N . In particular, the reductionÑ v := N × O S κ v is connected for v /∈ S.Let v /∈ S and let y v ∈ Y (O v ). Set X yv := X × Y Spec(O v ), the morphismsbeing given by π and by y v : Spec (O v ) → Y . It is an O v -scheme. Then itsreduction ˜X yv is a homogeneous space of the connected κ v -group Ñv over the

26 Mikhail Borovoi and Cyril Demarchefinite field κ v . By Lang’s theorem ([31], Theorem 2) ˜X yv has a κ v -point. ByHensel’s lemma X yv has an O v -point. This means that y v ∈ π(X (O v )). Thusπ(X (O v )) = Y (O v ) for all v /∈ S. It follows that the map X(A) → Y (A) isopen.4.3. Let X = H\G be a homogeneous space of a connected k-group G. Letx 1 ∈ X(k). Consider the map π x1 : G → X, g ↦→ x 1 .g, it induces a homomorphismπx ∗ 1: Br(X) → Br(G). Let Br 1 (X, G) x1 denote the subgroup of elements b ∈ Br(X)such that πx ∗ 1(b) ∈ Br 1 (G). The following lemma shows that Br 1 (X, G) x1 does notdepend on x 1 , so we may write Br 1 (X, G) instead of Br 1 (X, G) x1 . Note thatBr 1 (X, G) = Br 1,x1 (X, G) + Br(k).Lemma 4.4. The subgroup Br 1 (X, G) x1 ⊂ Br(X) does not depend on x 1 .Proof. We have a commutative diagramBr(X)/Br(k)π ∗ x 1 Br(G)/Br(k)Br(X)π ∗ x 1 Br(G) .We see that it suffices <strong>to</strong> prove that the kernel of π ∗ x 1does 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), henceπ x2 (g ′ ) = x 2 .g ′ = x 1 .gg ′ = π x1 (gg ′ ) = (π x1 ◦ l g )(g ′ ),where l g denotes the left translation on G by g. Thusπ ∗ x 2= l ∗ g ◦ π ∗ x 1.Since l g is an isomorphism of the underlying variety of G, we see that lg ∗ : Br(G) →Br(G) is an isomorphism, hence ker π ∗ x 2= ker π ∗ x 1, which proves the lemma.Lemma 4.5. Let X be a right homogeneous space of a unipotent k-group U overa field k of characteristic 0. Then X(k) is non-empty and is one orbit of U(k).Proof. By [5], Lemma 3.2(i), X(k) ≠ ∅. Let x ∈ X(k), H := Stab(x), then H isunipotent, hence H 1 (k, H) = 1, and therefore X(k) = x.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) isconnected.

<strong>Manin</strong> <strong>obstruction</strong> <strong>to</strong> strong approximation for homogeneous spaces 27Lemma 4.7. Let G be a unipotent k-group over a number field k. Let X be aright 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 ⊂ Gdenote the stabilizer of x 0 in G. We have X = H\G.Set g = Lie(G). Since g is a vec<strong>to</strong>r space and S ≠ ∅, by the classical strongapproximation theorem g is dense in g ⊗ k A S . Since char(k) = 0, we have theexponential map g → G, which is an isomorphism of k-varieties. We see thatG(k) is dense in G(A S ). It follows that x 0 G(k) is dense in x 0 G(A S ). Since H isunipotent, 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’slemma). Thus x 0 G(k) is dense in X(A S ), and X(k) is dense in X(A S ).5 Brauer groupWe are grateful <strong>to</strong> A.N. Skoroboga<strong>to</strong>v, E. Shustin, and T. Ekedahl for helping us<strong>to</strong> prove Theorem 5.1 below.Theorem 5.1. Let X be a smooth irreducible algebraic variety over an algebraicallyclosed field k of characteristic 0. Let G be a connected algebraic group(not necessarily linear) defined over k, acting on X. Then G(k) acts on Br(X)trivially.Proof. We write H i for Het í (étale cohomology). The Kummer exact sequence1 → µ n → G mn−−→ G m → 1of multiplication by n gives rise <strong>to</strong> a surjective mapH 2 (X, µ n ) ↠ Br(X) n ,where Br(X) n denotes the group of elements of order dividing n in Br(X). Sinceevery element of Br(X) is <strong>to</strong>rsion (because Br(X) embeds in Br(k(X)), cf. [20],II, Corollary 1.8), it is enough <strong>to</strong> prove the following Theorem 5.2.Theorem 5.2. Let X be a smooth irreducible algebraic variety over an algebraicallyclosed field k (of any characteristic). Let G be a connected algebraicgroup (not necessarily linear) defined over k, acting on X. Let A be a finiteabelian group of order invertible in k. Then G acts on Het í (X, A) trivially for alli.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

28 Mikhail Borovoi and Cyril DemarcheHB i (X, A). Since G is connected, the group G(C) is connected, hence we can connectg with the unit element e ∈ G(C) by a path. We see that the au<strong>to</strong>morphismof Xg ∗ : X → X, x ↦→ x.gis homo<strong>to</strong>pic <strong>to</strong> the identity au<strong>to</strong>morphisme ∗ : 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.Lemma 5.3. Let X, Y be smooth algebraic varieties over an algebraically closedfield k (of any characteristic). Let A be a finite abelian group of order invertiblein k. Consider the projection p Y : X × Y → Y . Then the higher direct imageR i (p Y ) ∗ A in the étale <strong>to</strong>pology is the pullback of the abelian group H i (X, A) consideredas a sheaf on Spec(k).Proof. Consider the commutative diagramX × k Yp XXp YYs Xs Y Spec (k) ,here X × k Y is the fibre product of X and Y with respect <strong>to</strong> the structure morphismss X and s Y . Clearly R i (s X ) ∗ A is the constant sheaf on Spec (k) withstalk H i (X, A). By [13], Th. finitude, Theorem 1.9(ii), the sheaf R i (p Y ) ∗ A on Yis the pullback of the constant sheaf R i (s X ) ∗ A on Spec (k) along the morphisms Y : Y → Spec (k), which concludes the proof.Let y ∈ Y (k). It defines a canonical morphism f y : X → X × k Y such thatp X ◦ f y = id X and p Y ◦ f y = y ◦ s X : X → Y.Lemma 5.4. Let X, Y be two smooth algebraic varieties over an algebraicallyclosed field k (of any characteristic). Let A be a finite abelian group of orderinvertible in k. For a closed point y ∈ Y consider the map f y : X → X ×Y definedabove. If Y is irreducible, then the mapdoes not depend on y.f ∗ y : H i (X × Y, A) → H i (X, A)

<strong>Manin</strong> <strong>obstruction</strong> <strong>to</strong> strong approximation for homogeneous spaces 29Proof. Let η ∈ H i (X × Y, A). Then η defines a global section λ(η) of the sheafR i (p Y ) ∗ A (via compatible local sections U ↦→ η| X×U ∈ H i (X × U, A) of thecorresponding presheaf, for all étale open subsets U → Y ). By Lemma 5.3 thesheaf R i (p Y ) ∗ A is a constant sheaf with stalk H i (X, A). It is easy <strong>to</strong> see thatf ∗ y (η) = λ(η)(y) ∈ H i (X, A).Since Y is irreducible, it is connected, hence the global section λ(η) of the constantsheaf 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 mapm: X × G → X, (x, g) ↦→ x.g(the action). Let ξ ∈ H i (X, A). Set η = m ∗ ξ ∈ H i (X × G, A). For a k-pointg ∈ G(k) consider the map f g : X → X × G defined by x ↦→ (x, g), as above. Sincex.g = m(x, g) = m(f g (x)), we haveg ∗ ξ = f ∗ g m ∗ ξ = f ∗ g η ∈ H i (X, A).By Lemma 5.4 f ∗ g η does not depend on g. Thus g ∗ ξ does not depend on g. Thismeans that G(k) acts on H i (X, A) trivially.6 Homogeneous spaces of simply connected groupsIn the proof of the main theorem we shall need a result about strong approximationin homogeneous spaces of semisimple simply connected groups with connectedstabilizers. If X = H\G is such a homogeneous space, since G is semisimple andsimply 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 connectedk-group over a number field k, and let H ⊂ G be a connected subgroup. SetX := H\G. Let S be a non-empty finite set of places of k such that G(k) isdense in G(A S ). Then the set of points x ∈ X(A) such that 〈b, x〉 = 0 for allb ∈ Br 1,x0 (X, G) = Br x0 (X) coincides with the closure of the set X(k).G(k S ) inX(A) for the adelic <strong>to</strong>pology.Proof. This is very close <strong>to</strong> a result of Colliot-Thélène and Xu, see [11], Theorem3.7(b). Since their result is not stated in these terms in [11], we give here a proofof Theorem 6.1, following their argument.We prove the nontrivial inclusion of the theorem. Let x ∈ X(A) be orthogonal<strong>to</strong> 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

30 Mikhail Borovoi and Cyril Demarchethere exists an open neighbourhood U G ⊂ G(A) of g such that for any g ′ ∈ U Gwe 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 thatg 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 <strong>to</strong>pology.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 setX(k).G(k S ).Lemma 6.2. Let X be a smooth geometrically integral variety over a number fieldk. Let b ∈ Br(X). Then the functionX(A) → Q/Z: x ↦→ 〈b, x〉is locally constant in x for the adelic <strong>to</strong>pology in X(A).Proof. Arguing as in [38], Proof of Lemma 6.2, we can reduce our lemma <strong>to</strong> thelocal case. In other words, it is enough <strong>to</strong> prove that for any completion k v of kthe functionφ b : X(k v ) → Br(k v ): x ↦→ b(x)is locally constant in x. This follows from non-published results from the thesis ofAn<strong>to</strong>ine Ducros, cf. [16], Part II, Propositions (0.31) and (0.33).Since those results of Ducros are not published, we give another proof of thisfact. We are grateful <strong>to</strong> J.-L. Colliot-Thélène and <strong>to</strong> the referee for this proof. Letx ∈ X(k v ). The problem is local, so we can replace X by an affine open subsetcontaining x. From now on, X is assumed <strong>to</strong> be affine over k v .To show that the map φ b is locally constant around x, we may replace b byb − b(x) ∈ Br(X). Now we have b(x) = 0 and we want <strong>to</strong> prove that φ b is zero ina <strong>to</strong>pological neighbourhood of x.Since X is a smooth affine k v -variety, by a result by Hoobler (see [28], Corollary1) there exists a class η ∈ H 1 (X, PGL n ) such that b is the image of η by the usualcoboundary map. The class η is represented by an X-<strong>to</strong>rsor f : Y → X underPGL n .For x ′ ∈ X(k v ) let η(x ′ ) ∈ H 1 (k v , PGL n ) denote the image of η under the map(x ′ ) ∗ : H 1 (X, PGL n ) → H 1 (k v , PGL n ). Consider the exact sequence1 = H 1 (k v , GL n ) → H 1 (k v , PGL n )∆−−→ Br(k v ) .It is clear that b(x ′ ) = ∆(η(x ′ )). From the exact sequence we see that b(x ′ ) = 0 ifand only if η(x ′ ) = 1. On the other hand, clearly η(x ′ ) is the class of the k v -<strong>to</strong>rsorf −1 (x ′ ) under PGL n . It follows that b(x ′ ) = 0 if and only if f −1 (x ′ ) contains ak v -point, i.e. x ′ = f(y) for some y ∈ Y (k v ). Hence the set of points x ′ ∈ X(k v )

<strong>Manin</strong> <strong>obstruction</strong> <strong>to</strong> strong approximation for homogeneous spaces 31such that b(x ′ ) = 0 is exactly the subset f(Y (k v )) of X(k v ). We now concludeby the implicit function theorem: since f : Y → X is a smooth morphism ofk v -schemes, the image f(Y (k v )) is an open subset of X(k v ). Therefore, φ b is zeroon the open neighbourhood f(Y (k v )) of x, which concludes the proof.7 Proof of the main theoremThroughout this section we consider X = H\G satisfying the assumptions ofTheorem 1.4. Let x ∈ X(A) be an adelic point, we write x = (x f , x ∞ ), wherex f ∈ X(A f ), x ∞ ∈ X(k ∞ ). Let S be a finite set of places of k containing allarchimedean places, and we set S f := S ∩ Ω f . Let U f X ⊂ X(Af ) be an openneighbourhood of x f . For v ∈ Ω ∞ , we set U X,v <strong>to</strong> be the connected componen<strong>to</strong>f x v in X(k v ). We set U X,∞ := ∏ v∈Ω ∞U X,v , then U X,∞ is the connectedcomponent of x ∞ in X(k ∞ ). We setandU X := U f X × U X,∞ ⊂ X(A)U ′ X := U X .G scu (k Sf ) = U X .G scu (k S ) = U ′ fX × U ′ X,∞ ,where U ′ fX = U f X .Gscu (k Sf ) and U ′ X,∞ = U X,∞ = U X,∞ .G scu (k ∞ ) (becauseG scu (k ∞ ) is a connected <strong>to</strong>pological group, see [34], Theorem 5.2.3). Then U Xand U ′ X are open neighbourhoods of x in X(A). We say that U X is the specialneighbourhood of x defined by U f X .For the sake of the argument it will be convenient <strong>to</strong> introduce Property (P S )of a pair (X, G):(P S ) For any point x ∈ X(A) orthogonal <strong>to</strong> Br 1 (X, G), and for any open neighbourhoodU f X of xf , 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 Sf )∩U X ) is non-empty, where U X is the specialneighbourhood of x defined by U f X .The nontrivial part of Theorem 1.4 precisely says that property (P S ) holds forany X, G and S as in the theorem.We start proving Theorem 1.4. The structure of the proof is somewhat similar<strong>to</strong> that of Theorem A.1 of [7].7.1. First reduction.We reduce Theorem 1.4 <strong>to</strong> the case G u = 1. Let X and G be as in the theorem.We represent G as an extension1 → G lin → G → G abvar → 1,where G lin is a connected linear algebraic k-group and G abvar is an abelian varietyover k. We use the notation of 1.1. Set G ′ := G/G u , Y := X/G u . We

32 Mikhail Borovoi and Cyril Demarchehave a canonical epimorphism ϕ: G → G ′ and a canonical smooth ϕ-equivariantmorphism ψ : X → Y . We have G ′ lin = G lin /G u , hence G ′u = 1. We haveG ′ 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 <strong>to</strong> Br 1 (X, G). Set y :=ψ(x) ∈ Y (A). By func<strong>to</strong>riality, y is orthogonal <strong>to</strong> Br 1 (Y, G ′ ). Let U f Xbe as in(P S ), and let U X , U X ′ be the special neighbourhoods of x defined by U f X . Notethat U X ′ = U X ′ f ×U X,∞ , where U X ′ f is an open subset of X(A f ). Indeed, G scu (k v )is connected for all v ∈ Ω ∞ (see [34], Theorem 5.2.3).Set U fY := ψ(U f X ) ⊂ Y (Af ), U Y := ψ(U X ) ⊂ Y (A) and U Y′ := ψ(U X ′ ) ⊂Y (A). Since G u is connected, by Lemma 4.2 U fY is open in Y (Af ) and U Y andU Y ′ are open in Y (A). Set U Y,v := ψ(U X,v ). By [7], Lemma A.2, for each v ∈ Ω ∞the set U Y,v is the connected component of y v in Y (k v ). Set U Y,∞ := ∏ v∈Ω ∞U Y,v .We have U Y = U fY × U Y,∞. We see that U Y is the special neighbourhood of ydefined by U fY. From the split exact sequence1 → G u → G scu → G ′ sc → 1 ,we see that the map G scu (k S ) → G ′ sc (k S ) is surjective. Note that G ′ scu = G ′ sc . Itfollows thatU ′Y = U Y .G scu (k S ) = U Y .G ′ sc (kS ) = U Y .G ′ scu (kS ).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 unipotentgroup G u . By Lemma 4.7, X y0 (k) ≠ ∅ and X y0 has the strong approximationproperty away from Ω ∞ : the set X y0 (k) is dense in X y0 (A f ). Consider the setV f := X y0 (A f ) ∩ U ′ fX . By Corollary 4.6, for any v ∈ Ω ∞ the set X y0 (k v ) isconnected, and by Lemma 4.5, X y0 (k v ) is one orbit under G u (k v ). Set V :=V f × X y0 (k ∞ ).Let v ∈ Ω ∞ . We show that X y0 (k v ) ⊂ U X,v . Since y 0 ∈ U ′Y= ψ(U X ′ ), thereexists a point x v ∈ U X,v such that y 0 = ψ(x v ). Clearly x v ∈ X y0 (k v ). SinceX y0 (k v ) is one orbit under G u (k v ), we see that X y0 (k v ) = x v .G u (k v ) ⊂ U X,v ,because G u (k v ) is a connected group. Thus X y0 (k ∞ ) ⊂ U X,∞ and V ⊂ U ′fX ×U X,∞ = U X ′ , hence V ⊂ X y 0(A) ∩ U X ′ .Since y 0 ∈ ψ(U X ′ ), the set V is non-empty. Since by Lemma 4.7 X y 0(k) is densein X y0 (A f ), there is a point x 0 ∈ X y0 (k) ∩ V . Clearly x 0 ∈ X(k) ∩ U X ′ . Thus thepair (X, G) has Property (P S ). We see that in the proof of Theorem 1.4 we mayassume that G u = 1.7.2. Second reduction.By [7], Proposition 3.1 we may regard X as a homogeneous space of anotherconnected group G ′ such that G ′u = {1}, G ′ ss is semisimple simply connected,

<strong>Manin</strong> <strong>obstruction</strong> <strong>to</strong> strong approximation for homogeneous spaces 33and 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 isa surjective homomorphism G abvar → G ′ abvar . Since by assumption X(G abvar ) isfinite, 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 <strong>to</strong> Br 1 (X, G), then it isorthogonal <strong>to</strong> Br 1 (X, G ′ ). More precisely, we prove that Br 1 (X, G ′ ) is a subgroupof Br 1 (X, G).By construction (see [7], proof of Proposition 3.1), there is an exact sequenceof connected algebraic groups1 → S → G ′ q−→ G1 → 1,where G 1 is the quotient of G by the central subgroup Z(G)∩H and S is a k-<strong>to</strong>rus.Consider the following natural commutative diagram Xπ ′ π 1π1 S qG ′ G 1 1where 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 <strong>to</strong>p row ofdiagram (11)):Br(X) π ∗π ∗ 10 = Pic(S) Br(G 1 )p ∗pGπ ′∗,q ∗ Br(G ′ )Br(G) .Therefore, the injectivity of the map q ∗ : Br(G 1 ) → Br(G ′ ) implies that the naturalinclusion Br 1,x0 (X, G 1 ) ⊂ Br 1,x0 (X, G ′ ) is an equality. And by func<strong>to</strong>rialityBr 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 followsthat if a point x ∈ X(A) is orthogonal <strong>to</strong> Br 1 (X, G), then it is orthogonal <strong>to</strong>Br 1 (X, G ′ ).Thus if Theorem 1.4 holds for the pair (X, G ′ ), then it holds for (X, G). Wesee that we may assume in the proof of Theorem 1.4 that G lin is reductive, G ss is

34 Mikhail Borovoi and Cyril Demarchesimply connected, and the stabilizers of the geometric points of X in G are linearand connected.Now in order <strong>to</strong> prove Theorem 1.4 it is enough <strong>to</strong> prove the following Theorem7.3.Theorem 7.3. Let k be a number field, G a connected k-group, and X := H\Ga homogeneous space of G with connected stabilizer H. Assume:(i) G u = {1},(ii) H ⊂ G lin , i.e. 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. Weassume 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 <strong>to</strong>r → G sab . Wefirst prove a crucial special case of Theorem 7.3.Proposition 7.4. With the hypotheses of Theorem 7.3, assume that H <strong>to</strong>r injectsin<strong>to</strong> G sab (i.e. H ∩ G ss = H ssu ), and that the homomorphism Br 1,e (G sab ) →Br 1,e (H <strong>to</strong>r ) is surjective. Then the pair (X, G) has Property (P S ).Construction 7.5. Set Y := X/G ss . Then Y is a homogeneous space of thesemiabelian variety G sab , hence it is a <strong>to</strong>rsor of some semiabelian variety G ′ . Wehave G ′ abvar = G abvar , hence X(G ′ abvar ) is finite. We have a canonical smoothmorphism 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.Proposition 7.6. Let G, X be as in Proposition 7.4. Let Y, ψ : X → Y be as inConstruction 7.5. Let x 0 ∈ X(k), y 0 := ψ(x 0 ), X y0 := ψ −1 (y 0 ) = x 0 .G ss . Thenthe natural pullback homomorphismis surjective.i ∗ : Br 1,x0 (X, G) → Br x0 (X y0 )Note that Br 1,x0 (X y0 , G ss ) = Br x0 (X y0 ) because Br(G ss ) = 0.Construction 7.7. Consider the map π x0 : G → X, g ↦→ x 0 .g. This map identifiesX (resp. X y0 ) with a quotient of G (resp. G ss ) by a connected subgroup H ′ (resp.H ′ ssu ), so we haveX = H ′ \G, X y0 = H ′ ssu \G ss .We define a k-variety Z byZ := H ′ ssu \G

<strong>Manin</strong> <strong>obstruction</strong> <strong>to</strong> strong approximation for homogeneous spaces 35and denote by z 0 ∈ Z(k) the image of e ∈ G(k). We have a commutative diagramof k-varieties:1111 H ′ ssu w H ′ vgH <strong>to</strong>rh j1 G ss pG π Zr π ′ πx x00Z f uq X iy0ψX YG sab 1 1where the first two rows and the last column are exact sequences of connectedalgebraic groups, and the other maps are the natural maps between the differenthomogeneous spaces.The following two lemmas are versions of exact sequence (9) of Theorem 2.8.Lemma 7.8. The following sequence is exact:Br 1,e (G sab ) −→ r∗Br 1,z0 (Z, G) −→ f ∗Br x0 (X y0 ) → 0.Proof. We use the func<strong>to</strong>riality of exact sequence (9) of Theorem 2.8 <strong>to</strong> get thefollowing commutative diagram with exact columns. Here the second column isthe exact sequence (9) for Z = H ′ ssu \G and the third column is exact sequence(9) applied <strong>to</strong> X y0 = H ′ ssu \G ss .1Pic(G ss ) = 0(29)Pic(H ′ ssu )Pic(H ′ ssu )Br 1,e (G sab )∆ G/Zr ∗ Br 1,z0 (Z, G)f ∗∆ G ss /Xy0 ∼ =Br x0 (X y0 )Br 1,e (G sab )π ∗ Zp ∗ Br 1,e (G) Br 1,e (G ss ) = 0 .

36 Mikhail Borovoi and Cyril DemarcheWe have Pic(G ss ) = 0 and Br 1,e (G ss ) = 0 by [38], Lemma 6.9(iv), because G ss issimply connected. From the bot<strong>to</strong>m row of diagram (11) of Corollary 2.12 we getan exact sequenceBr 1,e (G sab ) −→ p∗Br 1,e (G) −→ l∗Br 1,e (G ss ),where l : G ss → G is the canonical embedding. But Br 1,e (G ss ) = 0, therefore thehomomorphism p ∗ : Br 1,e (G sab ) → Br 1,e (G) is surjective. The composition r ◦ fbeing the trivial morphism, the second row of the diagram is a complex. A diagramchase in diagram (29) proves the exactness of the sequence of the lemma.Lemma 7.9. The sequenceis exact.Br 1,x0 (X, G) −→ q∗Br 1,z0 (Z, G) −→ g∗Br 1,e (H <strong>to</strong>r )Proof. We consider the following diagram, in which the middle column and thelast 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 bot<strong>to</strong>m row ofdiagram (11) of Corollary 2.12:Pic(H ′ )(30)Pic(H ′ )w ∗Pic(H ′ ssu )w ∗Pic(H ′ ssu )∆ G/XBr 1,x0 (X, G)∆ G/Zq ∗ Br 1,z0 (Z, G)g ∗∆ H ′ /H <strong>to</strong>rBr 1,e (H <strong>to</strong>r )Br 1,x0 (X, G)π ∗ x 0π ∗ Z Br 1,e (G)v ∗h ∗ Br 1,e (H ′ ) .We prove that the diagram is commutative. In this diagram the two first columnsdefine a commutative diagram by func<strong>to</strong>riality, 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 ∗ ◦πZ ∗ . Let us prove the commutativity of the square in the <strong>to</strong>p right-hand corner,i.e. let us prove that g ∗ ◦∆ G/Z = ∆ H′ /H<strong>to</strong>r. We observe that the following diagramof <strong>to</strong>rsors under H ′ ssu is cartesian:H ′ vhH <strong>to</strong>rgG π Z Z ,

<strong>Manin</strong> <strong>obstruction</strong> <strong>to</strong> strong approximation for homogeneous spaces 37i.e. π −1Z(H<strong>to</strong>r ) = H ′ . In other words, the H <strong>to</strong>r -<strong>to</strong>rsor H ′ is the pullback of theZ-<strong>to</strong>rsor G by the morphism H <strong>to</strong>r −→ g Z. Therefore, if1 → G m → H 1 → H ′ ssu → 1is a central extension representing an element p ∈ Pic(H ′ ssu ) via the isomorphismExt c k(H ′ ssu , G m ) ∼ = Pic(H ′ ssu ), we get a commutative diagram:H 1 (Z, H ′ ssu )∂ H1H 2 (Z, G m )g ∗H 1 (H <strong>to</strong>r , H ′ ssu ) ∂ H 1 H 2 (H <strong>to</strong>r , G m )g ∗such that g ∗ [G] = [H ′ ] in H 1 (H <strong>to</strong>r , H ′ ssu ). We deduce from this diagram thatg ∗ (∂ H1 ([G])) = ∂ H1 ([H ′ ]) in H 2 (H <strong>to</strong>r , G m ), i.e. that ∆ H ′ /H <strong>to</strong>r(p) = g∗ (∆ G/Z (p))in Br(H <strong>to</strong>r ). Therefore the <strong>to</strong>p right-hand square in diagram (30) is commutative.Returning <strong>to</strong> diagram (30), we see that its commutativity and the exactness ofthe last two columns and of the last row imply, via an easy diagram chase, thatthe 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 ofProposition 2.8:Proposition 7.10. Let k be a field of characteristic zero, and1 → H 1 → G → H 2 → 1be an exact sequence of connected linear algebraic groups over k. Let π : Z → Xand π ′ : Y → Z be two morphisms of algebraic varieties such that the compositeY → X is an X-<strong>to</strong>rsor under G, such that the restriction <strong>to</strong> H 1 of the action of Gon Y defines the structure of a Z-<strong>to</strong>rsor under H 1 on Y , and such that Z → X isa <strong>to</strong>rsor under H 2 via the induced action. Then we have a natural exact sequencePic(Z) −→ ϕ1Pic(H 2 ) ∆ Z/X−−−−→ Br 1 (X, Y ) −→ π∗Br 1 (Z, Y ) ϕ′ 2−→ Br 1,e (H 2 ) . (31)If in addition z ∈ Z(k), we have an exact sequencePic(Z) i∗ z−→ Pic(H 2 ) ∆ Z/X−−−−→ Br 1,x (X, Y ) −→ π∗Br 1,z (Z, Y ) i∗ z−→ Br 1,e (H 2 ) , (32)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 <strong>to</strong>p row of diagram (8) applied <strong>to</strong> the <strong>to</strong>rsor Z → X:Pic(Z) −→ ϕ1Pic(H 2 ) ∆ Z/X−−−−→ Br(X) −→ π∗Br(Z) m∗ −p ∗ Z−−−−−→ Br(H 2 × Z) . (33)

38 Mikhail Borovoi and Cyril DemarcheDefine a map ϕ ′ 2 : Br(Z, Y ) → Br 1,e (H 2 ) <strong>to</strong> be the compositionBr 1 (Z, Y ) −−→ m∗Br 1 (Z × H 2 , Y × G) p∗ Z +p∗ H←−−−−−2Br1,e (H 2 , G) ⊕ Br 1 (Z, Y )π H2−−→Br1,e (H 2 , G)where the morphism p ∗ Z + p∗ H 2is an isomorphism by Lemma 3.1, and π H2 is theprojection on<strong>to</strong> the first fac<strong>to</strong>r. Note that Br 1,e (H 2 , G) = Br 1,e (H 2 ) as a consequenceof the injectivity of the homomorphism Br(H 2 ) → Br(G), which comesfrom the exactness of the <strong>to</strong>p row of diagram (11) of Corollary 2.12 and the factthat Pic(G) → Pic(H 1 ) is on<strong>to</strong> (see [38], Remark 6.11.3).Consider the diagramBr 1 (Z, Y ) m∗ −p ∗ Z Br 1 (H 2 × Z, G × Y )ϕ ′ 2Br 1,e (H 2 )∼ =p ∗ Z +p∗ H 2i H Br 1,e (H 2 ) ⊕ Br 1 (Z, Y ) .(34)This diagram is commutative. Since i H is a canonical embedding, we see thatker ϕ ′ 2 = ker(m ∗ − p ∗ Z ).We prove that sequence (31) is exact. We use the exactness of (33). We knowfrom 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 exactvertical lines:00Pic(Z)Br 1 (X, Y )∆ Z/Xϕ 1 Pic(H 2 ) ∆ Z/X Br(X)π ∗π ∗Br 1 (Z, Y )Br(Z) m∗ −p ∗ Z Br(H 2 × Z)Br(Y ) Br(Y ) .(35)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, whichcompletes the proof.7.11. Alternative proof of Lemma 7.9. We consider the exact sequence of linear

<strong>Manin</strong> <strong>obstruction</strong> <strong>to</strong> strong approximation for homogeneous spaces 39algebraic groups1 → H ′ ssu → H ′ → H <strong>to</strong>r → 1and the <strong>to</strong>rsors π Z : G H′ssu−−→ Z and q : Z H<strong>to</strong>r−−→ X. The composition π x0 : G → Xis naturally a <strong>to</strong>rsor under H ′ . Therefore, we can apply Proposition 7.10 <strong>to</strong> getthe exact sequencePic(Z) −→ g∗Pic(H <strong>to</strong>r ) ∆ Z/X−−−−→ Br 1,x0 (X, G) −→ q∗Br 1,z0 (Z, G) −→ g∗Br 1,e (H <strong>to</strong>r ) ,which concludes the proof.7.12. Proof of Proposition 7.6. We have a commutative diagram iBr 1,x0 (X, G)∗ Br x0 (X y0 )q∗f∗Br 1,z0 (Z, G)r g ∗∗Br 1,e (G sab j ∗) Br 1,e (H <strong>to</strong>r ) 0 ,where the last row is exact by assumption, and the two slanted sequences are alsoexact by Lemmas 7.8 and 7.9. A diagram chase shows that the homomorphismBr 1,x0 (X, G) −→ i∗Br x0 (X y0 )is surjective, which completes the proof of Proposition 7.6.For the proof of Proposition 7.4 we need three lemmas.Lemma 7.13. Let G, X be as in Proposition 7.4 and Y := X/G ss . Let ψ : X → Ybe the canonical map. Let x 1 , x 2 ∈ X(k), y i := ψ(x i ), X i := X yi , (i = 1, 2). Letr i : Br 1 (X, G)/Br(k) → Br(X i )/Br(k) be the restriction homomorphisms. Thenthere exists a canonical isomorphism λ 1,2 : Br(X 1 )/Br(k) ∼ → Br(X 2 )/Br(k) suchthat the following diagram commutes:0Br 1 (X, G)/Br(k)idBr 1 (X, G)/Br(k)(36)r 1Br(X 1 )/Br(k)r 2λ 1,2 Br(X 2 )/Br(k) .

40 Mikhail Borovoi and Cyril DemarcheProof. Choose g ∈ G(k) such that x 1 = x 2 .g. Then we have ψ(x 1 ) = ψ(x 2 ).g,hence ψ −1 (ψ(x 1 )) = ψ −1 (ψ(x 2 )).g, thus X 1 = X 2 .g. We obtain commutativediagramsX 2gX 1Br(X)g ∗Br(X)X g X Br(X 1 )g ∗ Br(X 2 ) .By Theorem 5.1 g ∗ : Br(X) → Br(X) is the identity map. Therefore, the followingdiagramBr(X)/Br(k)idBr(X)/Br(k)r 1Br(X 1 )/Br(k)λ 1,2r 2Br(X 2 )/Br(k)µ 1Br(X 1 )µ 2g ∗ Br(X 2 )is commutative.Here r 1 and r 2 are surjective by Proposition 7.6, while µ 1 and µ 2 are injectiveby 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 <strong>to</strong>psquare of this new diagram is the desired diagram (36).Lemma 7.14. Let X := H\G, where G is a simply connected semisimple k-groupover a field k of characteristic 0, and H ⊂ G is a connected k-subgroup such thatH <strong>to</strong>r = 1. Then Br(X)/Br(k) is finite and the canonical homomorphismis injective.Br(X)/Br(k) → Br(X)Proof. Let x 0 denote the image of e ∈ G(k) in X(k). We have a canonical isomorphismBr(X)/Br(k) ∼ = Br x0 (X).By Theorem 2.8 we have a canonical exact sequencePic(G) → Pic(H) → Br 1,x0 (X, G) → Br 1,e (G),where by [38], Lemma 6.9(iv), we have Pic(G) = 0 and Br 1,e (G) = 0. Moreover,by [18] we have Br(G) = 0, hence Br 1,x0 (X, G) = Br x0 (X). We obtain a canonicalisomorphism Br x0 (X) ∼ = Pic(H), func<strong>to</strong>rial in k.

<strong>Manin</strong> <strong>obstruction</strong> <strong>to</strong> strong approximation for homogeneous spaces 41We have H = H ssu , hence Pic(H) ∼ = Pic(H ss ). In the commutative diagramBr(X)/Br(k)∼ =Pic(H ss )Br(X)∼ = Pic(H ss )the right vertical arrow is clearly injective, hence so is the left one.Lemma 7.15. Let X := H\G, where G is a connected k-group over a numberfield k, and H ⊂ G is a connected linear k-subgroup. Let M ⊂ Br 1 (X, G)/Br(k) bea finite subset. Let y := (y f , y ∞ ) ∈ X(A f )×X(k ∞ ) = X(A) be a point orthogonal<strong>to</strong> M with respect <strong>to</strong> <strong>Manin</strong> pairing. Let U ∞ denote the connected component ofy ∞ in X(k ∞ ). Then there exists an open neighbourhood U f ⊂ X(A f ) of y f suchthat U f × U ∞ is orthogonal <strong>to</strong> M.Proof. It is an immediate consequence of Lemma 6.2, using the finiteness of M.7.16. Recall that X = H\G. Let x 0 be the image of e ∈ G(k) in X(k). SetX 0 := x 0 .G ss . By assumption H ∩ G ss = H ssu . By Lemma 7.14 Br(X 0 )/Br(k) isa finite group. By Proposition 7.6 the map Br 1 (X, G)/Br(k) → Br(X 0 )/Br(k) issurjective. We choose a finite subset M ⊂ Br 1 (X, G)/Br(k) such that M surjectson<strong>to</strong> Br(X 0 )/Br(k).Now let x 1 ∈ X(k) be any other point. Set X 1 := x 1 .G ss . It follows fromLemma 7.13 that M ⊂ Br 1 (X, G)/Br(k) surjects on<strong>to</strong> Br(X 1 )/Br(k).7.17. Proof of Proposition 7.4. Let x ∈ X(A) be orthogonal <strong>to</strong> Br 1,x0 (X, G). LetU f X be an open neighbourhood of the Af -part x f of x. Let U X be the specialneighbourhood of x defined by U f X .Let M ⊂ Br 1 (X, G)/Br(k) be as in 7.16. Then x is orthogonal <strong>to</strong> M. By Lemma7.15 there exists an open neighbourhood U f of x f such that the correspondingspecial neighbourhood U of x in X(A) is orthogonal <strong>to</strong> M. We may assume thatU f X ⊂ U f , then U X ⊂ U , hence U X is orthogonal <strong>to</strong> 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 <strong>to</strong> Br 1,x0 (X, G), we see by func<strong>to</strong>riality thaty is orthogonal <strong>to</strong> Br 1,y0 (Y, G sab ). Clearly there is a semiabelian variety G ′ suchthat Y is a (trivial) principal homogeneous space of G ′ . We have a morphism ofpairs (Y, G sab ) → (Y, G ′ ), hence a homomorphism Br 1 (Y, G ′ ) → Br 1 (Y, G sab ). ButBr 1 (Y, G ′ ) = Br 1 (Y ), hence y is orthogonal <strong>to</strong> the group Br 1 (Y ). As in the Firstreduction, see 7.1, we define U fY:= ψ(U f X) and we construct the correspondingspecial open neighbourhood U Y of y. By [7], Lemma A.2, for any v ∈ Ω ∞ wehave ψ(U X,v ) = U Y,v . We see that ψ(U X ) is an open subset of Y (A) of theform U f × U ∞ , where U f ⊂ Y (A f ) is an open subset and U ∞ = ∏ v∈Ω ∞U Y,v ,where U Y,v ⊂ Y (k v ) is the connected component of y v . Then U ∞ is the connected

42 Mikhail Borovoi and Cyril Demarchecomponent of x ∞ in X(k ∞ ). Now since Y is a <strong>to</strong>rsor of the semiabelian varietyG ′ with finite Tate–Shafarevich group, by [22], Theorem 4, there exists a k-pointy 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 apoint x ′ = (x ′ v) ∈ V . In particular, X y1 (k v ) ≠ ∅ for any v ∈ Ω r . The varietyX y1 is a homogeneous space of G ss with geometric stabilizer H ∩ G ss = H ssu . Thegroup G ss is semisimple simply connected by (iii). The group H ssu is connectedand character-free, i.e. (H ssu ) <strong>to</strong>r = 1. By [4], Corollary 7.4, the fact that X y1has points in all real completions of k is enough <strong>to</strong> 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 isorthogonal <strong>to</strong> M, we see that V ⊂ U X is orthogonal <strong>to</strong> M. Since M surjects on<strong>to</strong>Br(X y1 )/Br(k), see 7.16, we see that V is orthogonal <strong>to</strong> Br(X y1 ). By Theorem 6.1(due <strong>to</strong> Colliot-Thélène and Xu) there is a point of the form x 1 .g S in V , wherex 1 ∈ X y1 (k) and g S ∈ G ss (k S ). It follows that the set V .G ss (k S ) contains a k-poin<strong>to</strong>f X y1 . Clearly V .G ss (k S ) ⊂ U X .G ss (k S ). Thus U X .G ss (k S ) contains a k-poin<strong>to</strong>f X, which shows that the pair (X, G) has Property (P S ). This completes theproof 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 <strong>to</strong>rusH <strong>to</strong>r , i.e. an exact sequence of k-<strong>to</strong>ri0 → H <strong>to</strong>r → P → Q → 0where P is a quasi-trivial <strong>to</strong>rus and Q is a coflasque <strong>to</strong>rus. Recall that a <strong>to</strong>rusis coflasque if for any field extension K/k we have H 1 (K, ̂Q) = 0, where ̂Q isthe character group of Q. Consider the k-group F := G × P . The group H mapsdiagonally in<strong>to</strong> F , and we can consider the quotient homogeneous space W := H\Fof F . There is a natural morphism t: W → X. We have F abvar = G abvar , henceX(F abvar ) is finite. We have a canonical homomorphism H <strong>to</strong>r → F sab , andthis homomorphism is clearly injective. Let us prove the following fact, which isnecessary <strong>to</strong> apply Proposition 7.4 <strong>to</strong> the homogeneous space W of F .Lemma 7.19. With the notation of Construction 7.18, the pullback homomorphismis surjective.Br 1,e (F sab ) → Br 1,e (H <strong>to</strong>r )

<strong>Manin</strong> <strong>obstruction</strong> <strong>to</strong> strong approximation for homogeneous spaces 43Proof. By definition, we have the following exact commutative diagram:00(37)G sabG sab0 H <strong>to</strong>r F sab = G sab × P S0 H <strong>to</strong>r P Q 000 0where S is defined <strong>to</strong> be the quotient F sab /H <strong>to</strong>r and all the maps are the naturalones. The group S is a semi-abelian variety. By assumption Q is a coflasque <strong>to</strong>rus.ThereforeH 3 (k, ̂Q) ∼ ∏= H 3 (k v , ̂Q) ∼ = ∏H 1 (k v , ̂Q) = 0 .v realv realDenote by M F := [0 → F sab ] (resp. M S := [0 → S]) the 1-motive (in degrees −1and 0) associated <strong>to</strong> the semi-abelian variety F sab (resp. S), and by M F ∗ (resp.M S ∗ ) its Cartier dual (see [25], Section 1 page 97 for the definition of the Cartierdual of a 1-motive). We call a sequence of 1-motives over k exact if the associatedsequence of complexes of fppf sheaves on Spec(k) is exact.Considering diagram (37) as an exact diagram in the category of 1-motives overk, we get a commutative exact diagram of 1-motives:0 [0 → H <strong>to</strong>r ] M FM S 00 [0 → H <strong>to</strong>r ] [0 → P ] [0 → Q] 0 .We can dualize this diagram <strong>to</strong> get the following commutative diagram of 1-motives:0 [ ̂Q → 0] [ ̂P → 0] [Ĥ<strong>to</strong>r → 0] 00 M S∗ M F∗ [Ĥ<strong>to</strong>r → 0] 0 .

44 Mikhail Borovoi and Cyril DemarcheThis diagram is exact as a diagram of complexes of fppf sheaves since the 1-motive [0 → H <strong>to</strong>r ] is associated <strong>to</strong> a k-<strong>to</strong>rus (see [2], Remark 1.3.4). Hence thisexact diagram induces a commutative exact diagram in hypercohomology:H 2 (k, ̂P )H 2 (k, Ĥ<strong>to</strong>r ) H 3 (k, ̂Q) = 0H 1 (k, M F ∗ ) H 2 (k, Ĥ<strong>to</strong>r ) H 2 (k, M S ∗ ) .Therefore the map H 1 (k, M F ∗ ) → H 2 (k, Ĥ<strong>to</strong>r ) is surjective. But by [26], beginningof Section 4, there are natural maps ι F : H 1 (k, M F ∗ ) → Br 1,e (G sab ) and ι H <strong>to</strong>r :H 2 (k, Ĥ<strong>to</strong>r ) → Br 1,e (H <strong>to</strong>r ) such that the second map is the canonical isomorphismof [38], Lemma 6.9(ii). Hence we get a commutative diagramH 1 (k, M F ∗ )H 2 (k, Ĥ<strong>to</strong>r )ι F∼ =ι H <strong>to</strong>rBr 1,e (G sab ) Br 1,e (H <strong>to</strong>r ) .Since the <strong>to</strong>p map is surjective, so is the bot<strong>to</strong>m one.Let x ∈ X(A) be a point, and assume that x is orthogonal <strong>to</strong> Br 1 (X, G). Themap t: W → X is a <strong>to</strong>rsor under a quasi-trivial <strong>to</strong>rus, and we want <strong>to</strong> lift x <strong>to</strong>some w ∈ W (A) orthogonal <strong>to</strong> Br 1 (W, F ). To do this, we need the followinglemma.Lemma 7.20. With the above notation, the <strong>to</strong>rsor t: W = H\F → X under thequasi-trivial <strong>to</strong>rus P induces a canonical exact sequence0 → Br 1,x0 (X, G) −→ t∗Br 1,w0 (W, F ) −→ ϕ Br 1,e (P ),where x 0 is the image of e ∈ G(k) and w 0 is the image of e ∈ F (k).Proof. We first define the map ϕ of the lemma. The pullback homomorphism:Br(W ) π∗ W−−→ Br(F ) sends the subgroup Br 1,w0 (W, F ) in<strong>to</strong> Br 1,e (F ). But F = G×P ,hence thanks <strong>to</strong> [38], Lemma 6.6, we have a natural isomorphism Br 1,e (F ) ∼ =Br 1,e (G) ⊕ Br 1,e (P ). We compose this map with the second projectionπ P : Br 1,e (G) ⊕ Br 1,e (P ) → Br 1,e (P ).So we define a morphism ϕ := pr P ◦π ∗ W : Br 1,w 0(W, F ) → Br 1,e (P ). The morphismt ∗ : Br 1,x0 (X, G) → Br 1,w0 (W, F ) in the lemma is induced by the morphism of pairst: (W, F ) → (X, G). By Theorem 2.8 we have an exact sequencePic(P ) → Br(X) −→ t∗Br(W ) .

<strong>Manin</strong> <strong>obstruction</strong> <strong>to</strong> strong approximation for homogeneous spaces 45The <strong>to</strong>rus P is quasi-trivial, therefore by Lemma 6.9(ii) of [38], the group Pic(P )is trivial, so the homomorphism t ∗ : Br(X) → Br(W ) is injective. In particular,the homomorphism t ∗ : Br 1,x0 (X, G) → Br 1,w0 (W, F ) in Lemma 7.20 is injective.Therefore, it just remains <strong>to</strong> prove the exactness of the sequence of the lemma atthe term Br 1,w0 (W, F ). Consider the following diagramPic(H) ∆ G/X Br 1,x0 (X, G)Br 1,e (G)Br 1,e (H)t ∗Pic(H) ∆ F/W Br 1,w0 (W, F )Br 1,e (F )Br 1,e (H)ϕBr 1,e (P ) Br 1,e (P )where the rows come from Theorem 2.8. The commutativity of this diagram isa consequence of the func<strong>to</strong>riality of the exact sequences of Theorem 2.8 and ofthe definition of the map ϕ. We conclude the proof of the exactness of the secondcolumn of the diagram by an easy diagram chase, using the exactness of the twofirst rows and that of the third column (see Corollary 2.12).Corollary 7.21. With the above notation, if x ∈ X(A) is orthogonal <strong>to</strong> Br 1 (X, G),then there exists w ∈ W (A) such that t(w) = x and w is orthogonal <strong>to</strong> Br 1 (W, F ).Proof. Consider the exact sequence of Lemma 7.20. Taking dual groups, we obtainthe dual exact sequenceBr 1,e (P ) D−−→ ϕDBr 1,w0 (W, F ) Dt ∗−−→ Br 1,x0 (X, G) D → 0, (38)Let m X,G (x) ∈ Br 1,x0 (X, G) D denote the homomorphism b ↦→ 〈b, x〉: Br 1,x0 (X, G)→ Q/Z. By assumption m G,X (x) = 0. We wish <strong>to</strong> lift x <strong>to</strong> some w ∈ W (A) suchthat m W,F (w) = 0.Since H 1 (k v , P ) = 0 for all v, we can lift x <strong>to</strong> some point w ′ ∈ W (A) such thatt(w ′ ) = x (we use also Lang’s theorem and Hensel’s lemma). Then t ∗ (m W,F (w ′ )) =m X,G (x) ∈ Br 1,x0 (X, G) D . Since m X,G (x) = 0, we see from (38) that m W,F (w ′ ) =ϕ D (ξ) for some ξ ∈ Br 1,e (P ) D ∼ = Br a (P ) D . Let p ∈ P (A). By Corollary 3.5 wehave〈b W , w ′ .p〉 = 〈b W , w ′ 〉 + 〈ϕ(b W ), p〉for any b W ∈ Br 1,w0 (W, F ). This means thatm W,F (w ′ .p) = m W,F (w ′ ) + ϕ D (m P (p))But we have seen that m W,F (w ′ ) = ϕ D (ξ) for some ξ ∈ Br a (P ) D . Now it followsfrom Lemma 4.1 that there exists p ∈ P (A) such that m P (p) = −ξ. Thenm W,F (w ′ .p) = ϕ D (ξ) + ϕ D (−ξ) = 0 .

46 Mikhail Borovoi and Cyril DemarcheWe set w := w ′ .p ∈ W (A), then m W,F (w) = 0 and t(w) = x.7.22. We can now resume the proof of Theorem 7.3. We use Construction 7.18.Let U f X ⊂ X(Af ) be an open neighbourhood of x f . Let U X ⊂ X(A) be thecorresponding special neighbourhood of x. SetU f W := t−1 (U f X ) ⊂ W (Af ).For v ∈ Ω ∞ let U W,v be the connected component of w v in W (k v ), then by[7], Lemma A.2, we have t(U W,v ) = U X,v . Set U W,∞ := ∏ v∈Ω ∞U W,v . SetU W := U f W × U W,∞, then U W is the special open neighbourhood of w defined byU f W , and t(U W ) ⊂ U X .The pair (W, F ) of F satisfies the hypotheses of Proposition 7.4 (see Lemma7.19), so by that proposition, there is a point w 1 ∈ W (k) ∩ U W .F sc (k S ). Notethat F sc = G sc . Set x 1 := t(w 1 ), then x 1 ∈ X(k) ∩ U X .G sc (k S ). Thus the pair(X, G) has Property (P S ).This completes the proofs of Theorem 7.3 and proves the nontrivial inclusionof Theorem 1.4, that is, that any element of (X(A) • ) Br1(X,G) lies in the closureof the set X(k).G scu (k Sf ). The argument in the proof of the trivial inclusion ofTheorem 6.1 also proves the trivial inclusion of Theorem 1.4, that is, that eachelement of this closure is orthogonal <strong>to</strong> Br 1 (X, G). This completes the proof ofMain Theorem 1.4.8 The algebraic <strong>Manin</strong> <strong>obstruction</strong>In this section we prove Theorem 1.7 about the algebraic <strong>Manin</strong> <strong>obstruction</strong> (“algebraic”means coming from Br 1 (X)). We prove this result without using theresult of Colliot-Thélène and Xu (Theorem 6.1 or [11], Theorem 3.7(b)).8.1. Before proving Theorem 1.7, we need <strong>to</strong> prove a special case – an analogueof Theorem 6.1. In [6], the first-named author defined, for any connected groupH over a field k of characteristic 0, a Galois module π 1 (H), an abelian groupHab 1 (k, H) and a canonical abelianization mapab 1 : H 1 (k, H) → H 1 ab(k, H)(see also [8] in any characteristic). These π 1 (H), H 1 ab (k, H) and ab1 are func<strong>to</strong>rialin H.Now let k be a number field. Set Γ := Gal(k/k), Γ v := Gal(k v /k v ). We regardΓ v as a subgroup of Γ.For v ∈ Ω f we defined in [6], Proposition 4.1(i), a canonical isomorphismλ v : H 1 ab(k v , H) ∼ → (π 1 (H) Γv ) <strong>to</strong>rs ,

<strong>Manin</strong> <strong>obstruction</strong> <strong>to</strong> strong approximation for homogeneous spaces 47where π 1 (H) Γv denotes the groups of coinvariants of Γ v in π 1 (H), and ( ) <strong>to</strong>rsdenotes the <strong>to</strong>rsion subgroup. Here we set λ ′ v := λ v .For v ∈ Ω ∞ we defined in [6], Proposition 4.2, a canonical isomorphismλ v : H 1 ab(k v , H) ∼ → H −1 (Γ v , π 1 (H)).Here we define a homomorphism λ ′ v as the compositionλ ′ v : H 1 ab(k v , H)λ v−−→ H −1 (Γ v , π 1 (H)) ↩→ (π 1 (H) Γv ) <strong>to</strong>rs .For any v ∈ Ω we define the Kottwitz map β v as the compositionβ v : H 1 (k v , H) −−→ ab1Hab(k 1 v , H)λ ′ v−−→ (π 1 (H) Γv ) <strong>to</strong>rs .This map β v is func<strong>to</strong>rial 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 andfunc<strong>to</strong>rial in H bijection β v : H 1 (k v , H) ∼ → (π 1 (H) Γv ) <strong>to</strong>rs .For any v ∈ Ω we define a map µ v as the compositionµ v : H 1 (k v , H)β v−−→ (π 1 (H) Γv ) <strong>to</strong>rscor v−−→ (π 1 (H) Γ ) <strong>to</strong>rs , (39)where cor v is the obvious corestriction map.We write ⊕ v H1 (k v , H) for the set of families (ξ v ) v∈Ω such that ξ valmost all v. We define a map= 1 forµ := ∑ v∈Ωµ v : ⊕ v∈ΩH 1 (k v , H) → (π 1 (H) Γ ) <strong>to</strong>rs .Proposition 8.2 (Kottwitz [30], Proposition 2.6, see also [6], Theorem 5.15).The kernel of the map µ is equal <strong>to</strong> the image of the localization map H 1 (k, H) →⊕v H1 (k v , H).Proposition 8.3. 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 <strong>to</strong>r =1). Set X := H\G. Let S be a finite set of places of k containing at least onenonarchimedean place. Then any orbit of G(A S ) in X(A S ) contains a k-point.Proof. Write M := π 1 (H). First we prove that the mapµ S = ∑ ∏µ v : ker[H 1 (k v , H) → H 1 (k v , G)] → (M Γ ) <strong>to</strong>rsv∈Sv∈Sis surjective. Since H is geometrically character-free, the group M = π 1 (H) isfinite, and therefore (M Γ ) <strong>to</strong>rs = M Γ and (M Γv ) <strong>to</strong>rs = M Γv . In this case themap cor v is the canonical map M Γv → M Γ , which is clearly surjective. Let

48 Mikhail Borovoi and Cyril Demarchew ∈ S be a nonarchimedean place, then the map β w is bijective. It followsthat 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 themap µ w : ker[H 1 (k w , H) → H 1 (k w , G)] → M Γ is surjective. Now it is clear thatthe map µ S = ∑ v∈S µ v is surjective.We prove the proposition. We must prove that the localization mapX(k)/G(k) → X(A S )/G(A S )is surjective. In the language of Galois cohomology, we must prove that the localizationmapker[H 1 (k, H) → H 1 (k, G)] → ⊕ v /∈Sker[H 1 (k v , H) → H 1 (k v , G)]is surjective.Let ξ S = (ξ v ) ∈ ⊕ v /∈S ker[H1 (k v , H) → H 1 (k v , G)]. Set s := ∑ v /∈S µ v(ξ v ) ∈π 1 (H) Γ . Since the map µ S is surjective, there exists an element ξ S in the product∏v∈S ker[H1 (k v , H) → H 1 (k v , G)] such that ∑ v∈S µ v(ξ v ) = −s. Setξ := (ξ S , ξ S ) ∈ ⊕ v∈Ωker[H 1 (k v , H) → H 1 (k v , G)] ⊂ ⊕ v∈ΩH 1 (k v , H),then µ(ξ) = ∑ v∈Ω µ v(ξ v ) = s + (−s) = 0. By Proposition 8.2 there exists aclass ξ 0 ∈ H 1 (k, H) with image ξ in ⊕ v∈Ω H1 (k v , H). Since the Hasse principleholds⊕for G, we have ξ 0 ∈ ker[H 1 (k, H) → H 1 (k, G)]. Since the image of ξ 0 inv∈Ω H1 (k v , H) is (ξ S , ξ S ), we see that the image of ξ 0 in ⊕ v /∈S H1 (k v , H) is ξ S .Thus ξ S lies in the image of ker[H 1 (k, H) → H 1 (k, G)].Theorem 8.4. Let G be a simply connected k-group over a number field k, andlet H ⊂ G be a connected geometrically character-free subgroup (i.e. H <strong>to</strong>r =1). Set X := H\G. Let S be a finite set of places of k containing at least onenonarchimedean place v 0 . Assume that G ss (k) is dense in G ss (A S ). Then X hasstrong 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. Thenthere 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 ). Moreprecisely, there exists y v0 ∈ X(k v0 ) such that the point x ′ ∈ X(A) defined byx ′ v 0:= y v0 and x ′ v := x v for v ≠ v 0 belongs <strong>to</strong> the closure of the set X(k).G(k S ) inX(A) for the adelic <strong>to</strong>pology.Proof. Set Σ := {v 0 }. We denote by x Σ ∈ X(A Σ ) and x S ∈ x(A S ) the correspondingprojections of x. By Proposition 8.3 applied <strong>to</strong> the finite set of placesΣ, there exists a k-point x ′ 0 ∈ X(k) ∩ x Σ .G(A Σ ). Let y v0 := (x ′ 0) v0 ∈ X(k v0 )

<strong>Manin</strong> <strong>obstruction</strong> <strong>to</strong> strong approximation for homogeneous spaces 49and define x ′ ∈ X(A) as in the theorem. Then there exists g ∈ G(A) such thatx ′ 0.g = x ′ in X(A). Let U ⊂ X(A) be an open neighbourhood of x ′ . Since theorbit x ′ 0.G(A) ⊂ X(A) is open (because H is connected) and contains x ′ , we mayassume that U ⊂ x ′ 0.G(A).By assumption G(k).G(k S ) is dense in G(A). It follows that there exists g 0 ∈G(k) and g S ∈ G(k S ) such that x ′′ := x ′ 0.g 0 .g S belongs <strong>to</strong> U . Set x 0 := x ′ 0.g 0 ∈X(k), then x ′′ = x 0 .g S . We see that x ′′ ∈ X(k).G(k S )∩U . Therefore, we concludethat x ′ lies in the closure of X(k).G(k S ).Concerning the infinite places, for v ∈ Ω ∞ ∩ S we have x 0 ∈ x v .G(k v ), becausex ′ 0 ∈ x v .G(k v ). Since G is simply connected, the group G(k v ) is connected (see[34], Theorem 5.2.3), hence the image of x 0 in X(k v ) is contained in the connectedcomponent of x v in X(k v ).8.5. Proof of Theorem 1.7. To prove this theorem, we can follow the proof ofTheorem 1.4 <strong>to</strong> make reductions, so that we may assume the following:(i) G u = {1},(ii) H ⊂ G lin , i.e. H is linear,(iii) G ss is simply connected,(iv) X(G abvar ) is finite.(v) the homomorphism H <strong>to</strong>r → G sab is injective.Set Σ ′ := Ω ∞ ∪ {v 0 }. Let UXΣ′ ⊂ X(AΣ′ ) be an open neighbourhood of theprojection x Σ′ ∈ X(A Σ′ ) of x. Set U f Σ′X:= UX× X(k v 0). Let U X be the specialopen neighbourhood of x in X(A) defined by U f X . Set Y := Gsab /H <strong>to</strong>r , andconsider the canonical morphism ψ : X → Y . Set y := ψ(x) ∈ Y (A), then y isorthogonal <strong>to</strong> the group Br 1 (Y ) for the <strong>Manin</strong> pairing. Hence by [22], Theorem 4,there exists y 0 ∈ Y (k)∩ψ(U X ). Set X y0 := ψ −1 (y 0 ) ⊂ X and V := X y0 (A)∩U X .Then V is open and non-empty since y 0 ∈ ψ(U X ). As in the proof of Proposition7.4, we know that X y0 is a homogeneous space of the semisimple simply connectedgroup G ss = G sc , with connected character-free geometric stabilizers, and with a k-point. Therefore Theorem 8.4 implies that X y0 (k).G sc (k S ) ∩ V ≠ ∅. In particular,the set X(k).G sc (k S ) ∩ U X is non-empty. Set S ′ := S {v 0 }, Sf ′ := S′ ∩ Ω f ,U {v0}X:= UXΣ′× U X,∞, then it follows that the set X(k).G sc (k S ′) ∩ U {v0}X⊂.Gsc (k S ′f), we obtain easilyX(A {v0} ) is non-empty. Since U {v0}Xthat the set X(k).G sc (k S ′f) ∩ U {v0}Xproof of Theorem 1.7..Gsc (k S ′) = U {v0}X⊂ X(A {v0} ) is non-empty. This completes theAcknowledgements. The first-named author is grateful <strong>to</strong> the Tata Instituteof Fundamental Research, Mumbai, where a part of this paper was written, forhospitality and good working conditions. The second-named author thanks DavidHarari for many helpful suggestions. Both authors thank the referee for his/hercomments and suggestions.

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