Arithmetics of homogeneous spaces over p$p$ ‐adic function fields

Let K$K$ be the function field of a smooth projective geometrically integral curve over a finite extension of Qp$\mathbb {Q}_p$ . Following the works of Harari, Scheiderer, Szamuely, Izquierdo, and Tian, we study the local–global and weak approximation problems for homogeneous spaces of SLn,K$\textrm {SL}_{n,K}$ with geometric stabilizers extension of a group of multiplicative type by a unipotent group. The tools used are arithmetic (local and global) duality theorems in Galois cohomology, in combination with techniques similar to those used by Harari, Szamuely, Colliot‐Thélène, Sansuc, and Skorobogatov. As a consequence, we show that any finite abelian group is a Galois group over K$K$ , rediscovering the positive answer to the abelian case of the inverse Galois problem over Qp(t)$\mathbb {Q}_p(t)$ . In the case where the curve is defined over a higher dimensional local field instead of a finite extension of Qp$\mathbb {Q}_p$ , coarser results are also given.

Let k be a field and let Ω be a set of places (i.e. equivalence classes of non-trivial absolute values) of k. For each v ∈ Ω, we denote by k v the corresponding completion. We say that a class of smooth k-varieties satisfy the local-global principle (with respect to Ω) if for every variety X in this class, Suppose that X is a k-variety with X(k) = ∅. For each v ∈ Ω, the set X(k v ) is equipped with a local topology induced by the topology of k v (see for example [Con12,Proposition 3.1]). We say that X has weak approximation in Ω if the diagonal embedding X(k) ֒→ v∈Ω X(k v ) has dense image (where the product space is equipped with the product of local topologies, hence it is enough to work with finite subsets S ⊆ Ω). For example, affine spaces have weak approximation (Artin-Whaples).
These arithmetic properties (local-global principle and weak approximation) are k-stable birational invariants of smooth complete k-varieties, thanks to Serre's generalized version of the implicit function theorem [Ser92, Part II, Chapter III, §10.2] combined with the theorem of Nishimura-Lang [Nis55]. The classical case is when k is a number field (or its geometric counterpart, a global function field). In this case, Manin [Man71] defined an obstruction to the existence of rational points using the Brauer-Grothendieck group Br X := H 2 ét (X, G m ) and the global reciprocity law (Albert-Brauer-Hasse-Noether). These were later used by Colliot-Thélène and Sansuc to define an obstruction to weak approximation, which is now known as the Brauer-Manin obstruction.
The present article is motivated by the recent interest in studying these two arithmetic problems over fields of cohomological dimension greater than 2. Of particular interest is the case of a function field K of a (smooth, complete, geometrically integral) variety V over a field k, and where the set of places is that of points v ∈ V of codimension 1. In this article, we focus on the case where k is a finite extension of Q p (except in section 4, where k is a higher-dimensional local field), and where V is a curve (of course, the classical case of global function fields corresponds to the case where k is finite). The corresponding function field K is then a field of cohomological dimension 3.
The main tool for our approach will be Galois cohomology. Over a p-adic function field K as above, the group H 3 ét (X, Q/Z(2)) (and its variants) would play the role of Br X in the classical case. Using (local and global) duality theorems for tori, the works of Harari, Scheiderer and Szamuely gave us a certain understanding of the local-global principle [HS16] and weak approximation [HSS15] problems for tori. Tian [Tia20,Tia21] extended their ideas to study weak approximation for connected reductive groups G with the property that the universal cover G sc of its derived subgroup G ss has weak approximation and contains a split maximal torus. Tian's work relies on the powerful machinery of Borovoi called abelianization of non-abelian Galois cohomology [Bor93,Bor96,Bor98]. Izquierdo also provided some results over function fields of curves over higher-dimensional local fields [Izq17, Théorème 0.1].
It should be noted that if we consider the completions coming from all the rank 1 discrete valuations on K (not just those coming from the closed points of the curve), then many other local-global principles have been established. For these results, the readers are invited to consult the recent survey of Wittenberg [Wit22,§4].
In this article, the varieties under consideration will be homogeneous spaces of SL n (or more generally, a special, simply connected semisimple algebraic group that has weak approximation). We shall see that in the heart of the proof of each of the main theorems, there lies the use of a Poitou-Tate style duality theorem and the Poitou-Tate sequence. Aside from them, the techniques used here are the reinterpretation of the Brauer-Manin pairing by Harari-Szamuely [HS08,HS16] for the existence of rational points, and by Colliot-Thélène-Sansuc-Skorobogatov [CTS87a,Sko99] for the weak approximation property.
Acknowledgements. The author is funded by a "Contrat doctoral spéficique normalien" from École normale supérieure de Paris. I am grateful to my PhD advisor, David Harari, for his guidance and support. I thank Jean-Louis Colliot-Thélène, Yisheng Tian, Haowen Zhang, and Olivier Wittenberg for the relevant discussions. I also thank the Laboratoire de Mathématiques d'Orsay, Université Paris-Saclay for the excellent working condition.

Statement of the main results
Here are the main results of the article.
Theorem A (Theorems 3.4 and 3.14). Let K be the function field of a smooth projective geometrically integral curve Ω over a p-adic field, X a homogeneous space of SL n,K whose geometric stabilizers are extensions of a group of multiplicative type by a unipotent group. Then the unramified first obstruction to the local-global principle for X (that is, the obstruction relative to the subgroup of H 3 (K(X), Q/Z(2)) consisting of elements unramified over K and whose restriction to H 3 (K v (X), Q/Z(2)) comes from H 3 (K v , Q/Z(2)) for all closed points v ∈ Ω) is the only one.
Theorem B (Theorems 3.13 and 3.15). Let K be the function field of a smooth projective geometrically integral curve Ω over a p-adic field, X a homogeneous space of SL n,K whose geometric stabilizers are extensions of a group of multiplicative type by a unipotent group. Assume that X(K) = ∅. Then the reciprocity obstruction to weak approximation for X, relative to the subgroup of H 3 (K(X), Q/Z(2)) consisting of elements unramified over K and whose restriction to H 3 (K v (X), Q/Z(2)) comes from H 3 (K v , Q/Z(2)) for all but finitely many closed points v ∈ Ω, is the only one.
Using Theorem B, we also give a positive answer to the abelian case of the inverse Galois problem over p-adic function fields (see Corollary 3.18).
Actually, for each of the two above theorems, we present two proofs. The first ones (Theorems 3.4 and 3.13) rely on the observation that the two arithmetic problems (the local-global principle and weak approximation) for homogeneous spaces of SL n is closely related to the question of universal torsors over their smooth compactifications. This is part of a "descent theory" for torsors under tori, which (over number fields) originated from the formidable work of Colliot-Thélène and Sansuc [CTS87a,§3], and to which Skorobogatov then made some complements [Sko01,§6]. Hence, in the course of proving Theorems A and B, we shall develop this descent theory for varieties over padic function fields. In particular, the following analogue of [CTS87a, Théorème 3.8.1, Proposition 3.8.7] (see also [Sko01, Corollary 6.1.3]) is proposed.
Theorem C (Theorem 2.1). Let K be the function field of a smooth projective geometrically integral curve Ω over a p-adic field. Let X be a smooth proper geometrically integral variety over K such that Pic X K is a finitely generated free abelian group. If the universal torsors Y → X (see its definition at the beginning of section 2) satisfy the local-global principle (resp. the local-global principle and weak approximation), then the reciprocity obstruction (that is, the obstruction relative to the subgroup of H 3 (K(X), Q/Z(2)) consisting of elements unramified over K) to the local-global principle (resp. the local-global principle and weak approximation) on X is the only one.
The second proofs of Theorems A and B (Theorems 3.14 and 3.15, respectively) are inspired by the following remark, communicated to the author by Jean-Louis Colliot-Thélène 1 : Every homogeneous space of SL n,K , whose geometric stabilizers are extensions of a group of multiplicative type by a unipotent group, is K-stably birational to a torus. This allows us to deduce Theorems 3.14 and 3.15 from [HS16, Theorem 0.2] and [HSS15, Theorem 1.2], respectively.
For function fields of curves over higher-dimensional local fields, we propose weaker versions of Theorems A and B. If k is a d-dimensional local field k (see paragraph 1.4 below), consider the following condition. k = C((t)); or d ≥ 1 and the 1-local field associated with k has characteristic 0. (⋆) In particular, k has characteristic 0. Note that the case where k is p-adic corresponds to d = 1.
Theorem D (Theorem 4.2). Let K be the function field of a smooth projective geometrically integral curve Ω over a d-dimensional local field k satisfying (⋆), X a homogeneous space of SL n,K with finite abelian geometric stabilizers. Then the adelic first obstruction to the local-global principle for X (that is, the obstruction relative to the subgroup of H d+2 (X, Q/Z(d+1)) consisting of elements whose restriction to H d+2 (X Kv , Q/Z(d + 1)) comes from H d+2 (K v , Q/Z(d + 1)) for all closed points v ∈ Ω) is the only one.
Theorem E (Theorem 4.3). Let K be the function field of a smooth projective geometrically integral curve Ω over a d-dimensional local field k satisfying (⋆), X a homogeneous space of SL n,K with finite abelian geometric stabilizers. Assume that X(K) = ∅. Then the generalized Brauer-Manin obstruction to weak approximation for X (that is, the obstruction relative to the subgroup of H d+2 (X, Q/Z(d + 1)) consisting of elements whose restriction to H d+2 (X Kv , Q/Z(d + 1)) comes from H d+2 (K v , Q/Z(d + 1)) for all but finitely many closed points v ∈ Ω) is the only one.
Theorems D and E do not hold for stabilizers of multiplicative type; counter-examples in the case where d = 0 and where the stabilizers are tori are given in Example 4.6.
The condition (⋆) is crucial to establish duality theorems for finite modules. It can be slightly weakened by allowing k (if d = 0) or the 1-dimensional local associated with k (if d ≥ 1) to have characteristic p > 0, provided that p does not divide the order of the stabilizers.
The article is organized as follows. In paragraphs 1.3, 1.4, and 1.5, we recall the notations, conventions, and known results that will be used in the proofs of our main theorems. In section 2, we develop a version of Colliot-Thélène-Sansuc descent theory for varieties over p-adic function fields and prove Theorem C. Theorems A and B shall be proved in section 3, as consequences of the results already established in section 2. Here we also prove that any finite abelian group is a Galois group over any p-adic function field. Finally, we prove Theorems D and E (which are results over function fields over higher-dimensional local fields) in section 4.

Notations and conventions
The following conventions shall be deployed throughout the article.
Cohomology. Unless stated otherwise, all (hyper-)cohomology groups will be étale or Galois. We abusively identify each object of an abelian category to the corresponding 1-term complex concentrated in degree 0. Over a field K, a K-variety is a separated scheme of finite type X → Spec K. We use K to denote a fixed separable closure of K, Γ K = Gal(K/K) to denote its absolute Galois group, and X = X × K K. We denote by D + (X) (resp. D + (K), resp. D + (Ab)) the bounded-below derived category of étale sheaves over X (resp. of discrete Γ K -modules 2 , resp. of abelian groups). If v is a place of K, K v denotes the corresponding completion, X v = X × K K v , and loc v : H i (X, −) → H i (X v , −) denotes the localization map in cohomology. Whenever K is the function field of a smooth projective geometrically integral curve Ω over a field k, Ω (1) denotes the set of closed points of Ω. For each v ∈ Ω (1) , K v (resp. O v ) denotes the v-adic completion of K (resp. of the local ring O Ω,v ), while K h v and O h v denote the corresponding henselizations, and k(v) denotes the residue field of v.
Motivic complexes. Let X be a smooth variety over a field K. Lichtenbaum defined arithmetic complexes Z(i) over X ét for i = 0, 1, 2 [Lic87, Lic90] (we shall write Z X (i) if we want to emphasize the variety X). We have quasi-isomorphisms Z(0) ∼ = Z and Z(1) ∼ = G m [−1]. The complex Z(2) is concentrated in degrees 1 and 2. Also, there is a pairing . We also use this as the definition of the sheaf Q/Z(i) for i / ∈ {1, 2}. Abelian groups. For a topological abelian group A (the topology is understood to be discrete if not specified), A D = Hom cts (A, Q/Z) denotes its Pontrjagin dual. For n ≥ 1, we denote by A n the n-torsion subgroup of A, and A tors = lim − →n A n . Tori. If G is a smooth algebraic group over a field K, we denote by G = Hom K (G, G m ) its Γ K -module of geometric characters, that is, G = Hom K (G, G m ) equipped with the Galois action defined by the formula If T is a torus over a field K, its dual torus is defined to be the torus T ′ whose character module is the cocharacter module .
in D + (K). We say that T is quasi-split if it is isomorphic to Res A/K G m,A for some étale K-algebra A, where Res A/K denotes the restriction of scalars à la Weil. This is equivalent to saying that T is a permutation module (i.e. it has a Γ K -invariant Z-basis). In this case, H 1 (K, T ) = 0 by Shapiro's lemma and Hilbert's Theorem 90. Also, T ′ is quasi-split.
Tate-Shafarevich groups. Let K be the function field of a smooth projective geometrically integral curve Ω over a field k. Let S ⊆ Ω (1) be a finite set of closed points, and C a complex of Γ K -modules. For i ∈ Z, we define the groups Unramified cohomology. Let X a smooth integral variety over a field K, n ≥ 1 an integer invertible in K, i ≥ 0, and j ∈ Z. Let µ ⊗j n := Hom K (µ ⊗(−j) n , Z/n) if j < 0 and µ ⊗0 n := Z/n. One defines the unramified part H i nr (K(X)/K, µ ⊗j n ) to be the subgroup of H i (K(X), µ ⊗j n ) consisting of elements A that lift to H i (O, µ ⊗j n ) for every discrete valuation ring O ⊇ K with field of fractions K(X) (such a lifting is necessarily unique by the injectivity property, see [CT95, Theorem 3.8.1]). If X is proper, this amounts to requiring that A comes from H i (O X,v , µ ⊗j n ) for every point v ∈ X of codimension 1 (see Theorem 4.1.1 in loc. cit.). The group H i nr (K(X)/K, µ ⊗j n ) is a K-stable birational invariant [CT95, Proposition 4.1.4]). We define the "evaluation" pairing for any overfield K ′ /K, as follows. By Bloch-Ogus theorem (Gersten's conjecture for étale cohomology) [BO74] (see also [CT95,Theorem 4 , and then define A(P ′ ) to be its image by the pullback . Assume in addition that K has characteristic 0 and that X is proper. For i ≥ 3, there is a natural map (for the case where i = 3, see [Kah12, Proposition 2.9] and [HSS15,(19) defined as follows. First, let Z(2) Zar be the complex concentrated in degrees ≤ 2 defined in a similar way to Z(2), but over the small Zariski site X Zar . Let α : X ét → X Zar denote the changeof-sites map. Then the adjunction Q(2) Zar → Rα * Q(2) is an isomorphism [Kah12, Théorème 2.6(c)]. Since Q(2) Zar is concentrated in degrees ≤ 2, one has R i α * Q(2) = R i+1 α * Q(2) = 0, hence R i+1 α * Z(2) ∼ = R i α * Q/Z(2). The Leray spectral sequence for α yields an edge map Algebraic groups. Let K be a field, X a smooth K-variety and G a smooth K-group scheme. The (non-abelian) étale cohomology (pointed) set H 1 (X, G) classifies X-torsors under G. Let f : Y → X be such a torsor, with class [Y ] ∈ H 1 (X, G). For each Galois cocycle c : Γ K → G(K), we may twist G to obtain an inner K-form c G of G, and a torsor f c : c Y → X under c G. For each P ∈ X(K), we have the following equivalence: (see [Sko01,p. 22]). If G is commutative, where π : X → Spec K denotes the structure morphism. The following continuity result is perhaps well-known. We present a proof here for the lack of a reference.
Lemma 1.1. Let K be a topological field (i.e. its addition, multiplication and inversion are continuous), for example, a field equipped with an absolute value. Let X be a smooth integral variety over K and consider the natural topology on X(K) [Con12, Proposition 3.1].
(i). Let G be a smooth, quasi-projective, commutative group scheme over K. Then for any i ≥ 0 . Suppose that K is complete with respect to an absolute value. Let G be a (not necessarily commutative) smooth K-group scheme. Then for any torsor Y → X under G with class Proof. We prove (i), the proof of (ii) being similar. Denote by π : Chapter III, Remark 3.11], there is an étale neighborhood f : X ′ → X of P 0 and a point P ′ 0 ∈ X ′ (K) such that f (P ′ 0 ) = P 0 and f * A = 0. Now, f induces a homeomorphism between a neighborhood (for the K-topology) U ′ ⊆ X ′ (K) of P ′ 0 and a neighborhood U ⊆ X(K) of P 0 . Each point P ∈ U (K) then has a factorization Spec K → X ′ f − → X, thus the evaluation-at-P factorizes through the pullback map Let us now show (iii). Fix a K-point P 0 ∈ X(K) and let c : Γ K → G(K) be a Galois cocycle representing [Y ](P 0 ) ∈ H 1 (K, G). Twisting by c yields a torsor c Y under the inner form c G, which contains a K-point lying over P 0 (in view of (1.3.6)). Since K is complete with respect to an absolute value, we may apply Serre's generalized version of the implicit function theorem [Ser92, Part II, Chapter III, §10.2]. This assures the existence of a neighborhood U ⊆ X(K) of P 0 , whose K-points P has can be lifted to K-points of c Y . In other words, [Y ](P ) = [c] = [Y ](P 0 ) (again, by (1.3.6)) for all P ∈ U .
Following [CTS07, §4.2], we shall say that G is special if any torsor under G over a K-variety is locally trivial for the Zariski topology. This implies H 1 (L, G) = 1 for any overfield L/K. It follows that if H ⊆ G is a Zariski closed subgroup and X = H\G (hence the projection G → X is a torsor under H), then the evaluation map induces a bijection X(L)/G(L) ∼ = H 1 (L, H). Serre proved that any special group is linear and connected [Ser58, §4.1, Théorème 1]. Examples of special groups are the general linear group GL n (or more generally, GL A for a central simple algebra A over K), the special linear group SL n , or the symplectic group Sp 2n .

Reciprocity obstruction
The generalized Weil reciprocity law is the key to defining the reciprocity obstructions to the local global principle and weak approximation for varieties over fields with arithmetico-geometric nature. In this paper, we work with function fields of curves over p-adic fields, and more generally, higher-dimensional local fields. We shall recall their definition and properties here. By definition, a 0-dimensional local field is either a finite field or the field C((t)) for some algebraically closed field C of characteristic 0. These fields have absolute Galois group Z (for C((t)), this is the celebrated Puiseux's theorem), hence have cohomological dimension 1. For d ≥ 1, a d-dimensional local field is a complete discretely valued field whose residue field is a (d − 1)-dimensional local field.
Proposition 1.2. Let k be a d-local field satisfying the condition (⋆) from page 4.
Proof. Note that the commutativity of (1.4.3) follows from that of (1.4.2), since the composite is the multiplication by [ℓ : k], and since Q/Z is divisible. We prove (i), (ii), and (iii) by induction on d. For d = 0, one has Γ k ∼ = Z, so (i) and (ii) are obvious. Furthermore, for any finite extension ℓ/k, the inclusion Γ ℓ ֒→ Γ k is the multiplication by [ℓ : k] on Z, which implies (iii). Suppose that d ≥ 1, and let κ = k d−1 denote the residue field of k (which is a (d − 1)-local field). We recall from [Ser94, Chapitre II, Annexe, (2.2)] that for any n ≥ 1, i ≥ 0 and j ∈ Z, the Hochschild-Serre spectral sequence yields an exact sequence where ∂ is the residue map. Since cd(κ) = d, setting i = j = d in (1.4.4) yields Furthermore, one has cd(k) = d + 1 by [Ser94, Chapitre II, §4.3, Proposition 12]. It remains to establish diagram (1.4.2). For this, we make use of the following standard property of ∂. If ℓ is a finite extension of k with residue field λ, then we have a commutative diagram where e is the ramification index (this follows easily from the description of the residue map in [Kat86, §1, (1.3)(i)]). Since [ℓ : k] = e · [λ : κ], we deduce the commutativity of (1.4.2) from that of the same diagram for the extension λ/κ.
From now on, let k be a d-dimensional local field satisfying the condition (⋆) from page 4 and let K be the function field of a smooth projective geometrically integral curve Ω over k. For each closed point v ∈ Ω (1) , the v-adic completion K v is a (d + 1)-dimensional local field, a field of cohomological dimension cd(K v ) = d + 2. The field K itself has cohomological dimension cd(K) ≤ d + 2. Indeed, by [Har17, Proposition 5.10], it is enough to show that cd(k(t)) ≤ d + 2. Let A be any torsion Γ k(t) -module. Since Γ k ∼ = Gal(k(t)/k(t)), we have a spectral sequence H p (k, H q (k(t), A)) ⇒ H p+q (k(t), A). But cd(k(t)) ≤ 1 by Tsen's theorem [GS17, Proposition 6.2.3, Theorem 6.2.8] and cd(k) = d + 1, hence H i (k(t), A) = 0 for i > d + 2, or cd(k(t)) ≤ d + 2. Proposition 1.3 (Generalized Weil reciprocity law). We have a complex where σ is the sum of the isomorphisms H d+2 (K v , Q/Z(d + 1)) ∼ = Q/Z from Proposition 1.2(ii).
Proof. We prove this by applying the reciprocity law for cycle modules.  Let X be a smooth geometrically integral K-variety such that v∈Ω (1) X(K v ) = ∅. The evaluation pairings defined in (1.3.3) for the extensions K v /K reassemble into a pairing H d+2 nr (K(X)/K, Q/Z(d + 1)) × v∈Ω (1) (the above sum is finite by [CTPS16, Proposition 2.5(i)], see also [HSS15,Lemma 5.1] for the case where k is p-adic). Thanks to (1.4.5), the above pairing vanishes on the image of H d+2 (K, Q/Z(d + 1)) in H d+2 nr (K(X)/K, Q/Z(d + 1)). Hence it induces a pairing H d+2 nr (K(X)/K,Q/Z(d+1)) which vanishes on the diagonal image of X(K) on v∈Ω (1) X(K v ) (by (1.4.5) again). The absence of a family (P v ) v∈Ω (1) v∈Ω (1) X(K v ) orthogonal to H d+2 nr (K(X)/K, Q/Z(d + 1)) is an obstruction to the existence of K-rational points on X. We refer to this as the reciprocity obstruction to the local-global principle for X. We are also interested in the following coarser obstruction. Restricting (1.4.6) to locally constant classes yields a map If X(K) = ∅, then ρ X = 0. We refer to the non-vanishing of ρ X as the (unramified) first obstruction to the local-global principle for X. Suppose that X(K) = ∅. By Lemma 1.1(ii), the pairing (1.4.6) vanishes on the closure (for the product of v-adic topologies) of X(K) in v∈Ω (1) X(K v ). The non-orthogonality to is then an obstruction to approximating this family by K-rational points. We refer to this as the reciprocity obstruction to weak approximation for X. We are also interested in the following coarser obstruction to weak approximation. For any finite set S ⊆ Ω (1) , (1.4.6) restricts to a pairing which vanishes on the closure of X(K). The non-orthogonality of a family (P v ) v∈S to the subgroup of "constant-outside-S" elements is then an obstruction to approximating this family by K-rational points.
We also note that in the case where d = 1 (for example, when k is p-adic), by the Gersten resolution (1.3.4), the pairing (1.4.6) induces a pairing Similarly, the map (1.4.7) induces a map (1.4.10) Actually, the present article only deals with the (unramified) reciprocity obstruction in the case where k is p-adic. For function fields of curves over higher-dimensional fields, we do not prove that the obstruction is unramified. Instead, we shall work with the following adapted version of the pairing (1.4.6). Again, let K be the function field of a smooth projective geometrically integral curve Ω over a d-dimensional local field k satisfying the condition (⋆) from page 4. Let X be a smooth geometrically integral K-variety. For a non-empty open subset U ⊆ Ω, we shall call a smooth, separated, finitely presented scheme X → U such that X × U K = X an integral model of X over U. Such an integral model exists when U is sufficiently small. For v ∈ U (1) , we have X (O v ) ⊆ X(K v ) by the valuative criterion for separatedness. We define the set X(A K ) of adelic points of X as the subset of families (P v ) v∈Ω (1) ∈ v∈Ω (1) X(K v ) such that P v ∈ X (O v ) for all but finitely many v ∈ U (1) . If X(A K ) = ∅, we consider the pairing ) by virtue of the generalized Weil reciprocity law (1.4.5). Thus, we obtain a pairing which vanishes on X(K) (again, by (1.4.5)). Restricting (1.4.11) to locally constant classes yields a map which vanishes whenever X(K) = ∅. We refer to the non-vanishing of ρ X as the (adelic) first obstruction to the local-global principle for X. Now, suppose that X(K) = ∅. For any finite set S ⊆ Ω (1) , (1.4.11) restricts to a pairing which vanishes on the closure (for the product of v-adic topologies) of X(K) in v∈S X(K v ) (this uses Lemma 1.1(i)). The non-vanishing of a family (P v ) v∈S to the group on the left-hand side of (1.4.13) is an obstruction to approximating this family by K-rational points. We refer to this as the generalized Brauer-Manin obstruction to weak approximation for X in S.

Arithmetic duality theorems
This paragraph reassembles the tools for proving our main theorems, namely the local and global duality theorems for Galois cohomology of curves over higher-dimensional local fields.
We start with the case where k is a d-dimensional local field satisfying the condition (⋆) from page 4. Recall that for n ≥ 1, we have an isomorphism H d+1 (k, µ ⊗d n ) ∼ = Z/n from (1.4.1). By [Mil06, Chapter I, Theorem 2.17], if F is a finite n-torsion Γ k -module and F ′ = Hom k (F, µ ⊗d n ), the cupproduct pairing is a perfect duality of finite groups for 0 ≤ i ≤ d + 1. Suppose that d ≥ 1 and that F extends to a group scheme F over the ring of integers O of k (thus F ′ also extends to F ′ = Hom O (F , µ ⊗d n )).
, and these subgroups are exact annihilators of each other under the above duality pairing [Izq16, Proposition 2.5].
Let K be the function field of a smooth projective geometrically integral curve Ω over k. If j : U ֒→ Ω is a non-empty open subset and F is a complex of sheaves on U ét , we define the hypercohomology groups with compact support where j ! denotes the extension by zero [Mil80,p. 93]. From the proof of [Izq14, Lemme 1.3], one has a canonical isomorphism H d+3 , one has the Yoneda product pairing for cohomology with compact support (see [Mil80,p. 168]) On the other hand, since F ∼ = Hom U (F ′ , Q/Z(d + 1)), the spectral sequence which is in fact a perfect duality of finite groups [Izq16, Proposition 2.1]. We refer to it as the Artin-Verdier duality pairing. This pairing induces a perfect duality pairing If U is sufficiently small, we may lift η to an element η U ∈ H i (U, F ) and α to an element α U ∈ H d+3−i (U, F ′ ). By the localization exact sequence for cohomology with compact support (which can be proved by exactly the same argument as in [Mil06, Chapter II, The non-degeneracy of (1.5.2) shall serve in the proof of Theorem D.
In order the prove Theorem E, one needs the exact sequence [Izq17, Lemme 1.2] for any finite subset S ⊆ Ω (1) and 1 ≤ i ≤ d + 1, which is established in the course of establishing the Poitou-Tate sequence for finite modules. Here, the map θ is defined by Let us now focus on the case where k is a p-adic field, i.e. a finite extension of Q p (hence d = 1). Let T be a K-torus. Recall that the dual torus T ′ is the torus with character module T ′ = T . For each v ∈ Ω (1) , local duality between the tori T and T ′ over the 2-dimensional local field K v asserts that the pairing (1.3.2) induces a cup-product pairing which is a perfect duality of finite groups. For a sufficiently small non-empty open subset U ⊆ Ω, where the first arrow is the Yoneda product pairing for cohomology with compact support [Mil80,p. 168], the second arrow is induced by the above pairing, and the isomorphism H 5 where the first arrow is induced by the natural morphism , and the second arrow is the edge map from the spectral sequence From (1.5.6) and (1.5.7), we obtain a pairing We refer to it as the Artin-Verdier pairing (for tori). Unlike in the finite case, this is not a perfect duality (nor are the concerning cohomology groups finite). Nevertheless, for i = 1 (and by exchanging T and T ′ ), this induces a perfect duality pairing of finite groups, as follows (see Theorem 1.3 and the proof of Theorem 4.1 in [HS16]). Let η ∈ X 2 (K, T ) and α ∈ X 1 (K, T ′ ). If U is sufficiently small, we may lift η to an element η U ∈ H 2 (U, T ) and α to an element α U ∈ H 1 (U, T ′ ). By the localization exact sequence starting from degree 1 [HS16, Corollary 3.2], η U comes from an element η c U ∈ H 2 c (U, T ). Then η, α PT = η c U , α U AV . In this article, we shall need the fact that (1.5.9) is also induced by (1.5.8) for i = 2 (without exchanging T and T ′ ). To see this, it suffices to show the following Lemma 1.4. We have a commutative diagram of pairings [HS16,(26)], for any n ≥ 1, one has two commutative diagrams of pairings (2)) and From the commutativity of the above diagrams, one has It follows that the pairing (1.5.9) has the following alternative description. Let η ∈ X 2 (K, T ) and α ∈ X 1 (K, T ′ ). Lift η to an element η U ∈ H 2 (U, T ) and α to an element α U ∈ H 1 (U, T ′ ) (shrinking U if necessary). By the localization sequence (1.5.10), α U comes from an element α c U ∈ H 1 c (U, T ′ ). Then η, α PT = η U , α c U AV . The non-degeneracy of (1.5.9) shall serve in the proof of Theorem A. As for Theorem C, one needs the following part of the Poitou-Tate exact sequence for tori [HSS15, Proposition 3.5]: (1.5.11) This is an exact sequence of topological abelian groups. The groups H 1 (K, T ) and H 1 (K, T ′ ) are discrete. The group P 1 (K, T ) is, by definition, the topological restricted product of the finite groups The map θ is defined by .
In order to prove Theorem B, we require the following exact sequence. Let S ⊆ Ω (1) be any finite set. Since any element of v∈S H 1 (K v , T ) can completed by 0 into an element of P 1 (K, T ), (1.5.11) restricts to the three last terms of the exact sequence of discrete abelian groups. Dualizing this sequence and exchanging T and T ′ , one obtains an exact sequence where the map θ is defined by We conclude this section with the following lemma, which is [HSS15, Lemma 4.2(a)], whose proof relies on Tate-Lichtenbaum duality for p-adic curves. It is crucial for the constructions used in the proofs of Theorems A, B, and C.
Lemma 1.5. If Q is a quasi-split torus over a p-adic function field K, then X 2 ω (K, Q) = 0.
This lemma is used not only to establish the unramified nature of the obstructions but also the Poitou-Tate sequence for tori. For K a function field of a curve over a d-dimensional local field (where d ≥ 2), one would need the vanishing of X d+2 Bloch [Blo86], or equivalently, the vanishing of X 2 (K, G m ) [Izq14, Lemme 3.15]. Unfortunately, this is not always the case (this problem was studied by Izquierdo in [Izq15, §4 and §5]).
Remark 1.6. An independent but interesting consequence of Lemma 1.5 is the following localglobal principle. Let X be a Severi-Brauer variety over the function field K of a p-adic curve Ω. If X(K v ) = ∅ for all but finitely many v ∈ Ω (1) , then X(K) = ∅.

Descent theory
This section is devoted to the proof of Theorem C.

Preliminary remarks
We recall some facts. Let X be a smooth geometrically integral variety over a field K of characteristic 0. Following Colliot-Thélène and Sansuc (cf. [Sko01, Theorem 2.3.4, Definition 2.3.5]), we define the elementary obstruction e X ∈ Ext 2 K (Pic X, K[X] × ) to be the inverse class of the 2-fold Conversely, for any given In the case where X(K) = ∅, we have e X = 0 and torsors of any type exist.
In particular, we may regard λ * e X as an element of H 2 (K, M). The equivalence between the vanishing of this element and the existence of X-torsors of type λ is part of the "fundamental exact sequence" of Colliot-Thélène and Sansuc (see Theorem 2.3.6 and Corollary 2.3.9 in loc. cit.), which reads If Pic X is finitely generated as an abelian group and M is the K-group of multiplicative type such that M = Pic X (that is, an isomorphism M ∼ = Pic X is fixed), we call a torsor Y → X under M universal if its type is the identity morphism of M 3 . Indeed, the existence of such a torsor is equivalent to e X = 0. Assume furthermore that Pic X is free, then the Néron-Severi torus of X is by definition the K-torus T such that T = Pic X. For example, this is case when X is projective and rationally connected (combine [BLR90, §8.4, Theorem 1], [Deb01, Corollary 4.18], and [Kle71, Théorème 5.1]), that is, for any algebraically closed overfield K ′ /K and two general points P 0 , P 1 ∈ X(K ′ ), there exists a morphism γ : P 1 K ′ → X K ′ such that γ(0) = P 0 and γ(1) = P 1 . Examples of such varieties are smooth compactifications of geometrically unirational varieties (such as homogeneous spaces of connected linear algebraic groups; indeed, a celebrated theorem of Chevalley asserts that connected linear algebraic groups are geometrically rational, even K-unirational [Che54]).
Before starting, let us restate the main result of this section (i.e. Theorem C).
Theorem 2.1. Let K be the function field of a smooth proper geometrically integral curve Ω over a p-adic field k, and let X be a smooth proper geometrically integral variety over K such that the abelian group Pic X is finitely generated and free (for example, X is projective and rationally connected). Let T be the Néron-Severi torus of X. There exists a homomorphism with the following properties. Suppose that v∈Ω (1) X(K v ) = ∅, then (i). universal X-torsors exist (that is, e X = 0) if and only if there exists a family of v∈Ω (1) X(K v ) which is orthogonal to u(X 1 (K, T ′ )) relative to the pairing (1.4.9); consisting of the families orthogonal to Im(u) relative to the pairing (1.4.9), then (iii). there are only finitely many isomorphism classes of universal torsors Y → X such that (iv). if the universal torsors Y → X satisfy the local-global principle (resp. the local-global principle and weak approximation), then the reciprocity obstruction (1.4.9) to the local-global principle (resp. the local-global principle and weak approximation) on X attached to Im(u) is the only one.
Let K be a field of cohomological dimension cd(K) ≤ 3 and let π : X → Spec K be a smooth proper geometrically integral variety such that the abelian group Pic X is finitely generated and free. Let T be the Néron-Severi torus of X (that is, T = Pic X). We construct a map as in the statement of Theorem 2.1, as follows. First, since K[X] × = K × , we have the following distinguished triangle in the category D + (K): (2.1.5) Applying the exact functor − ⊗ L G m yields a distinguished triangle be the map induced by the pairing (1.3.1). Next, denote by θ 2 the composite • the second arrow is the canonical "base change" morphism constructed in [Fu11,p. 306], • the third arrow is induced by the natural map τ ≤1 Rπ * G m,X → Rπ * G m,X , • and the last arrow is induced by the adjunction π * Rπ * G m,X → G m,X .

Existence of universal torsors
In this paragraph, we prove Theorem 2.1(i). Let K be the function field of a smooth projective geometrically integral curve Ω over a p-adic field k. The point is to relate the first obstruction (1.4.10) and the global Poitou-Tate duality pairing (1.5.9), as in the following analogue of [CTS87a, Lemme 3.3.3] (see also [Sko01, (6.4)]).
Proposition 2.2. Let π : X → Spec K be a smooth proper geometrically integral variety such that the abelian group Pic X is finitely generated and free. Let T be the Néron-Severi torus of X. Assume in addition that v∈Ω (1) X(K v ) = ∅. In particular, the class η ∈ H 2 (K, T ) corresponding to the elementary obstruction e X ∈ Ext 2 K ( T , G m ) (under the identification (2.1.2)) belongs to X 2 (K, T ). Then, for all α ∈ X 1 (K, T ′ ), one has the equality Here, the map ρ X was defined in (1.4.10), and −, − PT is the pairing (1.5.9).
Proof. We follow the argument in [HS08,§3] and [HS16,Proposition 5.3]. First, we inspect the pairing η, − PT . By [BvH09, Lemma 2.3], the object τ ≤1 Rπ * G m,X in (2.1.5) is represented by the complex [K(X) × div −→ Div X] concentrated in degree −1 and 0. It follows that the class −e X ∈ Ext 2 Let π U : X → U be an integral model of X over some non-empty open subset U ⊆ Ω. We may assume that T extends to a U-torus T . Then T (resp. T ′ ) extends to the finitely presented locally constant group scheme T = Hom U (T , G m ) (resp. the U-torus T ′ = T ⊗ G m ). For U sufficiently small, π U * G m,X = G m and R 1 π U * G m,X = T , hence one has a distinguished triangle in D + (U), which extends (2.1.5). The inverse class e U ∈ Ext 2 U ( T , G m ) of the morphism T → G m [2] associated with (2.2.1) is a lifting of e X ∈ Ext 2 K ( T , G m ). We claim that there is a commutative diagram The two left triangles of (2.2.2) are obtained by applying H 2 (U, −) to the commutative diagram in D + (U). The map r 1 , r 2 , r 3 in (2.2.2) are the edge maps from the spectral sequences The two middle squares of (2.2.2) commute by the functoriality of these spectral sequences. The map γ 1 , γ 2 , γ 3 in (2.2.2) are induced by the respective pairings and (1.3.1). The other triangle and square of (2.2.2) obviously commute. Since r 1 is an isomorphism by [Sko01, Lemma 2.3.7], the class e U ∈ Ext 2 where means the Yoneda product, and where the top square commutes thanks to the construction of the cup-product (Artin-Verdier) pairing for cohomology with compact support (see (1.5.6) and (1.5.7)). Hence, we have the following equality for all α c U ∈ H 1 c (U, T ′ ): Let α ∈ X 1 (K, T ′ ). By the localization sequence (1.5.10), when U is sufficiently small, α lifts to an element α c U ∈ H 1 c (U, T ). The right hand side of (2.2.5) is η, α PT by the discussion following the construction of (1.5.9) in paragraph 1.5. The next step is to inspect the element ρ X (u(α)). Consider the commutative diagram Im H 4 (Kv,Z(2)) , (2.2.6) with exact rows (each map H 4 (K v , Z(2)) → H 4 (X v , Z(2)) is injective since X(K v ) = ∅). Since α ∈ X 1 (K, T ′ ), u(α) lies in the kernel of the right vertical arrow in (2.2.6). Let β ∈ v∈Ω (1) H 4 (Kv,Z(2)) Im H 4 (K,Z(2)) be its image by the snake lemma construction.

Description of the obstruction using universal torsors
In this paragraph, we prove Theorem 2.1(ii). When universal torsors exist, they give an explicit description of the map u in (2.1.4) as in the following analogue of [CTS87a, Lemme 3.5.2] (see also [Sko99, Lemma 3]).
Proposition 2.4. Let π : X → Spec K be a smooth proper geometrically integral variety such that the abelian group Pic X is finitely generated and free. Let T be the Néron-Severi torus of X. Suppose that Y → X is a universal torsor 4 under T . Then the map u constructed in (2.1.4) is equal to the composite . We claim that there is a commutative diagram where means the Yoneda product, and where the maps θ 1 , θ 2 were defined in the course of constructing the map θ from (2.1.7). To see this, let us consider the four rectangles of (2.3.2) from the top to the bottom. As for the first rectangle, the left bottom horizontal arrow is induced by the natural map τ ≤1 Rπ * G m,X → Rπ * G m,X (keeping in mind that Ext 1 K ( T , Rπ * G m,X ) = Ext 1 X ( T , G m )), and the right bottom horizontal arrow is induced by the map from triangle (2.1.5). The commutativity of this rectangle and the established isomorphisms are well-known; see for example [HS13, Proof of Proposition 8.1, Appendix B]. The second rectangle commutes by the functoriality of − ⊗ L − (bearing in mind that Ext 1 K (T ′ , Rπ * (G m,X ⊗ L G m,X )) = Ext 1 X (T ′ , G m ⊗ L G m )). As for the third rectangle, the left square obviously commutes (noting that, of course, Ext 3 K (T ′ , Rπ * Z(2)) = Ext 3 X (T ′ , Z(2))), and the right square is induced by diagram (2.1.8). As for the fourth rectangle, the left square obviously commutes, and the right square is obtained by taking cohomology of (2.3.1).
Let Y → X be a universal torsor. By our convention, its type is the identity of T . Hence, the image of [Y ] ∈ H 1 (X, T ) in H 4 (K, Z X/K (2)) by (2.3.2) is precisely λ * α. Under the identification (2.1.9), this is the same as u(α). Thus, in order to prove Proposition 2.4, it remains to show that the image of [Y ] in H 4 (X, Z(2)) by (2.3.2) is precisely [Y ] ∪ π * α. To this end, we argue as in the proof of Proposition 2.2 to obtain a commutative diagram  . This yields the identity θ 1 * γ 2 (ε) π * α = [Y ] ∪ π * α ∈ H 4 (X, Z(2)), which is exactly what we need. Proposition 2.4 is hence proved.
Proof of Theorem 2.1(ii). We start with the inclusion "⊆". Suppose that there exists a family (P v ) v∈Ω (1) orthogonal to Im(u) relative to the pairing (1.4.9). By (i), we know that there exists a universal torsor f : Y → X. In the light of Proposition 2.4, we have by the valuative criterion for properness. Furthermore, shrinking U if necessary, we may assume that Y extends to a torsor Y → X under a U-torus T extending T . Thus, . By virtue of (2.3.5) and the exact sequence (1.5.11), there exists t ∈ H 1 (K, T ) such that loc v (t) = [Y ](P v ) for all v ∈ Ω (1) . Twisting by a Galois cocycle representing t yields a torsor f ) (see (1.3.6)) for all v ∈ Ω (1) . The torsor t Y is again universal by the fundamental exact sequence (2.1.3). This proves the inclusion "⊆". Conversely . This obviously implies the identity (2.3.5), which means (P v ) v∈Ω (1) orthogonal to Im(u) by Proposition 2.4. This proves the inclusion "⊇".

End of the proof of Theorem C
In this paragraph, we finish the proof of Theorem 2.1 (i.e. Theorem C).
Proof of Theorem 2.1(iii). Suppose that there is a universal torsor Y → X (otherwise, there would be nothing to prove). In view of the fundamental exact sequence (2.1.3) 5 , we have to show that there are only finitely many classes t ∈ H 1 (K, T ) for which v∈Ω (1) t Y (K v ) = ∅. Equivalently, by (1.3.6), we have to show that the property (2.4.1) holds for only finitely many classes t ∈ H 1 (K, T ). Let X → U be a proper integral model of X over some non-empty open subset U ⊆ Ω. Shrinking U if necessary, we may assume that T extends to a U-torus T and Y extends to a torsor Y → X under T . Suppose that t ∈ H 1 (K, T ) satisfies (2.4.1). For all v ∈ U (1) , since X(K v ) = X (O v ) by the valuative criterion for properness, the class On the other hand, we have an exact sequence Proof of Theorem 2.1(iv). Suppose that the universal X-torsors under T satisfy the local-global principle. If there exists a family (P v ) v∈Ω (1) ∈ v∈Ω (1) X(K v ) orthogonal to Im(u) relative to the pairing (1.4.9), then it follows from (ii) that there exists a universal torsor f : Y → X such that v∈Ω (1) Y (K v ) = ∅. By our assumption that Y satisfies the local-global principle, one has Y (K) = ∅, a fortiori X(K) = ∅. Suppose that the universal X-torsors under T satisfy the local-global principle and weak approximation. Let (P v ) v∈Ω (1) ∈ v∈Ω (1) X(K v ) be a family orthogonal to Im(u), S ⊆ Ω (1) a finite set of closed points, and U v ⊆ X(K v ) a v-adic neighborhood of P v for each v ∈ S. By (ii), there exists a universal torsor f : Y → X and a family (Q v ) v∈Ω (1) ∈ v∈Ω (1) Y (K v ) such that f (Q v ) = P v for all v ∈ Ω (1) . By our assumption on Y , there exists a point Q ∈ Y (K) which belongs to We conclude this section with the following interesting Remark 2.5. In fact, the image of the map u from (2.
Consequently, the statements (i), (ii), and (iv) of Theorem 2.1 can be refined as follows (i). Universal X-torsors exist if and only if the map ρ X from (1.4.12) is the zero map.
(iv). If the universal torsors Y → X satisfy the local-global principle (resp. the local-global principle and weak approximation), then the obstruction to the local-global principle (resp. the local-global principle and weak approximation) on X, defined by the pairing (1.4.11), is the only one.

Local-global principle and weak approximation
This section is devoted to the proof of Theorems A and B. For each of these results, we offer two proofs. The first proofs invoke the results from section 2 (Theorem 2.1(i) for the local-global principle and Proposition 2.4 for the weak approximation). They are presented in paragraphs 3.1 and 3.2 respectively. The second proofs, which use the fibration methods, rely on an observation communicated to the author by Jean-Louis Colliot-Thélène. They shall be presented in paragraph 3.3. We also discuss some questions related to weak approximation in paragraph 3.4. In particular, we show that any finite abelian group is a Galois group over any p-adic function field, rediscovering the positive answer to the abelian case of the inverse Galois problem over Q p (t).

Local-global principle for stabilizers of type umult
We establish Theorem A in this paragraph. First, recall some facts. As a rule, the problem of the existence of rational points on homogeneous spaces is harder than that of weak approximation. It requires the general machinery of liens (or bands, kernels) and non-abelian Galois cohomology, which has been systematically studied in the last 30 years [Bor93,DLA19]. We refer to [FSS98, §1] for a complete exposition. Let X be a homogeneous space of a smooth algebraic group G over a field K. Let H denote the stabilizer of a K-point of X, which is supposed to be smooth. If H is commutative, it has natural K-form H. Otherwise, H need not be defined over K. Nevertheless, one can always define the associated Springer Klien L X (grosso modo, it is the K-group H equipped with a natural outer Galois action, i.e. a Galois action modulo conjugation), the set H 2 (K, L X ) of non-abelian Galois 2-cohomology, and the Springer class η X ∈ H 2 (K, L X ). The class η X is neutral if and only if X is dominated by a principal homogeneous space of G (if H 1 (K, G) = 1, for example, when G is special, this is equivalent to X(K) = ∅).
Since  (K, H), and its only neutral class is 0. Finally, a morphism L → L ′ of algebraic K-liens induces a relation H 2 (K, L) ⊸ H 2 (K, L ′ ). This turns out to be a map if either the underlying K-group of L ′ is commutative or the underlying morphism between K-groups is surjective.
The following description of the Picard groups of homogeneous spaces is due to Popov [Pop74, Corollary to Theorem 4], see also [BDH13] and [BvH12, Theorem 5.8].
Lemma 3.1. Let X be a homogeneous space a smooth, simply connected semisimple linear algebraic group G over a field K, with smooth geometric stabilizers H. Then, as a Γ K -module, Pic X is isomorphic to the character group of H (the Galois action on this group was defined above). The isomorphism is given by pushing forward the class [G] ∈ H 0 (K , H 1 (X, H)) of the torsor G → X under H.
Proof. Here we used the fact that Proposition 3], and Pic G = 0 since G is simply connected semisimple [Vos98, §4.3].
Let K be the function field of a smooth projective geometrically integral curve Ω over a padic field k, and let X be a homogeneous space of a simply connected semisimple linear algebraic group G over K, with geometric stabilizers H of type umult, hence an extension of a group M of multiplicative type by a unipotent group U. Since U (the unipotent radical of H) is characteristic in H, we have a natural Galois action on M = H/U (hence a K-form M of M). Since U does not have any non-trivial characters, the character module of H is just M , hence Pic X = M by Lemma 3.1. Let X c be a smooth projective compactification of X. Since X c is smooth, projective and geometrically unirational, the abelian group Pic X c is finitely generated and free (see the discussion preceding Theorem 2.1). There is an exact sequence where Div ∞ X c denotes the group of Weil divisors on X c supported in X c \ X (it is a permutation Proposition 3]. Let T (resp. Q) be the K-torus with character module Pic X c (resp. Div ∞ X c ). Then Q is quasi-split. We have exact sequences [Q ′ → T ′ ]) concentrated in degrees 0 and 1.
Remark 3.2. If M = F is finite abelian, the map T → Q on cocharacter modules is injective. Hence there is an exact sequence where F ′ = Hom K (F, Q/Z(2)). In this case, we have a quasi-isomorphism M ′ ∼ = F ′ .
(3.1.5) Furthermore, since the torus Q ′ is quasi-split, one has H 1 (L, Q ′ ) = 0 for any overfield L/K and X 2 ω (K, Q ′ ) = 0 by Lemma 1.5. The long exact sequence associated with (3.1.3) gives More generally, one has for any finite set S ⊆ Ω (1) . We define the map τ in (3.1.4) as the composite of (3.1.5), the map u : H 1 (K, T ′ ) → H 4 (X c ,Z(2)) Im H 4 (K,Z(2)) constructed in (2.1.4), and (3.1.6). This map τ shall serve as an obstruction to the local-global principle and weak approximation for X. .
We can now state Theorem 3.4 (Theorem A). Let K be the function field of a smooth projective geometrically integral curve Ω over a p-adic field k, and X a homogeneous space of a special, simply connected semisimple algebraic group G over K, with geometric stabilizers H of type umult. We keep the above notations; in particular, there is a map τ as in (3.1.4). If there exists a family (P v ) v∈Ω (1) ∈ v∈Ω (1) X(K v ) orthogonal to τ (X 2 (K, M ′ )) relative to the pairing (1.4.6), then X(K) = ∅. In particular, the unramified first obstruction (1.4.7) to the local-global principle for X is the only one.
First, we deal with unipotent stabilizers using the following well-known result. It shall also serve in the proof of Theorem B.
Lemma 3.5. Let K be a field of characteristic 0 and G a special algebraic group over K. Then homogeneous spaces of G with unipotent geometric stabilizers have K-rational points. They have weak approximation if G does.
Proof. Let X be such a homogeneous space. The Springer class η X is neutral by [Dou76, Chapitre IV, Théorème 1.3] (see also [Bor93,Corollary 4.2]), hence X is dominated by a principal homogeneous space of G. Since G is special, this means there exists a G-equivariant morphism φ : G → X. In particular, X(K) = ∅.
Let S be a finite set of places of K, Lemme 1.13]. If G has weak approximation, we find a point Q ∈ G(K) which belongs to v∈S φ −1 (U v ). Then φ(Q) ∈ X(K) belongs to v∈S U v .  H 1 (X, M )), i.e. the torsor Z → X L has type id. The construction Y → Z defines a morphism G X → G of algebraic K-gerbes, thus H 2 (K, L X ) → H 2 (K, M) maps η X to η.
Return to the proof of Theorem 3.4. Since (2.1.1) is functorial in X, the map Ext 2 It follows that the map H 2 (K, M) → H 2 (K, T ) sends η to an element η c corresponding to e X c (under the identification H 2 (K, T ) ∼ = Ext 2 K ( T , G m ) of (2.1.2)).
is a family orthogonal to τ (X 2 (K, M ′ )), then, as a family in v∈Ω (1) X c (K v ), it is orthogonal to u(X 1 (K, T ′ )) by the construction of τ (where u is the map constructed in (2.1.4)). Theorem 2.1(i) then implies that e X c = 0, or η c = 0. On the other hand, since H 1 (K, Q) = 0 (the torus Q being quasi-split), the long exact sequence associated with (3.1.2) assures that H 2 (K, M) → H 2 (K, T ) is injective. It follows that η = 0. By Lemma 3.6, the map where • X 1 is a homogeneous space of G × K SL n with Springer lien L X 1 ∼ = L X and Springer class η X 1 = η X , • X 2 = M\ SL n for some K-embedding M ֒→ SL n and some n, • φ is a torsor under SL n , • the fibres of ψ are homogeneous spaces of G with geometric stabilizers Ker(H → M ) = U .

Weak approximation for stabilizers of type umult
In this paragraph, we establish Theorem B. We start by recalling the following well-known result, which already appeared in [Che95] (see also [Har07,LA14] for finite subgroups of SL n ). We give a proof here for the sake of reference.
Lemma 3.7. Let G be a smooth algebraic group over a field K, H a smooth Zariski closed subgroup of G, and X = H\G. The projection G → X is then a torsor under H. For any finite set S of places of K, if a family (P v ) v∈S ∈ v∈S X(K v ) lies in the closure (for the product of local topologies) of the diagonal image of X(K), then ([G](P v )) v∈S belongs to the image of the localization H 1 (K, H) → v∈S H 1 (K v , H). The converse holds if G is special and has weak approximation.
Proof. Suppose that (P v ) v∈S lies in the closure of X(K). By Lemma 1.1(iii) Conversely, suppose that there exists h ∈ H 1 (K, H) such that loc v (h) = [G](P v ) for all v ∈ S. Since G is special, the evaluation-at- [G] map X(L) → H 1 (L, H) induces a bijection X(L)/G(L) ∼ = H 1 (L, H) for any overfield L/K. In particular, we may write h = [G](P ) for some P ∈ X(K). For each v ∈ S, let U v ⊆ X(K v ) be a neighborhood of P v , and let V v denote its preimage by the continuous map Under the hypothesis that G has weak approximation, there exists g ∈ G(K) which belongs to v∈S V v . Then P · g ∈ X(K) belongs to v∈S U v .
Remark 3.8. Taking G = SL n in 3.7, we see that weak approximation for the quotient H\ SL n is an intrinsic property of the algebraic K-group H (independent of the embedding H ֒→ SL n ). Indeed, if H ֒→ SL n and H ֒→ SL m are two embeddings, the quotient varieties H\ SL n and H\ SL m are K-stably birational by the "no-name lemma" [CTS07, §3.2].
Let K be the function field of a smooth projective geometrically integral curve Ω over a p-adic field k. Let H be a K-group of type umult, hence an extension of a K-group of multiplicative type M by a unipotent group U. Let X = H\G for some embedding H ֒→ G into a simply connected semisimple linear algebraic group G over K. By Lemma 3.1, Pic X = H = M as Γ K -modules. As in paragraph 3.1, let X c be a smooth projective compactification of X, T the K-torus with T = Pic X c , and Q = T /M (it is a quasi-split K-torus). Finally, let M ′ = M ⊗ L Z(1) (it is represented by a 2-term complex of tori).
Since X c (K) = ∅, by the fundamental exact sequence (2.1.3), there exists a universal torsor Y c → X c under T . By Proposition 2.4, the map τ from (3.1.4) is the composite where the subscript X c denotes pullback along the structure morphism X c → Spec K, and the last arrow is induced by (1.3.4). Using the alternative description (3.2.1), we shall prove the following "functoriality" property of τ , which is required when we apply the fibration method.
Lemma 3.9. The map τ enjoys the following properties.
(i). τ does not depend on the choice of the universal torsor Y c .
(ii). τ does not depend on the choice of the smooth projective compactification X c .
(iii). Let G, G 1 be simply connected semisimple linear algebraic groups over K. Let H, H 1 be Kgroups of type umult, equipped with respective embeddings into G, G 1 , and let X = H\G, X 1 = H 1 \G 1 . Let M, M 1 be the respective parts of multiplicative type of H, H 1 , and let τ, τ 1 be the respective maps constructed in (3.1.4). Assume that there exists a morphism ϕ : G → G 1 and a ϕ-equivariant dominant morphism φ : X → X 1 . Then there is a commutative diagram Im H 3 (K,Q/Z(2)) .
T ) for some t ∈ H 1 (K, T ) by virtue of the fundamental exact sequence (2.1.3) 6 . This yields a map Since t ∪ α is an element of H 4 (K, Z(2)) ∼ = H 3 (K, Q/Z(2)), by the description (3.2.1), we see that τ remains unchanged when Y is replaced by Y 1 .
We prove (ii). Let X c 1 be a second smooth projective compactification of X. Let T 1 be the K-torus with T 1 = Pic X c 1 , and Y c 1 → X c 1 a universal torsor. Then there is an exact sequence where Q 1 is a quasi-split torus. Thus there are K-tori R, R 1 , T 2 such that R, R 2 are quasi-split and T × K R ∼ = T 1 × R 1 ∼ = T 2 , and the diagram commutes. Let Y → X, Y 1 → X denote the respective restrictions of Y c → X c and Y c 1 → X c 1 , which are torsors whose type are respectively given by the bottom arrows of (3.2.2). Since X(K) = ∅, by the fundamental exact sequence (2.1.3), there exists a torsor Y 2 → X under T 2 whose type T 2 → Pic X is given by either of the two composites in (3.2.2). By the very same sequence, the image of [Y 2 ] in H 1 (X, T ) (resp. in H 1 (X, T 1 )) has the form [Y ] + t X (resp. [Y 1 ] + t 1,X ) for some t ∈ H 1 (K, T ) (resp. t 1 ∈ H 1 (K, T 1 )). Twisting the X c -torsor Y c (resp. Y c 1 ) by t (resp. by t 1 ) yields another universal torsor; this is something allowed by (i). Hence, we may assume that the image of [Y 2 ] in H 1 (X, T ) (resp. in H 1 (X, T 1 )) is precisely [Y ] (resp. [Y 1 ]). One then obtains a commutative diagram (the two squares on the right-hand side are (1.3.5)). By the description (3.2.1), this shows that the compactifications X c and X c 1 yield the same map τ . We prove (iii). By Nagata's Theorem [Nag63], there exists a compactification φ c : X c → X c 1 of φ. Let T (resp. T 1 ) be the K-torus with T = Pic X c (resp. T 1 = Pic X c 1 ). Then φ c induces a morphism ψ : T → T 1 of K-tori. Let Y c → X c (resp. Y c 1 → X c 1 ) be a universal torsor under T (resp. under T 1 ). Then both the contracted product Y c × T K T 1 and the pullback Y c 1 × X c 1 X c are X c -torsors under T 1 of type ψ * : T 1 → T . Again, by the fundamental exact sequence (2.1.3), Twisting the X c 1 -torsor Y c 1 by t 1 (which does not modify the map τ , thanks to (i)), we may assume that these images coincide. Cup-products with this common value then yield the oblique arrow in the diagram Proof. Let Y c → X c be a universal torsor 7 and let Y → X be its restriction to X, which is a torsor under T whose type is the map T → M = Pic X from (3.1.1). Since the (universal) torsor G → X under M has type id by Lemma 3.1, the fundamental exact sequence (2.1.3) assures that the map H 1 (X, M) → H 1 (X, T ) sends [G] to [Y ] + t X for some t ∈ H 1 (K, T ). Twisting Y c by t (which does not change the map τ , by Lemma 3.9(i)), we may assume that the image of [G] in Since the pairing (3.2.3) is functorial, the diagram commutes (except the left triangle, which commutes up to a sign). Indeed, the square on the right-hand side is (1.3.5). By the description (3.2.1) of τ , the Lemma is proved.
The next step is to establish following analogue of the exact sequences (1.5.4) and (1.5.12).
Lemma 3.11. For each finite set S ⊆ Ω (1) of closed points, there is an exact sequence Here the local cup-products Proof. Consider the commutative diagram with exact rows (the two top rows are the exact sequences associated with (3.1.2), noting that H 1 (L, Q) = 0 for any overfield L/K since Q is quasi-split). The two bottom horizontal arrows are isomorphisms since H 1 (L, Q ′ ) = 0 for any overfield L/K and X 2 (K, Q ′ ) = X 2 S (K, Q ′ ) = 0 by Lemma 1.5. That (3.2.5) commutes follows from the functoriality of (3.2.3). The right column is exact because it is (1.5.12). The left vertical arrow has dense image (the torus Q has weak approximation since it is K-rational). For each v ∈ S, the map Q(  1.1(iii). A diagram chasing then shows that the left column (i.e. the sequence (3.2.4)) is exact.
Theorem 3.12. Let K be the function field of a smooth projective geometrically integral curve Ω over a p-adic field k, M a K-group of multiplicative type, and let X = M\G, where M ֒→ G is some embedding into a special, simply connected semisimple algebraic group G over K that has weak approximation. Let M ′ = M ⊗ L Z(1), τ the map constructed in (3.1.4), and S ⊆ Ω (1) a finite set of closed points. Then any family (P v ) v∈S ∈ v∈S X(K v ) orthogonal to τ (X 2 S (K, M ′ )) relative to the pairing (1.4.8) lies in the closure of the diagonal image of X(K). Moreover, X has weak approximation in S if and only if X 2 S (K, M ′ ) = X 2 (K, M ′ ). Proof. Let (P v ) v∈S ∈ v∈S X(K v ) be a family orthogonal to τ (X 2 S (K, M ′ )) relative to the pairing (1.4.8). This means v∈S A(P v ) = 0 for any α ∈ X 2 S (K, M ′ ) and any lifting A ∈ H 3 nr (K(X)/K, Q/Z(2)) of τ (α). By virtue of Lemma 3.10, this is equivalent to Any family (P v ) v∈S ∈ v∈S X(K v ) is orthogonal to τ (X 2 (K, M ′ )) since this group consists of everywhere locally constant classes and X(K) = ∅. It follows that X has weak approximation in S whenever . By Lemma 3.7, the family (P v ) v∈S does not lie in the closure of X(K), thus X fails approximation in S. This concludes the proof of the theorem.
Finally, we extend Theorem 3.12 to the main theorem of this section by allowing the stabilizers to have a unipotent part. The proof uses the fibration method.
Theorem 3.13 (Theorem B). Let K be the function field of a smooth projective geometrically integral curve Ω over a p-adic field k, H a linear algebraic K-group extension of a group multiplicative type M by a unipotent group U, and X = H\G for some K-embedding H ֒→ G into a special, simply connected semisimple linear algebraic group G over K that has weak approximation. Let M ′ = M ⊗ L Z(1), τ the map constructed in (3.1.4), and S ⊆ Ω (1) a finite set of closed points. Then any family (P v ) v∈S ∈ v∈S X(K v ) orthogonal to τ (X 2 S (K, M ′ )) relative to the pairing (1.4.8) lies the closure of the diagonal image of X(K). In particular, the reciprocity obstruction to weak approximation for X is the only one. Moreover, X has weak approximation in S if and only if X 2 S (K, M ′ ) = X 2 (K, M ′ ). Proof. Choose an embedding M ֒→ SL n for some n. The embedding H ֒→ G and the composite H → M ֒→ SL n yield a diagonal embedding H ֒→ G × K SL n . We have a diagram where X 1 = H\(G × K SL n ), X 2 = M\ SL n , φ is a torsor under SL n , and the fibres of ψ are homogeneous spaces of G with geometric stabilizers Ker(H → M ) = U . Let τ, τ 1 , τ 2 be the respective maps associated with X, X 1 , X 2 via the construction (3.1.4). By Lemma 3.9(iii), diagram (3.2.6) yields a commutative diagram Since H 1 (K(X), SL n ) = 1 by a variant of Hilbert's Theorem 90, the generic fibre of φ has a section. The extension K(X 1 )/K(X) is thus purely transcendental, hence φ * is an isomorphism.
. By Serre's generalized version of the implicit function theorem [Ser92, Part II, Chapter III, §10.2], we find for Since X(K) = ∅, any family (P v ) is orthogonal to τ (X 2 (K, M ′ )) (a subgroup consisting of everywhere locally constant classes). It follows that X has weak approximation in S whenever X 2 S (K, M ′ ) = X 2 (K, M ′ ). Conversely, suppose that X has weak approximation in S. We show that it is also the case for . By our assumption on X, there is a K-point P ∈ X(K) belonging to v∈S U v . Then the fibre φ −1 (P ) contains a family (Q ′ v ) v∈S ∈ v∈S ψ −1 (V v ). Since G is special and has weak approximation, the fibre φ −1 (P ) has a K-point Q ∈ v∈S ψ −1 (V v ). Then ψ(Q) ∈ X 2 (K) belongs to v∈S V v . Hence X 2 has weak approximation in S. By Theorem 3.12, one has X 2 S (K, M ′ ) = X 2 (K, M ′ ).

Alternative proofs
The idea of the "fibration method" at the end of the proofs of Theorems 3.4 and 3.13 can be applied in an alternative way. They can be used to show that any homogeneous space of a special, Krational algebraic group is K-stably birational to a K-torsor under a torus. If such a homogeneous space has a K-rational point, it is K-stably birational to a torus. This observation allows us to obtain Theorems 3.14 and 3.15 in this paragraph, which are variants of Theorems 3.4 and 3.13 respectively (of course, they also imply Theorems A and B respectively).
Theorem 3.14 (Theorem A). Let K be the function field of a smooth projective geometrically integral curve Ω over a p-adic field k, and X a homogeneous space of a special, K-rational algebraic group G, with geometric stabilizers H of type umult. Let M denote the natural K-form of the part of multiplicative type M of H and M ′ = M ⊗ L Z(1). Then X is K-stably birational to a K-torsor under a torus. Moreover, there exists a map nr (K(X)/K,Q/Z(2)) Im H 3 (K,Q/Z(2)) (3.3.1) with the following property. If there exists a family (P v ) v∈Ω (1) ∈ v∈Ω (1) X(K v ) orthogonal to τ 1 (X 2 (K, M ′ )) relative to the pairing (1.4.6), then X(K) = ∅. In particular, the unramified first obstruction (1.4.7) to the local-global principle for X is the only one.
The action of T on Z (by multiplication in T (K) = Z(K)) makes Z a homogeneous space of Q with geometric stabilizers M . We shall show that the Springer lien L Z is isomorphic to lien(M), and the Springer class η Z ∈ H 2 (K, M) is precisely η. To this end, we invoke the description of L Z and η Z in terms of cocycles (see [DLA19, §2.2.2] or [FSS98,§5]). Indeed, for each σ ∈ Γ K , its action on Q(K) restricts to the usual Galois action on M(K), so that L Z = lien(M). Next, fix the point 1 ∈ T (K) = Z(K). Then (3.3.3) yields where · : Z×Q → Z denotes the action of Q on Z induced by that of T . We have q σ q σ τ q −1 στ = ι(m σ,τ ) for all σ, τ ∈ Γ K , hence η Z is represented by the 2-cocycle {m σ,τ } σ,τ , i.e. η Z = η.
To conclude, the map H 2 (K, L X ) → H 2 (K, M) sends η X to η Z . Applying [DLA19, Theorem 3.4], we obtain a diagram X 1 where • X 1 is a homogeneous space of G × K Q with Springer lien L X 1 ∼ = L X and Springer class η X 1 = η X , • φ is a torsor under Q, • the fibres of ψ are homogeneous spaces of G with geometric stabilizers Ker(H → M ) =: U .
Since G is special, φ has a K-rational section. Since G is K-rational, the extension K(X 1 )/K(X) is purely transcendental. Thus X is K-stably birational to X 1 . Next, by Lemma 3.5, the generic fibre of ψ is isomorphic to U\G K(Z) , where U ⊆ G K(Z) is a unipotent Zariski closed subgroup. Since both U and G K(Z) are K(Z)-rational (for U, this is because its exponential map is a biregular isomorphism onto an affine space), the field extension K(G)/K(X 1 ) and K(G)/K(Z) are purely transcendental. It follows that X 1 (hence also X) is K-stably birational to Z. In particular, φ and ψ induce isomorphisms between H 3 nr (K(X)/K, Q/Z(2)), H 3 nr (K(X 1 )/K, Q/Z(2)), and H 3 nr (K(Z)/K, Q/Z(2)).
For the problem of weak approximation, the above proof is actually simpler, since the involved homogeneous spaces already have K-rational points.
Theorem 3.15 (Theorem B). Let K be the function field of a smooth projective geometrically integral curve Ω over a p-adic field k, H a K-group of type umult, and X = H\G for some K-embedding H ֒→ G into a special, K-rational algebraic group G. Let M denote the part of multiplicative type of H, and M ′ = M ⊗ L Z(1). Then X is K-stably birational to a torus. Furthermore, let τ 1 be the map X 2 ω (K, M ′ ) → H 3 nr (K(X)/K,Q/Z(2)) Im H 3 (K,Q/Z(2)) constructed as in (3.3.1), and S ⊆ Ω (1) a finite set of closed points. Then any family (P v ) v∈S ∈ v∈S X(K v ) orthogonal to τ 1 (X 2 S (K, M ′ )) relative to the pairing (1.4.8) lies the closure of the diagonal image of X(K). In particular, the reciprocity obstruction to weak approximation for X is the only one. Moreover, X has weak approximation in S if and only if X 2 S (K, M ′ ) = X 2 (K, M ′ ).
Proof (after J.-L. Colliot-Thélène). Following the proof of Theorem 3.14, let M ֒→ Q be an embedding into a quasi-split torus. The embedding H ֒→ G and the composite H → M ֒→ Q yield a diagonal embedding H ֒→ G × K Q. This gives us a diagram of the map (3.3.5) coincides with the construction using "flasque resolution" by Harari, Scheiderer and Szamuely, which serves in the proof of their theorem on weak approximation for K-tori [HSS15, Theorem 5.2]. Actually, combining this with Theorem 4.3(a) in loc. cit. gives us a more precise statement, that any family (P v ) v∈S ∈ v∈S T (K v ) orthogonal to τ 2 (X 2 S (K, T ′ )) lies in the closure of the diagonal image of T (K). Moreover, T has weak approximation in S if and only if X 2 S (K, T ′ ) = X 2 (K, T ′ ). By repeating the fibration argument as in the proof of Theorem 3.13, one sees that the map τ 1 defined in (3.3.1) has the stated property. Moreover, X has weak approximation in S if and only if X 2

Examples
Let us discuss some corollaries to Theorem 3.13. As always, let K be the function field of a smooth projective geometrically integral curve Ω over a p-adic field k. Recall that when M is a K-group of multiplicative type, M ′ = M ⊗ L Z(1) is quasi-isomorphic to a 2-term complex of tori fitting into the distinguished triangle (3.1.3). If M = F is finite abelian, then M ′ is quasi-isomorphic to F ′ = Hom K (F, Q/Z(2)) (see Remark 3.2). In what follows, if H is a K-group of type umult, then M denotes its part of multiplicative type.
Corollary 3.16. Let H be a K-group of type umult and X = H\ SL n for some embedding H ֒→ SL n . Then X has weak approximation if and only if X 2 ω (K, M ′ ) = X 2 (K, M ′ ).
Proposition 3.17. Let H be a K-group of type umult and X = H\ SL n for some embedding H ֒→ SL n . There is an infinite set S 0 ⊆ Ω (1) in which X has weak approximation.
Proof. Let L/K be a finite extension splitting the torus T ′ in the triangle (3.1.3). It corresponds to a branched cover f : Ω ′ → Ω of smooth projective geometrically integral curves over k. For any non-empty open subset U ⊆ Ω, a result of Poonen [Poo01, Corollary 2] assures the existence of a closed point w ∈ f −1 (U) such that k(w) = k(f (w)). Hence, there are infinitely many points w ∈ Ω ′ having this property. If moreover f is unramified over f (w), then L w = K f (w) . Thus, the set S 0 of closed points v ∈ Ω for which there exists a point w ∈ f −1 (v) with L w = K v is infinite.
We claim that X 2 S (K, M ′ ) = X 2 (K, M ′ ) for any finite set S ⊆ S 0 (which would conclude the proof by virtue of Theorem 3.13). Indeed, let α ∈ X 2 S (K, M ′ ), that is, loc v (α) = 0 for any v / ∈ S. For v ∈ S, we have by definition of S 0 a closed point w ∈ Ω ′ lying over v such that L w = K v . We deduce from (3.1.6) and Hilbert's Theorem 90 that X 2 ω (L, It follows that α ∈ X 2 (K, M ′ ).
A theorem of Harbater [Har87] says that every finite group is a Galois group over Q p (t). His original proof involves the technique of patching. There are several other proofs by Liu [Liu95], Colliot-Thélène [CT00], and Kollár [Kol00,Kol03]. As remarked by Colliot-Thélène himself, the inverse Galois problem for a group G (viewed as a finite constant group) over number fields can be reduced to the question of weak weak approximation (see below) on G\ SL n . Using this idea, we show Corollary 3.18. Any finite abelian group is a Galois group over K.
Proof. Let F be a finite abelian group, which can be seen as a finite constant K-group scheme. By Proposition 3.17, the variety F \ SL n (for some embedding F ֒→ SL n ) has weak approximation in some infinite set S 0 ⊆ Ω (1) . By Lemma 3.7, this means the localization map Let v : F → S 0 be any injective map. For each x ∈ F , choose a continuous homomorphism c x : Γ K v(x) → F whose image contains x; this is possible since we have surjections Γ K v(x) ։ Γ k(v(x)) ։ Z (the residue field of k(v(x)) being a finite field, its absolute Galois group is Z), then it is enough to inflate the continuous homomorphism Z → F mapping 1 to x. Each homomorphism c x is an element of H 1 (K v(x) , F ). Then, there exists c ∈ H 1 (K, F ) such that loc v(x) (c) = c x for all x ∈ F . Thus c is a continuous homomorphism Γ K → F whose image contains every element of F , i.e. it is surjective. Its kernel is Γ L for some finite Galois extension L of K, and Gal(L/K) = Γ K /Γ L ∼ = F .
Nevertheless, the abelian case of the regular inverse Galois problem over Q (hence over any field of characteristic 0 by a base change argument) is known long before Harbater's theorem (see, e.g., [Ser07, §4.2]) 8 .
We say that a smooth K-variety X (with X(K) = ∅) satisfies the weak weak approximation property (resp. countable weak weak approximation property) if it has weak approximation away from a finite (resp. countable 9 ) set S 0 ⊆ Ω (1) . This means X has weak approximation in every finite set S ⊆ Ω (1) with S ∩ S 0 = ∅.
Corollary 3.19. Let H be a K-group of type umult and X = H\ SL n for some embedding H ֒→ SL n . Then X satisfies the weak weak approximation property (resp. countable weak weak approximation property) if and only if X 2 ω (K, M ′ ) is finite (resp. countable).
Conversely, suppose that X has weak approximation away from a finite (resp. countable) set S 1 ⊆ Ω (1) . By Theorem 3.13, we have X 2 S (K, M ′ ) = X 2 (K, M ′ ) for any finite set S ⊆ Ω (1) disjoint from S 1 . We deduce from this the exactness of the bottom row of the diagram (induced by the distinguished triangle (3.1.3)) for all v ∈ V (1) precisely when it comes from H 2 (V, F ′ ). Since the group H 2 (V, F ′ ) is finite, there exists a finite set S ⊆ V (1) , a closed point v 0 ∈ V (1) , and an element α ∈ X 2 are exact annihilators of each other under the cup-product pairing In view of the exact sequence (1.5.4), this family does not come from H 1 (K, F ), hence F fails the hyperweak approximation property.
The following vanishing result was communicated to the author by Jean-Louis Colliot-Thélène. Recall that a K-group of multiplicative type M is said to be split over a finite extension L/K if Γ L acts trivially on M . A finite group is said to be metacyclic if all of its Sylow subgroups are cyclic.
Proposition 3.22. If F is a finite Γ K -module split by a metacyclic extension, then X 2 ω (K, F ) = 0.
Proof. Let L/K be a finite Galois extension splitting F , with metacyclic Galois group G = Gal(L/K). The "coflasque resolution" [CTS87b, Lemma 0.6] provides an exact sequence of finitely generated G-modules, where T and Q are free as abelian groups, where Q is permutation, and where T is coflasque (that is, H 1 (H, T ) = 0 for all subgroups H ⊆ G). Since G is metacyclic, a theorem of Endo-Miyata [EM75, Theorem 1.5] says that T is a direct factor of a permutation module. By dualizing, we obtain an exact sequence where Q is a quasi-split torus and T is a direct factor of a quasi-split torus. Since H 1 (K ′ , T ) = 0 for any overfield K ′ /K and X 2 ω (K, Q) = 0 by Lemma 1.5, one has X 2 ω (K, F ) = 0.
Thus, we are interested in the following question, to which a negative answer is expected.
Question 3. Let K be a p-adic function field. Is the group X 2 ω (K, µ ⊗2 2 m ) trivial for m ≥ 3?
4 Curves over higher-dimensional local fields In this section, we work with homogeneous spaces over function fields over higher-dimensional local fields. The main results here are Theorems D and E. Our approach is similar to that in paragraphs 3.1 and 3.2. The price we have to pay is that the obtained results are much coarser than those in the case of p-adic function fields. First, the geometric stabilizers are supposed to be finite abelian; the case of toric stabilizers is not treated because we do not have the corresponding duality theorems for tori at our disposal. Second, the constructed obstruction is not shown to be unramified.
Theorem 4.2 (Theorem D). Let K be the function field of a smooth projective geometrically integral curve Ω over a d-dimensional local field k satisfying the condition (⋆) from page 4, G a special, simply connected semisimple algebraic group G over K, and X a homogeneous space of G with finite abelian geometric stabilizers F . Let F be the natural K-form of F , F ′ = Hom K (F, Q/Z(d + 1)), and r (d) the map defined in (4.2.1). If there is a point of X(A K ) orthogonal to r (d) (X d+1 (K, F ′ )) relative to the pairing (1.4.11), then X(K) = ∅. In particular, the first adelic obstruction (1.4.12) to the local-global principle for X is the only one.
Theorem 4.3 (Theorem E). Let K be the function field of a smooth projective geometrically integral curve Ω over a d-dimensional local field k satisfying the condition (⋆) from page 4, F a finite Γ K -module, equipped with an embedding F ֒→ G into a special, simply connected semisimple algebraic group over K that has weak approximation, and X = F \G. Let F ′ = Hom K (F, Q/Z(d+1)), r (d) the map defined in (4.2.1), and S ⊆ Ω (1) a finite set. Then any family (P v ) v∈S ∈ v∈S X(K v ) orthogonal to r (d) (X d+1 S (K, F ′ )) relative to the pairing (1.4.13) lies in the closure of the diagonal image of X(K). In particular, the generalized Brauer-Manin obstruction to weak approximation for X is the only one. Moreover, X has weak approximation in S if and only if X d+1 S (K, F ′ ) = X d+1 (K, F ′ ).
Proof. Let e X ∈ Ext 2 K ( F , G m ) denote the elementary obstruction of X (see the discussion at the beginning of paragraph 2.1). By Lemma 3.6, the isomorphism (2.1.2) sends η X to e X . By the same argument as at the beginning of the proof of Proposition 2.2, −e X is represented by a morphism F → G m [2] associated with the distinguished triangle (4.2.2) in D + (K). Let π U : X → U be an integral model of X over some non-empty open subset U ⊆ Ω. We may assume that F extends to a locally constant finite étale U-group scheme F . Then F (resp. F ′ ) extends to the locally constant finite étale U-group scheme F = Hom U (F , G m ) (resp. F ′ = F ⊗ Q/Z(d) = Hom (F , Q/Z(d + 1))). We have π U * G m,X = G m (since the canonical map G m → π U * G m,X induces isomorphisms on the stalks of geometric points, by Rosenlicht's in D + (U), with distinguished rows (the middle row being (4.2.4)). The existence of this diagram implies that the map H 2 (U, F ) → Ext 2 U ( F, Q/Z(1)) from (4.2.5) effectively sends η U to the inverse class of the morphism F → Q/Z(1)[2] associated with triangle (4.2.4).
Return to the proof of Proposition 4.4. Just like in the proof of Proposition 4.1, we have a commutative diagram H 2 (U, Hom U ( F, Q/Z(1)))