Local-global principles for homogeneous spaces of reductive groups over global function fields

Let $K$ be a global field of positive characteristic. We prove that the Brauer-Manin obstructions to the Hasse principle, to weak approximation and to strong approximation are the only ones for homogeneous spaces of reductive groups with reductive stabilizers. The methods involve abelianization techniques and arithmetic duality theorems for complexes of tori over K.


Introduction
Let K be a global field of characteristic p ≥ 0 and let A K denotes the ring of adèles of K. Let G be a reductive group over K, and X be a homogeneous space of G.We are interested in rational points on X, and more precisely, on various local-global principles associated to X: does X satisfy the Hasse principle, i.e. does X(A K ) = ∅ imply X(K) = ∅?If not, can we explain the failure using the so-called Brauer-Manin obstruction to the Hasse principle?Assuming that X(K) = ∅, can we estimate the size of X(K) by studying the so-called weak and strong approximation on X (with a Brauer-Manin obstruction if necessary), i.e. the closure of the set X(K) in the topological space X(A S K ), where S is a (not necessarily finite) set of places of K and A S K is the ring of S-adèles (with no components in S)?The answer to those questions is known in the case where K is a number field, provided that the stabilizers of points in X are connected (see [Bor3] and [BD]).
In the case of a global field of positive characteristic, the answer is known for semisimple simply connected groups (thanks to works by Harder, Kneser, Chernousov, Platonov, Prasad), but the general case is essentially open (see [Ros2], Theorem 1.9 for some related results).In this paper, we deal with these questions when both G and the stabilizers are smooth, connected and reductive.
Several new ingredients are needed to obtain our results: • To show that the Brauer-Manin obstruction to the Hasse principle is the only one (see Theorem 2.5 for the precise result), one has to use Poitou-Tate duality for complexes of tori in positive characteristic (proven in [DH2]) and a (non straightforward) compatibility result between Brauer-Manin and Poitou-Tate pairings.
• The statement on weak approximation (Theorem 4.1) relies on some part of Poitou-Tate exact sequence (which is established in [DH2]) for a certain complex of tori, and on abelianization techniques (namely Lemma 4.3).Beforehand we define in section 3 a new abelianization map associated to a homogeneous space X = G/H as above, and prove a rather intricate compatibility formula (Theorem 3.7).
• Theorem 5.8 presents the obstruction to strong approximation.As in the number field case (settled in [BD]), it is related to the Brauer-Manin pairing, but there are two important differences.The first one is that there is an additional term in the exact sequence describing the obstruction, which reflects the fact that the global reciprocity map of class field theory is not surjective in positive characteristic.The second difference is that a fibration method like in loc.cit.would probably not work here (see for instance section 5 in loc.cit.).Therefore, one should again rely on abelianization techniques (in particular the compatibility formula of Theorem 3.7 in its full generality, and not only for elements of the algebraic Brauer group Br 1 X).An important role is also played by duality theorems for complex of tori, some of them extending results of [DH2].
Notation and conventions.
In the whole article (except in section 3, where k is an arbitrary field), we consider a finite field k and a projective, smooth k-curve E. We set K = k(E), which is a global function field of characteristic p, and fix a separable closure K s of K.The absolute Galois group Gal(K s /K) of K is denoted by Γ K .Denote by Ω K the set of all places of K; for every v ∈ Ω K , we will identify the Brauer group Br K v of the completion K v to Q/Z thanks to local class field theory.For every K-variety X, we set X = X × K K s .The (cohomological) Brauer group of X is denoted Br X, and we set Br 1 X := ker[Br X → Br X].
We still denote by Br K the image of Br K in Br X, even though the map Br K → Br X is not necessarily injective if X has no rational point.Notation like H i (K, C) for a commutative K-group scheme C (resp. a bounded complex of commutative K-group schemes) always denotes ffpf cohomology (resp.fppf hypercohohomology) of C. It coincides with étale (=Galois) cohomology when C (resp.every group scheme occurring in C) is smooth.For every finite set of places S of K, we set where the direct limit runs over all finite subsets S of Ω K .The Pontryagin dual A D of a topological group A is the group of continuous homomorphism from A to Q/Z (if topology is not specified, we assume that A is discrete).
Let G be a reductive group (always meaning: smooth connected reductive) over K. Let G ss denote the derived subgroup of G and G sc the simply connected cover of G ss , together with the obvious morphism ρ : G sc → G. Set G tor = G/G ss (it is the maximal toric quotient of G).Let T sc ⊂ G sc and T G ⊂ G be maximal tori such that ρ(T sc ) ⊂ T G .Let C G be the complex C G := [T sc ρ − → T G ], with T G in degree 0. Following Borovoi (cf. [Bor2] in characteristic zero), we have a natural map of Galois (hyper)cohomology sets: which is functorial in K.There is an exact sequence of K-group schemes where µ G is a finite K-group scheme of multiplicative type.It induces an exact triangle We denote by Z G the center of a reductive group G.

Hasse principle for homogeneous spaces
We start with extending a well-known result on the abelianization maps to the positive characteristic case.
Proposition 2.1.There is a natural exact sequence of groups

Proof:
Let F be K or K v .By construction, one has a short exact sequence of groups and pointed sets For F = K, following [Har2], Satz A, the set H 1 (F, G sc ) is trivial, hence the map ab 1 G has trivial kernel and the first sequence in the statement is exact.A twisting argument implies that the map ab 1 G is even injective.For the local statement (F = K v ), the injectivity (and exacteness of the first sequence) follows from [BT],Theorem 4.7.(ii).
The proof of the surjectivity is an adaptation of the proofs of [Bor2], Theorems 5.4 and 5.7, except that the existence of anisotropic maximal tori over local fields of positive characteristic is provided by [DeB], Lemma 2.4.1 (see also [Ros1], Proposition 4.4).
We are interested in the Hasse principle for homogeneous spaces under G, with reductive (recall that by definition this includes smoothness and connectedness) geometric stabilizers.Following Raynaud (see [Ray], definition VI.1.1 and Proposition VI.1.2),if G is a smooth group scheme over K, a homogeneous space of G is a smooth K-scheme X with an action of G, such that for any x ∈ X(K s ) (such an x exists since X is smooth), the stabilizer of x in G := G × K K s is a finite type (over K s ) subgroup scheme H x of G and X is isomorphic to the quotient G/H x .Definition 2.2.Let X be a homogeneous space of a reductive group G with reductive stabilizer H = H x .Let L X be the K-kernel defined by X and σ X ∈ H 2 (K, L X ) be the Springer class, that is the class of the gerbe associated to X (cf.[FSS], §5.2. or [Bor1], §7.7 in characteristic zero).By assumption, the stabilizers are reductive, hence following loc.cit., there exists a K-torus T , which is a K-form of H tor , and a natural map of marked sets which is functorial in K. On the other hand we have the class constructed by Borovoi in [Bor4] (where characteristic zero is assumed, but not used in the definition of η X ).There is an exact sequence in Galois (hyper)-cohomology

Proof:
The method is similar to the one used in the proof of [BvH2], Th. 9.6.We start with the case when G itself is a torus.Then T is a subtorus of G and Assume now that G is an arbitrary reductive group but satisfies the additional hypothesis: (*) The canonical morphism T → G tor is injective.
Then Y := X/G ss is a homogenous space of G tor with stabilizer (defined over K in this case) T .The Springer class σ Y ∈ H 2 (K, T ) is (by construction) just ab 2 X (σ X ), and it is also the image of η Y ∈ H 1 (K, [T → G tor ]) in H 2 (K, T ) by the first case.But η Y = η X by functoriality of the class η X ([Bor4], §1.6), whence the result when (*) is satisfied.
We now deal with the general case.By [BvH2], Prop 9.9 (whose proof is identical in characteristic p thanks to the assumption that G and H are reductive), there exists a homogeneous space Z of a K-group F = G × P , where P is a quasi-trivial torus, such that: the homogeneous space Z satisfies (*) and there is a K-morphism π : Z → X, compatible with the respective actions of F , G (via the projection F → G), which makes Z an X-torsor under P .In particular the G-homogeneous space Z still has geometric stabilizer H, with L Z = L X and σ Z = σ X .Since Proposition 2.3 holds for Z (because its satisfies (*)), it also holds for X by functoriality of the class η X and commutativity of the diagrams Remark 2.4.The previous proposition holds over an arbitrary field.Recall also that the existence of a K-point on X implies that the class σ X is neutral, as well as the vanishing of η X .
Theorem 2.5.Let G be a reductive group over K. Then the Brauer-Manin obstruction to the Hasse principle is the only one for homogeneous spaces of G with reductive stabilizers.More precisely, such a torsor X has a rational point if and only if X has an adelic point orthogonal to the subgroup for the Brauer-Manin pairing : Recall that by global class field theory, the sum v∈ΩK α(P v ) is zero for every element α ∈ Im[Br K → Br X], hence the Brauer-Manin pairing (2) is well defined.Also, for α ∈ B(X), the element α, (P v ) BM is independent of the choice of (P v ) ∈ X(A K ), because the localisation α v ∈ Br X Kv of α is a constant element (i.e. it comes from Br K v ) for every place v.

Proof:
Fix a point x ∈ X(K s ) and let H = H x be the stabilizer of x in G. Up to replacing G by a flasque resolution ( [CT]) where S is a flasque torus (which is central in G ′ ) and G ′ is a quasi-trivial group (that is: extension of a quasi-trivial torus by a semisimple simply connected group), we can assume that the group G itself is quasitrivial.Indeed X is also a homogenous space of G ′ such that the stabilizer of x is connected (it is an extension of H x by S K s ).In particular Pic G = 0 and the group of characters G tor of G tor is a permutation Galois module ( [CT], Prop 2.2).
Since H 1 (K, G tor ) = 0 (the torus G tor being quasi-trivial), we have by exact sequence (1) that the canonical map H 1 (K, [T → G tor ]) → H 2 (K, T ) is injective.Besides the class η X (viewed as an element of H 2 (K, T )) is just ab 2 X (σ X ) by Proposition 2.3.Assume that X(A K ) = ∅.Then the class σ X is neutral at every place v of K, which implies that η X = ab 2 X (σ X ) ∈ X 2 (K, T ).The Brauer-Manin pairing defines a morphism β By [BDH], §4, there is a complex UPic (X) (up to a shift this is the complex UPicX of [BvH2]; we will recover this complex in section 3) such that Br 1 X/Br K is isomorphic to H 1 (K, UPic (X)) (this is valid over any field K such that H 3 (K, G m ) = 0).Moreover by [BvH2], Th. 5.8 (whose proof is still valid in characteristic p thanks to the additional assumption that G and H are reductive), the complex UPic (X) is quasi-isomorphic to [ G tor → T ] (recall that Pic G = 0).This induces a natural morphism λ : T → UPic (X) of Γ K -modules.Since H 1 (K, G tor ) = 0 (indeed G tor is a permutation Galois module), we obtain an injective morphism of abelian groups λ * : H 1 (K, T ) → Br 1 X/Br K, which induces an injective morphism of abelian groups ψ X := λ * : X 1 (K, T ) ֒→ B(X).
By Theorem 5.2 in [DH2], there is a Poitou-Tate perfect duality of finite groups This pairing and the class η X = ab 2 (σ X ) define a morphism α X : Lemma 2.6.The following diagram is commutative (up to sign).

Proof:
By [BvH2], Th. 9.6.(whose proof is identical in characteristic p thanks to our additional assumptions), the class η X ∈ H 1 (K, [T → G tor ]) ֒→ H 2 (K, T ) coincides (up to a sign) with the element of Ext 2 (UPic (X), G m ) given by the map w : UPic (X) → G m [2] of exact sequence (1) in [HSk].Therefore the class η X viewed in [HSk].Now Theorem 3.5. of loc.cit.(whose proof in characteristic p is identical) shows that for every a ∈ X 1 ( T ), we have or in other words : This concludes the proof of the lemma.
We can now finish the proof of Theorem 2.5.Assuming that X(A K ) B(X) = ∅, we have β X = 0, hence α X = 0 by the lemma.The exactness of the pairing (3) and Proposition 2.3 imply that ab 2 X (σ X ) = η X = 0. Now, the analogue of [Bor1], Propostion 6.5 for global fields of positive characteristic (the proof of which is the same, except that Lemma 5.7 in [Bor1] is replaced by Proposition 2.1) implies that the map ab 2 X has "trivial kernel", hence η X is neutral.As a consequence, there exists a G-equivariant morphism P → X defined over K, where P is a K-torsor under G. Since G is quasi-trivial, H 1 (K, G) is trivial (see [Har2], Satz A), hence 3 Abelianization of homogeneous spaces and a compatibility formula Let H be a reductive subgroup of a reductive group G over an arbitrary field k.Set X = G/H and denote by e ∈ X(k) the image of the neutral element of G in X.Consider the complex of Galois-modules (or of commutative smooth k-group schemes) defined by with T G in degree 0. In other words, we have Denote by ab 0 X (see [Dem4], §2.6, where the characteristic zero assumption is not needed thanks to our additional assumptions that G and H are smooth and reductive) the abelianization map Following [Dem3], we set Br which is a subgroup of Br X containing Br 1 X.We also consider the subgroup Br 1,e (X, G) ≃ Br 1 (X, G)/Br k of Br 1 (X, G) consisting of those elements α such that α(e) = 0, and use a similar notation for Br 1,e X ≃ Br 1 X/Br k.The goal of this section is to prove (see Theorem 3.7 below) that there is a natural isomorphism and that the following diagram where T H is in degree 0).Here ab ′ X is an abelianization map, which we are going to define by changing a little the map ab 0 X .In [Demeio], Demeio proves such a compatibility in characteristic zero for ab 0 X with a long and impressive cocyle computation.It is quite likely that the maps ab 0 X and our new map ab ′ X actually coincide, but we did not succeed in proving this.The modified map ab ′ X seems more suitable to check the required compatibility.

Abelianization map over a field
From now on, let k be a field, X a smooth geometrically integral k-variety and π : Y → X a torsor under a reductive k-group H.We fix a point y 0 ∈ Y (k) and let x 0 := π(y 0 ) ∈ X(k).
Following [Dem3], we define Z to be Y /Z H , z 0 to be the image of y 0 and UPic ′ (π) to be the following complex of Galois modules: with k(Z) × in degree −1 (here Pic ′ (Z/X) is the relative Picard group of Z over X).We also define UPic (π with the obvious natural exact sequence of complexes: We will sometimes need the following pointed version of those complexes, which are canonically quasi-isomorphic to the previous ones: and UPic (π where k(Z) × 0 (resp.k(Z) × 0,1 ) denotes the subgroup of rational functions defined at z 0 (resp.taking the value 1 at z 0 ) and Div (Z) 0 is the group of divisors D such that z 0 is not contained in the support of D. We also have the classical complexes UPic UPic ′ (Z) etc. (corresponding to the case when π : Z → Z is the identity map).
The construction of UPic ′ (π) is contravariant in π (see for instance Proposition 2.2 in [Dem3]), and for any x ∈ X(k), the natural morphism k× Therefore, one gets a well-defined map where Hom k denotes the set of morphisms in the derived category of bounded complexes of Galois modules.
In addition, when a point y 0 ∈ Y (k) is given, one gets a natural splitting UPic (π) → UPic ′ (π), hence it defines the required map When Y = X and π = id X , we denote ab ′ π by ab ′ X .We want to think about ab ′ π as an abelianization map for the set of rational points of X, which is a replacement of ab 0 X when X = G/H.We now study the dévissage of UPic (π) in terms of Y and H : Lemma 3.1.There is a natural exact triangle

Proof:
This is essentially the proof of Corollary 3.3 in [Dem3].
Let us now prove that α −1 is an isomorphism (the proof of this fact in [Dem3] is too sketchy): using the exact sequences 4) proves that the result follows from the commutativity of the following diagram: where ∆ : H → Pic (X) is the map defined by χ → χ * [Y ] and the isomorphism is constructed in [Dem3].To prove the required commutativity, given χ ∈ H, functoriality of the various maps implies that it is sufficient to consider the case H = G m (and and let Y 0 be the fiber of π at x 0 .Then the restriction of f at Y 0 and the point y 0 ∈ Y 0 (k) induce a morphism f 0 : G m → G m .Making explicit the maps α −1 and β, the required commutativity boils down to the natural equalities π * (D) = div(f ) in Div (Y D ) and f 0 = id.
We now want to compare the map ab ′ π defined earlier with the maps ab 0 H and ab 1 H defined by Borovoi in [Bor2] for reductive k-groups H.We first prove the following Lemma 3.2.With the above notation, we have a commutative diagram (up to a sign) with exact rows where the second line comes from Lemma 3.1 and the isomorphisms come from [BvH], Theorem 4.8 and Corollary 4.9.
In addition, when Y = G is a reductive k-group, H ⊂ G is a reductive subgroup and

Proof:
The commutativity of the central square is a consequence of the functoriality of the map ab ′ π with respect to morphisms of torsors.Let us now prove the commutativity of the left hand side square : using the functoriality of the map ab π and the fact that the morphism H → Y , given by the action on ) is defined as ab ′ id H .There exists a coflasque resolution (see [CT], Proposition 4.1) where P is a quasi-trivial k-torus and H 1 is an extension of a (coflasque) torus T by a semisimple simply connected k-group H ss .Then functoriality of the map ab ′ H and Hilbert 90 imply that it is enough to prove the required compatibility ab ′ T = ab 0 T for the k-torus T .By definition, the map ab ′ T : , and it clearly coincides with the map ab 0 T : T (k) → T (k), composed with the natural identification T (k) ∼ − → Hom k ( T , k× ).It concludes the proof of the commutativity of the left hand side square.Note that this also proves the last statement in Lemma 3.2.
Let us now prove the remaining commutativity, concerning the right hand side square.Let x ∈ X(k) and consider the torsor π x : Y x → Spec k defined as the pullback of π by x (Y x denotes the fiber of Y at x).We have a commutative diagram UPic (π) where the vertical maps are quasi-isomorphisms (see the proof of Lemma 3.1).We now prove that the diagram cone is commutative.The only non obvious square is the right hand side one, but it comes from the functoriality of the cone in the category of complexes, together with the definition of ∂ Yx .
The two previous commutative diagrams imply that the square commutes, where ab ′ 1 H maps the cohomology class of a torsor W → Spec k under H to the rightmost morphism in the natural exact triangle with the canonical identification UPic (W ) ∼ − → UPic (H).We conclude the proof using [BvH], Theorem 5.5, which proves that ab ′ 1 H = −ab 1 H .
We need to prove other properties of the map ab ′ π : Proposition 3.3.Let X = G/H be the quotient of a reductive group G by a reductive subgroup H. Let π : G → X be the quotient map (pointed by e ∈ G(k)).Then for all g ∈ G(k) and In particular, if G is semi-simple and simply connected, we have ab ′ π (g • x) = ab ′ π (x) .

Proof:
Consider id G : G → G as a torsor under the trivial group, and id × π : G × k G → G × k X as a natural torsor under H. Then we have natural morphisms of torsors : that induce, by functoriality of UPic (π) and by Lemma 3.1, a commutative diagram in the derived category, where the rows are exact triangles (see [BvH], Lemma 5.1 for the third vertical map) : The five-lemma implies that the norphism By functoriality of the construction of ab ′ π , the morphism of torsors which concludes the proof.The non trivial commutativity in this last diagram is that of the triangle at the bottom, which we explain now: recall that we are given a point x 0 = π(e) ∈ X(k).
Consider the following commutative diagram of morphisms of torsors: where the bottom horizontal maps are defined by ι x0 (g) = (g, x 0 ) and ι 1 (x) = (1, x).If we denote by ̟ G (resp.̟ X ) the projection from UPic (G)⊕UPic (π) to UPic (G) (resp.UPic (π)), then we deduce from the previous diagram that therefore we get that the required triangle commutes.
Finally, if G is assumed to be semi-simple and simply connected, then the complex UPic (G) is quasi-isomorphic to 0, hence the map π ′ is trivial, which implies the required result.
Remark 3.4.It is worth noting that the construction of the map ab π depends on the choice of a k-point y 0 ∈ Y (k).But one can prove, using the same kind of arguments as in the proof of Lemma 3.3, that the map ab π depends only on the image x 0 of y 0 in X(k).More precisely, two points y 0 , y ′ 0 ∈ Y (k) such that π(y 0 ) = π(y ′ 0 ) define the same map ab π , or equivalently, the construction of this map depends only on the choice of a point x 0 ∈ π(Y (k)).Lemma 3.2 and Proposition 3.3 imply the following proposition : Proposition 3.5.If k is a non archimedian local field and X = G/H is the quotient of a reductive group G by a reductive subgroup H, then the map ab ′ π : The same holds when k is a global field of positive characteristic.

Proof:
In the proof, k denotes either a non archimedian local field, or a global field of positive characteristic.
Consider the following commutative diagram with exact rows (see Lemma 3.2): Then diagram chasing proves that surjectivity of ab ′ π is a consequence of surjectivity of ab 1 H , injectivity of ab 1 G , surjectivity of ab 0 G (those properties follows from Proposition 2.1), and of Proposition 3.3.
Let us now prove the second part of the Proposition: let x, x ′ ∈ X(k) such that ab ′ π (x) = ab ′ π (x ′ ).Using the previous diagram and the injectivity of ab 1 H (see Proposition 2.1), we get that x and x ′ have the same image in H 1 (k, H).Therefore, there exists g ∈ G(k) such that g • x = x ′ .Applying Lemma 3.3, we get that π ′ (ab 0 G (g)) = 0, hence by Proposition 2.1, g lifts to G sc (k), which concludes the proof.
Let X = G/H, with G and H reductive.Let us now construct a canonical isomorphism φ X : C X → UPic (π) in the derived category, inspired by [Dem3], sections 4.1.2and 4.1.3.
By construction, C X is the cone of the morphism of complexes where T H is in degree 0. Consider the following commutative diagram of complexes, where the vertical maps are either obvious or defined in [BvH], section 4): All the vertical maps, except the ones between the last two lines, are quasi-isomorphisms.Hence this diagram induces a natural isomorphism φ ′ X : C X → Cone(ϕ), which we can compose with the quasi-isomorphism α : UPic (π)[1] → Cone(ϕ) induced by the two last lines of the previous diagram (see the proof of Lemma 3.1), to get a natural isomorphism φ X : C X → UPic (π) [1].
By construction, this isomorphism fits into the following commutative diagram of exact triangles in the derived category: where the vertical maps are isomorphisms (morphisms φ H and φ G are defined in [BvH], section 4, and also in the first four lines of the previous big diagram).
In addition, Lemma 3.1 in [BDH] implies that the natural morphism ) is an isomorphism.These facts lead to the following: Definition 3.6.We denote by ab ′ X : X(k) → H 0 (k, C X ) the map defined by where π : G → X is the quotient morphism.
In particular, Lemma 3.2, Definition 3.6 and diagram (6) imply that the following useful diagram is commutative (up to sign): where the unnamed maps are the natural ones.

The compatibility result
We can now prove the main compatibility result of this section, which can be seen as a generalization of [Dem3], Theorem 4.14 and of [BDH], Theorem 6.2: Theorem 3.7.Let k be a field and G be a reductive group.Let H ⊂ G be a reductive ksubgroup and X := G/H.Let π : G → X be the quotient map (pointed by e ∈ G(k) and its image x 0 := π(G)).The canonical isomorphism φ X : C X → UPic (π) in the derived category induces an isomorphism and the following diagram is commutative, up to a universal sign (independent of all the data).

Proof:
Following [Dem3], there is a natural morphism UPic G).Together with the isomorphism φ X : C X → UPic (π), we get the required isomorphism.Let us now prove the commutativity.
For any x ∈ X(k), one can define a natural splitting ab ′ (x) of k× → τ ≤2 Rπ * G m induced by x.
We can decompose the diagram above as the composition of the following diagrams: By construction, and by functoriality of cup-products, this last diagram is commutative (up to sign).
Remark 3.8.Let X = G/H as in Definition 3.6 and Theorem 3.7.Up to replacing G by a flasque resolution G 1 , one can realise X as the quotient of the quasi-trivial group G 1 by a reductive subgroup H 1 .Since Pic (G 1 ) = 0, one gets a natural isomorphism where G 1 is a permutation Galois module.Assuming the group G is quasi-trivial will be very useful in the next two sections.

The abelianization map over an arbitrary base
In this section, we extend the definition of the map ab ′ X for homogeneous spaces of reductive group schemes defined over an arbitrary base scheme S. It will be useful in the next sections in order to take integral points into account (case S = Spec (O v )).
Let S be an integral regular noetherian scheme and H be a reductive group scheme over S. Let π : Y → X be a torsor under H.For any S-scheme W , let p W : W → S denote the structure morphism.
Let Z := Y /Z H and ̟ : Z → X be the associated H ′ := H/Z H -torsor.We define UPic ′ (π) to be the following complex of étale sheaves over S : where the sheaves K × Z/X and Div Z/X are defined in [HSk], Appendix A. By loc.cit., there is a natural exact sequence of étale sheaves over Applying p X * , one gets a natural morphism (G m ) S → UPic ′ (π), and we define UPic (π) to be the cone of this morphism, whence an exact triangle Let y 0 ∈ Y (S) be a point, and let x 0 := π(y 0 ).Denote by D(S) the derived category of bounded complexes of étale sheaves over S. Following the construction in section 3.1, we get a natural map, functorial in S and X, Let now G be a reductive group scheme over S and H ⊂ G a reductive subgroup scheme.Taking Y = G and X = G/H, we can apply the previous constructions.From now on, we assume that H and G admit compatible maximal tori T H ⊂ T G .The diagram (5), as a diagram of étale sheaves over S, still holds.All vertical morphisms, except the bottom ones, are quasiisomorphisms of étale sheaves over S, since it is true over any separably closed field (see after diagram (5)).Similarly, since the result holds over separably closed fields, diagram (5) induces an isomorphism of complexes of étale sheaves φ X : C X → UPic (π) [1].
As a conclusion, composing the map ab ′ π : X(S) → Hom D(S) (UPic (π), (G m ) S ) with the isomorphism φ X , one gets the required map: that is functorial in S and (H, G), and that coincides with the definition of section 3.1 in the case S is the spectrum of a field.

Weak approximation
From now on, the setting is the following: K = k(E) is again the function field of a projective, smooth curve E over a finite field k.We consider a reductive linear algebraic group G over K, and X is a homogeneous space of G.We assume that X(K) = ∅.Let e ∈ X(K) and let H ⊂ G be the stabilizer of e in G. Then X = G/H (with e identified to the image of the neutral element of G in X) and we still suppose that H is a reductive subgroup of G. Set We are interested in the closure of X(K) in X(K Ω ), for the product topology.We define B ω (X) as the subgroup of Br 1,e X ≃ Br 1 X/Br K consisting of those elements α such that their localization α v ∈ Br 1,e X Kv is zero for all but finitely many v.For each finite set of places S of K, we set K S = v∈S K v and we define In particular B(X) (cf.section 2) identifies to B ∅ (X).For all α ∈ B ω (X), (P v ) ∈ X(K Ω ), the Brauer-Manin pairing : α, is well-defined.For every subgroup B of B ω (X), we denote by X(K Ω ) B ⊂ X(K Ω ) the orthogonal of B for the Brauer-Manin pairing.
Theorem 4.1.Let G be a reductive group over K and H a reductive subgroup.Then the Brauer-Manin obstruction to weak approximation on X = G/H associated to B ω (X) is the only one, i.e.X(K) is dense in X(K Ω ) Bω (X) .More precisely, for any finite set S of places of K, the Brauer-Manin pairing induces a surjective map X(K S ) → (B S (X)/B(X)) D , whose kernel is exactly the closure of X(K) inside X(K S ).
Remark 4.2.Assume that X admits a regular compactification, i.e. there exists a regular proper K-variety X c and an open immersion X → X c .Then the group B ω (X) is exactly the algebraic Brauer group of X c (see for example [BDH], Prop 4.1).

Proof:
Up to replacing G by a flasque resolution, one can assume that G is quasi-trivial.Using an elementary instance of the fibration method, we see that G satisfies weak approximation, since G sc (by [Pra]) and the quasi-trivial (hence K-rational) torus G tor do satisfy weak approximation.In addition, we have H 1 (K, G) = 1 by [Har2] and Hilbert's 90 (the latter shows that H 1 (K, G tor ) = 0 thanks to Shapiro's lemma).
Let C := [H tor → G tor ] and C X := Cone(C H → C G ) (cf. section 3).We have the Cartier duals C = [ G tor → H tor ] and By construction, we have a natural commutative diagram of exact triangles of complexes: Since G is assumed to be quasi-trivial, we have G ss = G sc , hence µ G = 0 and C G is quasi-isomorphic to G tor , which is a quasi-trivial torus.Taking Cartier duals, we get an exact triangle : whence (using H 1 (K, G tor ) = 0) an exact sequence : and similarly replacing K with a completion K v .As G tor is a permutation Galois module, we have X 2 S (K, G tor ) = 0 by Shapiro's lemma and Čebotarev's Theorem.Thus which yields an isomorphism X 1 S (K, C) → X 1 S (K, C X ).Summing up, we get isomorphisms As recalled before (cf.proof of Th. 2.5.), the group Br 1,e X ≃ Br 1 X/Br K is isomorphic to H 1 (K, UPic (X)) = H 1 (K, C) (and this is true over any field).Therefore X 1 S (K, C) (resp.X 1 (K, C)) identifies to B S (X) (resp.to B(X)), whence a Brauer-Manin pairing : On the other hand, local duality for complex of tori ([Dem1], Th. 3.1) induces a map We also have a map in the derived category of Galois modules which induces (for ecah completion K v ) a cup-product pairing The latter is compatible (in an obvious sense) with local duality for the complexes C H , C H via the maps the coboundary map, which (by composing with the abelianization map) yields a morphism of pointed sets Similarly we have (for every completion . By Lemma 3.2, the map ∂ ab v can also be obtained by composing the abelianization map ab Lemma 4.3.There is a commutative diagram with exact rows : Proof: The exactness of rows follows from the triviality of H 1 (K, G) and H 1 (K v , G) (recall that G is quasi-trivial) combined with Proposition 2.1.The only non-trivial remaining point is the commutativity of the bottom square.By functoriality of the cup-product and Theorem 3.7, there is a commutative diagram : where the left vertical map is given by the local evaluation pairing and the right vertical map by local duality for the complex . By the previous diagram, we have where which yields the required commutativity.
Remark 4.4.Actually we used the difficult compatibility proven in Theorem 3.7 only for those elements of ), so Theorem 6.2. of [BDH] (whose proof is much easier) would be sufficient at this stage.However, we will definitely need Theorem 3.7 in its full generality in section 5.
We resume the proof of Theorem 4.1.Let us now prove that the right hand side column of diagram ( 9) is exact.Consider the following commutative diagram: Using [DH2], Theorem 5.7, the second row is exact.And by construction, the columns are exact.Hence an easy diagram chase implies that the bottom row is exact.Therefore, the right hand side column in ( 9) is exact.In addition, we know that G(K) is dense in G(K S ).Therefore, an easy diagram chase in ( 9), together with the comparison Theorem 6.2 in [BDH], implies that the map is surjective, and that the inverse image of 0 is exactly the closure of X(K), which concludes the proof.
Theorem 4.1 can be slightly refined when the homogeneous space X is a reductive group: Corollary 4.5.Let L be a reductive group over K. Let C L = [T sc → T L ] be the complex of tori associated to L. Then there is an exact sequence of groups

Proof:
We can view L as a homogeneous space L = G/H, with G semi-simple and simply connected, and H reductive.By Theorem 4.1, there is an exact sequence of pointed sets By Lemma 4.3, the Brauer-Manin map Therefore the Brauer-Manin map is a morphism of groups.By [Bor2] (paragraph 3.10), there is (for every completion K v ) an exact sequence of pointed sets It is now sufficient to show that the sequence of abelian groups is exact.This is done by observing that by [Dem1], Th. 3.1. and [DH2], Th. 5.2, this sequence is the dual of the exact sequence of discrete abelian groups Remark 4.6.The results and proofs of sections 2, 3 and 4 are still valid over a number field.The analogues of Theorems 2.5 and 4.1 were previously known in this context: they are proven by Borovoi in [Bor3] via more geometric techniques (namely fibration methods); the case of principal homogeneous spaces is due to Sansuc [San].Another approach (over an arbitrary global field) is to use flasque resolutions, see [Tha], Th. 3.9.In the next section, we will see that the situation is slightly different for strong approximation.

Strong approximation
Notation is as in the previous section.We set A ⋆ := Hom(A, Z) for every abelian group A. An abelian group is said to be in the class E (cf.Definition 3.10 in [DH2]) if it is an extension of a finitely generated group by a profinite group.
Since the degree of a principal divisor on the projective curve E is zero, this pairing is trivial on the subgroup H 0 (K, C) × H −1 (K, C).
Lemma 5.1.a) The kernel P of the map H Proof: a) Up to replacing C by a quasi-isomorphic complex, one can assume that T 1 is quasi-trivial.This yields a commutative diagram with exact rows and surjective left vertical map: The snake lemma now implies that the kernel of ⋆ by a closed subgroup, and this kernel is profinite by [Ros1], Proposition 5.7.5.Hence ker b) The exact sequence and Lemma 3.12 a) of [DH2] show that i is surjective.Since has exact first line by definition, and exact third line because it is obtained by completing the first line and H −1 (K, C) ⋆ is a lattice, so [DH2], Lemma 3.12 b) applies.To prove the injectivity of i, it is sufficient (by diagram chasing) to show that the second line is exact as well.Let π : induces (again by [DH2], Lemma 3.12 b) an exact sequence whence an exact sequence But the exact sequence H 0 (K, C) → π −1 (P ) → P → 0 induces (as P is profinite) a surjective map u : π −1 (P ) ∧ /H 0 (K, C) ∧ → P and j factorizes through u, hence the second line of the diagram is exact, as required.
; thus one can reduce to the case of a quasi-trivial torus.By Shapiro's Lemma, this case reduces to the known case of G m .Hence the map H 11) is commutative with exact rows, the second point follows.
Proposition 5.2.a) There is an exact sequence where the morphism ∂ is given (after completion) by pairing (10) for C. b) There is an exact sequence Observe that for C = G m , we have C = Z[1] and b) is just the classical exact sequence of global class field theory.

Proof:
By Theorems 5.7 and 5.10 in [DH2], there is a commutative diagram with exact rows : whence exactness of the sequence The exactness of sequence (12) now follows from Lemma 5.1, c), and part b) of the proposition follows from its part a) and diagram (13).
We now consider a homogenous space X = G/H with G and H reductive and we assume further that G is quasi-trivial.As in the previous sections, we define C = [H tor → G tor ] and We denote by C and C X their respective duals (cf.section 4).We have the analogue of the pairing (10) with C (resp.C) replaced by C X (resp.C X ), and again the pairing is trivial on H 0 (K, C X ) × H −1 (K, C X ).
We observe that H −1 (K, C) = H −1 (K, C X ) (and similarly over every completion K v of K) thanks to the exact triangle For every finite and non empty set of places S of K, we set Br S X = ker[Br e X → v∈S Br X Kv ] (not to be confused with the groups B S (X) of section 4) and Br S (X, G) := Br S X∩Br 1,e (X, G), Br 1,S X := Br S X ∩ Br 1,e X.We will also use a smooth model G (resp.H, X = G/H) of G (resp.H, X) over some non empty Zariski open subset U = E of the curve E. Shrinking U if necessary, we can assume that H and G admit compatible maximal tori T H ⊂ T G .We have the corresponding complexes C H , C = [H tor → G tor ], C X defined over U .For every bounded complex F of flat commutative finite type group schemes over U , the compact support hypercohomology groups H i c (U, F ) are defined as in [DH2], §2 and we set where F is the generic fibre of F over K.For v ∈ U , we denote by and if S is a finite set of places that does not meet U , we set (by convention, P i ∅ (U, C X ) is denoted by P i (U, C X )).

Proof:
Using the same method as in [Dem4], Th. 2.18, it is sufficient (via the version of Lemma 3.2 over O v ) to show that H 1 ab (O v , H) = 0 and ab 0 : H) follows by devissage from the fact that for an O v -torus T , we have H 1 (O v , T ) = H 2 (O v , T ) = 0 (see for example [HSz1], proof of Th. 2.10).Finally ) is trivial by Steinberg's Theorem (here G sc is the reduction mod.v of the reductive group scheme G sc ; it is a connected linear group scheme over the residue field F v of the curve E at v).This implies that the abelianization map ) is surjective because by definition of the abelianization map, there is an exact sequence We need now to extend a few duality results of [DH2] to the three-term complex C X : Proposition 5.4.a) The group which implies that H 1 c (U, C X ) is an extension of H 1 c (U, C) by the finite (cf.[DH1], Th. 1.1) group DH2], Prop 3.13), so is H 1 c (U, C X ).As H 3 (U, µ H ) = 0 and H 2 (U, µ H ) is finite ([DH1], Th. 1.1 and Cor.4.9), we also get that H 0 (U, C X ) is an extension of H 0 (U, C) (which is of finite type by [DH2], Prop 3.6.b) by a finite group, hence it is also of finite type.
b) The finiteness of D 0 (U, C X ) follows from that of D 0 (U, C) (see [DH2], Lemma 4.15) and that of H 2 (U, µ H ) ([DH1], Cor.4.9) thanks to the commutative diagram with exact rows: and D 1 (U, C) is finite by [DH2], Th. 5.2).c) We first show that Artin-Verdier duality induces an isomorphism To prove this, we use a devissage given by the triangle (14).By [DH2], Theorem 4.11 b), the discrete torsion group The second line is exact as the dual of an exact sequence of discrete torsion groups.The first line is exact as well thanks to Lemma 3.12 of [DH2] because H 3 c (U, µ H ) is finite and H 1 c (U, C X ) belongs to the class E (hence it injects into its completion).Now Artin-Mazur-Milne duality for µ H (see [DH1], Theorem 1.1) and for C (see [DH2], Theorem 4.11 b)) yield that the first, second, and fourth vertical maps are isomorphisms.The five lemma implies that the third one is also an isomorphism, as required.
The argument to prove the isomorphism H 1 c (U, C X ) ∧ ≃ H 1 (U, C X ) D , is similar, using the commutative diagram with exact rows: Indeed the first and fourth vertical maps are isomorphisms by [DH2], Th. 4.9 a), and so is the third by [DH1], Theorem 1.1.Proposition 5.5.a) There is an exact sequence b) There is an exact sequence and the orthogonal of H Proof: a) We extend the proof of the exactness of the second line of Poitou-Tate exact sequence in Theorem 5.7 in [DH2].Following the same method, it is sufficient to extend the first part of Lemma 5.6 a) in [DH2] to the complex C X .Namely, it remains to show the exactness of The proof is exactly the same as for C, applying Lemma 2.2 in [DH2] to the complex C X and using the three following facts (proven in Proposition 5.4): for every non empty Zariski open subset V ⊂ U , the group H 1 c (V, C X ) is in the class E, the group D 0 (V, C X ) is finite, and Artin-Verdier duality induces an isomorphism Similarly, it is sufficient to extend the second part of Lemma 5.6 b) in [DH2], that is to show the exactness of Applying again Lemma 2.2 of [DH2] to C X , one just has to check the following properties : -for every non empty Zariski open subset V ⊂ U , we have This holds thanks to Proposition 5.4 (the finite type group H 0 (V, C X ) has same dual as H 0 (V, C X ) ∧ ).
-the group D 1 (U, C X ) is finite, which is also proven in Proposition 5.4.c) Using the exact triangle ( 14) and the vanishing of H 3 (K v , µ H ) ( [Mil], Prop.III.6.4), there is a commutative diagram with exact rows (the completed first line remains exact because H 2 (K v , µ H ) is finite by [Mil], Ex.III.6.7, and the second line is obtained by dualizing an exact sequence of discrete torsion groups): Since the first, second, and fourth vertical map are isomorphisms by [Mil], Th.III.6.10 and [Dem1], Th. 3.1.,so is the third vertical map.
It remains to show that the map H There is a commutative diagram with exact lines: The fourth column is exact and the map H Th. 3.1. and 3.3).Since µ H is a finite group scheme, we have H [Mil], Th.III.6.10.The required result follows by diagram chasing.
Proposition 5.6.There is an exact sequence ) where the last non trivial map is defined via the natural map Proof: We consider diagram (8) comparing C X and C. Applying cohomology, we get a commutative diagram ) Local duality for µ H (cf. [Ces], Proposition 4.10 b) for instance) and Proposition 5.6 imply that the first row and the last one are exact.The exact triangles of (8) implies that the first column is exact.
The second column is exact, being the dual of the obvious exact sequence Now an easy diagram chase implies the exactness of the second line of ( 16).The exactness of (15) then follows from Proposition 5.5, a).Proposition 5.7.a) The following sequence: is exact.b) Let S be a finite set of places that does not meet U .Set Then the sequence c) There is an exact sequence follows immediately from Proposition 5.6.Applying Proposition 5.5 b) and taking into account that H 1 (K, C X ) is torsion, we get an exact sequence , Remark 5.9).Dualizing this exact sequence now yields the result, thanks to Proposition 5.5, c).
b) Dualizing the exact sequence of discrete torsion groups and taking into account the local duality isomorphism where the top line (by a) and the columns are exact (the left one because H 0 (K v , C X ) is in the class E by [DH2], Prop 3.13 and exact triangle (14); so the sequence remains exact after completion thanks to loc.cit., Lemma 3.12).A simple diagram chase implies that the bottom line is exact, which proves b).c) For v ∈ U and i ∈ Z, set (and similarly for C X ).Consider the commutative diagram: The middle column is obviously exact.The right column is exact as well, because by Proposition 5.5 c), it is the dual of the sequence of discrete groups which is exact by [DH2], Prop.2.1.It follows immediately from Proposition 5.5 b) that the second line of the diagram is exact.Since the map p is obviously surjective, a diagram chase shows that the first line of the diagram is exact as well.Dualizing it (and observing that H 0 (U, C X ) is an abelian group of finite type by Proposition 5.4, a) yields (thanks to Proposition 5.5, c) the exactness of H 0 (U, C X ) ∧ → P 0 S (U, C X ) ∧ → H 1 S (K, C X ) D .Since the canonical map P 0 S (U, C X ) ∧ → H 1 S (K, C X ) D factors through (H 1 S (K, C X )/X 1 (K, C X )) D , the result is proven.
Recall that for a finite (possibly empty) set of places S of K, we have the Brauer-Manin pairing BM : X(A S K ) × Br X → Q/Z, ((P v ) v ∈S , α) → v ∈S α(P v ).
By global class field theory, elements of X(K) ⊂ X(A S K ) are orthogonal to Br S (X) for this pairing; in particular when S = ∅, elements of X(K) are orthogonal to Br e X (hence to Br X).
By continuity of the pairing, the same holds for the closure X(K) S of X(K) in X(A S K ) for the strong topology.The following theorem gives various converse statements.
Theorem 5.8.Let X = G/H be a homogeneous space of a reductive group G with H reductive.Set U (X) := K[X] * /K * .Assume that G sc satisfies strong approximation outside S 0 (finite set of places).
1.There is a natural exact sequence of pointed topological spaces: In particular, X(A K ) Br X = ρ G sc (K) • G sc (K S0 ) • X(K) and X(A S0 K ) Br X ⊂ X(K) S0 .
2. If S is a non empty finite set of places, there is a natural exact sequence of pointed topological spaces: In particular, X(A S K ) Br S X = ρ(G sc (K S0 )) • X(K) S .

Proof:
As earlier, one can assume that the group G is quasi-trivial, up to replacing G by a flasque resolution 1 → S → G ′ → G → 1.
Indeed Pic S = 0 (since S is a torus), hence Br G ֒→ Br G ′ , which implies Br 1 (X, G) = Br 1 (X, G ′ ) by [San], Prop.6.10.Throughout the proof, we use Theorem 3.7 to translate results concerning complexes of tori and cup-products to results concerning homogeneous spaces and Brauer-Manin obstructions.
1.By [Dem3], Th. 4.14, the group U (X) is isomorphic to H −1 (K, C X ).By Theorems 3.7 and 5.6, there is a commutative diagram with exact second row: By Proposition 3.5 and Lemma 5.3, the maps ab ′ K and ab ′ AK are surjective.By diagram chasing, the sequence of three last non-trivial terms on the first line is also exact.Besides, every element x of X(K) ⊂ X(A K ) satisfies BM(x) = 0 by class field theory, and the same holds for an element of X(A K ) of the form (g v .x)with (g v ) ∈ ρ(G sc (A K )) thanks to the commutativity of the diagram and the property ab ′ (g v .x)= ab ′ (x) for every place v of K (Proposition 3.3).It remains to show conversely that every (P v ) ∈ X(A K ) such that BM((P v )) = 0 comes from ρ G sc (K) • G sc (K S0 ) •X(K) The diagram and the surjectivity of ab ′ K imply that there exists x ∈ X(K) such that for every place v.By Proposition 3.5, there exists for each v an element g v ∈ ρ(G sc (K v )) such that P v = g v .x,and we can assume that g v ∈ G sc (O v ) for v ∈ S, where S ⊃ S 0 is some finite set of places.Since G sc satisfies strong approximation outside S 0 , we finally obtain that (P v ) belongs to ρ G sc (K) • G sc (K S0 ) • X(K) as required.The two other assertions in 1. follow immediately.
Up to shrinking U , we can assume that it does not meet (S 0 ∪ S).We consider the following commutative diagram: