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Weak approximation for homogeneous spaces over some two‐dimensional geometric global fields

Weak approximation for homogeneous spaces over some two‐dimensional geometric global fields INTRODUCTIONGiven an algebraic variety Z$Z$ over a number field K$K$, we say that Z$Z$ satisfies weak approximation if Z(K)$Z(K)$ is dense in ∏v∈ΩZ(Kv)$\prod _{v\in {\Omega }}Z(K_v)$ with respect to the product of the v$v$‐adic topologies, where Kv$K_v$ denotes the v$v$‐adic completion and Ω$\Omega$ denotes the set of places of K$K$. Weak approximation is satisfied for simply connected groups, and when it is not satisfied, we want to characterize the closure Z(K)¯$\overline{Z(K)}$ of the set of rational points in ∏v∈ΩZ(Kv)$\prod _{v\in {\Omega }}Z(K_v)$ as some subset of elements satisfying certain compatibility conditions, for example, as the Brauer–Manin set Z(KΩ)Br$Z(K_{\Omega })^{\operatorname{Br}}$ (cf. [14, Section 8.2]), and this gives an obstruction to weak approximation.Going beyond number fields, recently there has been an increasing interest in studying analogous questions over some two‐dimensional geometric global fields. As in [11], we consider a field K$K$ of one of the following two types:(a)the function field of a smooth projective curve C$C$ over C((t))$\mathbb {C}((t))$;(b)the fraction field of a local, henselian, two‐dimensional, excellent domain A$A$ with algebraically closed residue field of characteristic 0 (for example, any finite extension of C((x,y))$\mathbb {C}((x,y))$, the field of Laurent series in two variables over the field of complex numbers).In case (a), one can consider the set Ω$\Omega$ of valuations coming from the closed points of the curve C$C$. Colliot‐Thélène/Harari (cf. [7]) proved the following exact sequence describing the obstruction to weak approximation for the case of tori T$T$:1.0.1where Б(Z)$\Be (Z)$ (resp. Бω(Z)$\Be _\omega (Z)$) is defined to be the subgroup of Br1(Z)/Br(K)$\operatorname{Br}_1(Z)/\operatorname{Br}(K)$ containing elements vanishing in Br1(Zv)/Br(Kv)$\operatorname{Br}_1(Z_v)/\operatorname{Br}(K_v)$ for all places (resp. almost all places) v∈C(1)$v\in C^{(1)}$, and AD$A^{D}$ denotes the group of continuous homomorphisms from A$A$ to Q/Z(−1)$\mathbb {Q}/\mathbb {Z}(-1)$.In case (b), one can take Ω$\Omega$ to be the set of valuations coming from prime ideals of height one in A$A$. Izquierdo (cf. [13]) proved the exact sequence 1.0.1 in such situations for tori T$T$.Using some dévissage arguments as in [7], we can first generalize this result to a connected linear group G$G$ over K$K$.1.1TheoremTheorem 3.1We keep the notation as above. For a connected linear group G$G$ over K$K$ of the type (a) or (b), there is an exact sequence1.1.1In particular, the obstruction to weak approximation is controlled by the Brauer set . However, such an exact sequence does not generalize to homogeneous spaces, and we do find counterexamples:1.2TheoremProposition 4.1Let Q$Q$ be a flasque torus such thatH1(K,Q)→⊕v∈ΩH1(Kv,Q)→Шω1(Q̂)D$$\begin{equation*} H^1(K,Q)\rightarrow \oplus _{v\in \Omega }H^1(K_v,Q)\rightarrow \Sha ^1_{\omega }(\widehat{Q})^D \end{equation*}$$is not exact (for example, the one constructed in [7, Corollary 9.16]). Embed Q$Q$ into some SLn$\operatorname{SL}_n$ and let Z:=SLn/Q$Z:=\operatorname{SL}_n/Q$. Then .In order to better understand the obstruction to weak approximation for homogeneous spaces over fields of the type (a) or (b), we should somehow combine the Brauer–Manin obstruction with the descent obstruction, another natural tool used in the study of such questions, as done by Izquierdo and Lucchini Arteche in [11] for the study of obstruction to rational points. We present the following result which applies in particular to homogeneous spaces under connected stabilizers:1.3TheoremProposition 4.5For G$G$ a connected linear group, we consider a homogeneous space Z$Z$ under G$G$ with geometric stabilizer H¯$\bar{H}$ such that Gss$G^{{\rm ss}}$ is simply connected and H¯torf$\bar{H}^{{\rm torf}}$ is abelian. Definewhere f$f$ runs over torsors W→Z$W\rightarrow Z$ under quasi‐trivial tori T$T$. Then .NOTATION AND PRELIMINARIESThe notation will be fixed in this section and used throughout this article. The setting is pretty much the same as in [11] where they treated the problem of Hasse principle for such varieties.CohomologyThe cohomology groups we consider are always in terms of Galois cohomology or étale cohomology.FieldsThroughout this article, we consider a field K$K$ of one of the following two types:(a)the function field of a smooth projective curve C$C$ over C((t))$\mathbb {C}((t))$;(b)the field of fractions of a two‐dimensional, excellent, henselian, local domain A$A$ with algebraically closed residue field of characteristic zero. An example of such a field is any finite extension of the field of fractions C((X,Y))$\mathbb {C}((X, Y))$ of the formal power series ring C[[X,Y]]$\mathbb {C}[[X,Y]]$.Both these two types of fields share a number of properties that hold also for totally imaginary number fields (cf. [10, Theorem 5.5], Theorem 1.2 and Theorem 1.4 of [6]):(i)the cohomological dimension is two,(ii)index and exponent of central simple algebras coincide,(iii)for any semisimple simply connected group G$G$ over K$K$, we have H1(K,G)=1$H^1(K,G)=1$,(iv)there is a natural set Ω$\Omega$ of rank one discrete valuations (that is, with values in Z$\mathbb {Z}$), with respect to which one can take completions. For type (a), let Ω$\Omega$ be the set of valuations coming from C(1)$C^{(1)}$ the set of closed points of the curve C$C$. For type (b), let Ω$\Omega$ be the set of valuations coming from prime ideals of height one in A$A$.Weak approximation is satisfied for semisimple simply connected groups over such fields (cf. [6, Theorem 4.7] for type (b), and [7, section 10.1 ] for type (a)).Sheaves and abelian groupsFor i>0$i&gt;0$, we denote by μn⊗i$\mu _n^{\otimes i}$ the i$i$‐fold tensor product of the étale sheaf μn$\mu _n$ of n$n$‐th roots of unity with itself. We set μn0=Z/nZ$\mu _n^0=\mathbb {Z}/n\mathbb {Z}$ and for i<0$i&lt;0$, and we define μn⊗i=Hom(μn⊗(−i),Z/nZ)$\mu _n^{\otimes i}=\operatorname{Hom}(\mu _n^{\otimes (-i)},\mathbb {Z}/n\mathbb {Z})$. Over the fields K$K$ we consider, since K$K$ contains an algebraically closed field and thus all the roots of unity, we have a (non‐canonical) isomorphism μn≃Z/nZ$\mu _n\simeq \mathbb {Z}/n\mathbb {Z}$. Denote by Q/Z(i)$\mathbb {Q}/\mathbb {Z}(i)$ the direct limit of the sheaves μm⊗i$\mu _m^{\otimes i}$ for all m>0$m&gt;0$. By choosing a compatible system of primitive n$n$‐th roots of unity for every n$n$ (for example, ξn=exp(2πi/n)$\xi _n=\exp (2\pi i/n)$), we can identify Q/Z(i)$\mathbb {Q}/\mathbb {Z}(i)$ with Q/Z$\mathbb {Q}/\mathbb {Z}$.For an abelian group A$A$ (that we are always supposed to be equipped with the discrete topology if there is no other topology defined on it), we denote by AD$A^D$ the group of continuous homomorphisms from A$A$ to Q/Z(−1)$\mathbb {Q}/\mathbb {Z}(-1)$. The functor A↦AD$A\mapsto A^D$ is an anti‐equivalence of categories between torsion abelian groups and profinite groups.Weak approximation and Brauer–Manin obstructionGiven a set Ω$\Omega$ of places of K$K$, let Z(KΩ):=∏v∈ΩKZ(Kv)$Z(K_\Omega ):=\prod _{v\in \Omega _K}Z(K_v)$ be the topological product where each Z(Kv)$Z(K_v)$ is equipped with the v$v$‐adic topology, and Kv$K_v$ denotes the completion at v$v$. We say that a K$K$‐variety Z$Z$ satisfies weak approximation with respect to Ω$\Omega$ if the set of rational points Z(K)$Z(K)$, considered through the diagonal map as a subset of Z(KΩ)$Z(K_\Omega )$, is a dense subset. Weak approximation is a birational invariant of smooth geometrically integral varieties, which is a consequence of the implicit function theorem for Kv$K_v$ (cf. Theorem 9.5.1 of [8] and the argument in [8, Proposition 12.2.3]).There are varieties for which weak approximation fails, thus we try to introduce obstructions that give a more precise description of the closure of Z(K)$Z(K)$ inside Z(KΩ)$Z(K_\Omega )$, thus explaining such failures.For a variety Z$Z$, the cohomological Brauer group Br(Z)$\operatorname{Br}(Z)$ is defined to be H2(Z,Gm)$H^2(Z,\mathbb {G}_m)$. Define also the following:Br0(Z):=Im(Br(K)→Br(Z))$_0(Z):=\operatorname{Im}(\operatorname{Br}(K)\rightarrow \operatorname{Br}(Z))$.Br1(Z):=ker(Br(Z)→Br(ZK¯))$_1(Z):=\ker (\operatorname{Br}(Z)\rightarrow \ Br(Z_{\bar{K}}))$.Bra(Z):=Br1(Z)/Br0(Z)$_{{\rm a}}(Z):=\operatorname{Br}_1(Z)/\operatorname{Br}_0(Z)$.Let w∈Z(K)$w\in Z(K)$ be a K$K$‐point of Z$Z$, we set Brw(Z):=Ker(w∗:Br(Z)→Br(K))$\operatorname{Br}_w(Z):=\operatorname{Ker}(w^*:\operatorname{Br}(Z)\rightarrow \operatorname{Br}(K))$ and Br1,w:=Ker(w∗:Br1(Z)→Br(K))$\operatorname{Br}_{1,w}:=\operatorname{Ker}(w^*:\operatorname{Br}_1(Z)\rightarrow \operatorname{Br}(K))$. We have a canonical isomorphism Br1,w(Z)≃Bra(Z)$\operatorname{Br}_{1,w}(Z)\simeq \operatorname{Br}_{{\rm a}}(Z)$.Б(Z):=ker(Bra(Z)→∏v∈ΩBra(ZKv))$\Be (Z):=\ker (\operatorname{Br}_{{\rm a}}(Z)\rightarrow \prod _{v\in \Omega }\operatorname{Br}_{{\rm a}}(Z_{K_{v}}))$.БS(Z):=ker(Bra(Z)→∏v∈Ω∖SBra(ZKv))$\Be _S(Z):=\ker (\operatorname{Br}_{{\rm a}}(Z)\rightarrow \prod _{v\in \Omega \backslash S}\operatorname{Br}_{{\rm a}}(Z_{K_{v}}))$ where S$S$ is a finite subset of Ω$\Omega$.Бω(Z)$\Be _\omega (Z)$: subgroup of Bra(Z)$\operatorname{Br}_{{\rm a}}(Z)$ containing the elements vanishing in Bra(ZKv)$\operatorname{Br}_{{\rm a}}(Z_{K_v})$ for almost all places v∈Ω$v\in \Omega$.Similar to the exact sequence from Class Field Theory for number fields, there is also an exact sequence (cf. Proposition 2.1.(v) of [7] and Theorem 1.6 of [13])Br(K)→⨁v∈ΩBr(Kv)→Q/Z→0$$\begin{equation*} \operatorname{Br}(K)\rightarrow \bigoplus _{v\in \Omega }\operatorname{Br}(K_v) \rightarrow \mathbb {Q}/\mathbb {Z}\rightarrow 0 \end{equation*}$$and a Brauer–Manin pairingZ(KΩ)×Бω(Z)→Q/Z((Pv),α)↦∑v∈Ω<Pv,α>$$\begin{align*} Z(K_\Omega )\times \Be _\omega (Z)&\rightarrow \mathbb {Q}/\mathbb {Z}\\ ((P_v),\alpha )& \mapsto \sum _{v\in \Omega }&lt;P_v,\alpha &gt; \end{align*}$$with the following property: Z(K)$Z(K)$ is contained in the subset of Z(KΩ)$Z(K_\Omega )$ defined as {(Pv)∈Z(KΩ):((Pv),α)=0forallα∈Бω(Z)}$\lbrace (P_v)\in Z(K_\Omega ):((P_v),\alpha )=0\text{ for all }\alpha \in \Be _{\omega }(Z)\rbrace$: the set of points that are orthogonal to Бω(Z)$\Be _{\omega }(Z)$. When Z(K)$Z(K)$ is non‐empty and , we say that the Brauer–Manin obstruction with respect to Бω$\Be _\omega$ is the only one to weak approximation.Tate–Shafarevich groupsFor a Galois module M$M$ over the field K$K$ and an integer i⩾0$i\geqslant 0$, we define the following Tate–Shafarevich groups:Шi(K,M):=ker(Hi(K,M)→∏v∈ΩHi(Kv,M))$\Sha ^i(K,M):=\ker (H^i(K,M)\rightarrow \prod _{v\in \Omega }H^i(K_v,M))$.ШSi(K,M):=ker(Hi(K,M)→∏v∈Ω∖SHi(Kv,M))$\Sha _S^i(K,M):=\ker (H^i(K,M)\rightarrow \prod _{v\in \Omega \backslash S}H^i(K_v,M))$ where S$S$ is a finite subset of Ω$\Omega$.Шωi(K,M):$\Sha _{\omega }^i(K,M):$ subgroup of Hi(K,M)$H^i(K,M)$ containing the elements vanishing in Hi(Kv,M)$H^i(K_v,M)$ for almost all places v∈Ω$v\in \Omega$.Algebraic groups and homogeneous spacesFor a linear algebraic K$K$‐group G$G$, the following notation will be used:D(G)$D(G)$: the derived subgroup of G$G$,G∘$G^{\circ }$: the neutral connected component of G$G$,Gf:=G/G∘$G^{{\rm f}}:=G/G^{\circ}$ the group of connected components of G$G$, which is a finite group,Gu:$G^{{\rm u}}:$ the unipotent radical of G∘$G^{\circ}$,Gred:=G∘/Gu$G^{{\rm red}}:=G^{\circ} /G^{\rm u}$ which is a reductive group,Gss:=D(Gred)$G^{{\rm ss}}:=D(G^{{\rm red}})$ which is a semisimple group,Gtor:=Gred/Gss$G^{{\rm tor}}:=G^{{\rm red}}/G^{{\rm ss}}$ which is a torus,Gssu:=ker(G∘→Gtor)$G^{{\rm ssu}}:=\ker (G^{\circ }\rightarrow G^{{\rm tor}})$ which is an extension of Gss$G^{{\rm ss}}$ by Gu$G^{\rm u}$,Gtorf:=G/Gssu$G^{{\rm torf}}:=G/G^{{\rm ssu}}$ which is an extension of Gf$G^{\rm f}$ by Gtor$G^{{\rm tor}}$,Ĝ$\widehat{G}$: the Galois module of the geometric characters of G$G$.A unipotent group over K$K$ a field of characteristic 0 is isomorphic to an affine space AKn$\bm {\mathrm{A}}_K^n$, thus K$K$‐rational and satisfies weak approximation.A torus T$T$ is said to be quasi‐trivial if T̂$\widehat{T}$ is an induced Gal(K¯/K)$\operatorname{Gal}(\bar{K}/K)$‐module.WEAK APPROXIMATION FOR CONNECTED LINEAR GROUPSThe aim of this section is to prove the following theorem which concerns the Brauer–Manin obstruction to weak approximation for connected linear groups:3.1TheoremLet K$K$ be a field of the type (a) or (b) and let G$G$ be a connected linear group over K$K$. Then there is an exact sequence1→G(K)¯→G(KΩ)→Бω(G)D→Б(G)D→1.$$\begin{equation*} 1\rightarrow \overline{G(K)}\rightarrow G(K_\Omega )\rightarrow \Be _\omega (G)^D\rightarrow \Be (G)^D\rightarrow 1. \end{equation*}$$We complete the proof by a series of lemmas.3.2LemmaSuppose that Theorem 3.1 holds for Gn$G^n$ where n$n$ is a positive integer, then it also holds for G$G$.ProofIt follows from the fact that all of the operations in the exact sequence above take products to products. (For example, the closure of a product is the product of closures.)□$\Box$An exact sequence1→F→H×KP→G→1,$$\begin{equation*} 1\rightarrow F\rightarrow H\times _K P\rightarrow G\rightarrow 1, \end{equation*}$$where H$H$ is semi‐simple simply connected, P$P$ is a quasitrivial K$K$‐torus and F$F$ is finite and central is called a special covering of the reductive K$K$‐group G$G$. For any reductive K$K$‐group G$G$, there exists an integer n>0$n&gt;0$ such that Gn$G^n$ admit a special covering. By the lemma above, we can suppose without loss of generality that G$G$ admits a special covering itself.3.3LemmaLet S$S$ be a finite set of places in Ω$\Omega$. The following sequence is exact1→G(K)¯→∏v∈SG(Kv)→БS(G)D,$$\begin{equation*} 1\rightarrow \overline{G(K)}\rightarrow \prod _{v\in S}G(K_v)\rightarrow \Be _S(G)^D, \end{equation*}$$where G$G$ is a reductive connected linear group.ProofThe proof is exactly the same as in [7, Lemma 9.6], using the special covering and the fact that H$H$ and P$P$ both satisfy weak approximation (so does their product) and H1(K,H)=H1(K,P)=H1(K,H×KP)=0$H^1(K,H)=H^1(K,P)=H^1(K,H\times _K P)=0$. We have a commutative diagram where the rows are exact and the columns are complexes.The exactness of the last row comes from the fact that H1(K,F̂)≃Ker(Bra(G)→Bra(H×KP))$H^1(K,\widehat{F})\simeq \operatorname{Ker}(\operatorname{Br}_{{\rm a}}(G)\rightarrow \operatorname{Br}_{{\rm a}}(H\times _K P))$ and Bra(H×KP)≃H2(K,P̂)$\operatorname{Br}_{{\rm a}}(H\times _K P)\simeq H^2(K,\widehat{P})$ (Corollary 7.4 and Lemma 6.9 of [15]), with the latter isomorphism inducing an isomorphism БS(H×P)≃ШS2(P̂)$\Be _S(H\times P)\simeq \Sha ^2_S(\widehat{P})$ by the same argument as in [7, Section 8.1]. The commutativity of the right‐bottom square can be deduced from [15, Lemma 8.11].□$\Box$3.4LemmaLet S$S$ be a finite set of places in Ω$\Omega$. The following sequence is exact:1→G(K)¯→∏v∈SG(Kv)→БS(G)D,$$\begin{equation*} 1\rightarrow \overline{G(K)}\rightarrow \prod _{v\in S}G(K_v)\rightarrow \Be _S(G)^D, \end{equation*}$$where G$G$ is a connected linear group over K$K$.ProofWe use the resolution 1→Gu→G→Gred→1$1\rightarrow G^{{\rm u}}\rightarrow G\rightarrow G^{{\rm red}}\rightarrow 1$ and the induced diagramtaking into account the vanishing of H1(K,Gu)$H^1(K,G^{{\rm u}})$. The unipotent radical Gu$G^{{\rm u}}$ satisfies weak approximation (see Section 2). The exactness of the rightmost column is known from the previous lemma, then a diagram chasing gives the result.□$\Box$3.5LemmaLet S$S$ be a finite set of places in Ω$\Omega$. The sequence3.5.1∏v∈SG(Kv)→БS(G)D→Б(G)D$$\begin{equation} \prod _{v\in S}G(K_v)\rightarrow \Be _S(G)^D\rightarrow \Be (G)^D\end{equation}$$is exact, where G$G$ is a connected linear group over K$K$.ProofAs in the previous lemmas, we can first treat the case where G$G$ is reductive. Suppose now that G$G$ is reductive, we use a coflasque resolution1→P→G′→G→1,$$\begin{equation*} 1\rightarrow P\rightarrow G^\prime \rightarrow G\rightarrow 1, \end{equation*}$$where P$P$ is a quasitrivial torus and G′$G^\prime$ fits into the exact sequence1→Gsc→G′→T→1,$$\begin{equation*} 1\rightarrow G^{{\rm sc}}\rightarrow G^\prime \rightarrow T\rightarrow 1, \end{equation*}$$where T$T$ is a coflasque torus and Gsc$G^{{\rm sc}}$ is a semisimple simply connected group. Suppose that the exactness of the sequence in question is known for G′$G^\prime$ (replacing G$G$). Then we can prove the exactness for G$G$ by chasing in the following diagram:The leftmost arrow is an isomorphism by [7, Proposition 2.6]. The last two rows are exact, and this can be proved by chasing the following diagram, as done in the proof of [3, Lemma 4.4].(The exactness of the rows can be deduced from [15, Proposition 6.10] and the fact that Pic(P)=H1(K,P̂)=0$\operatorname{Pic}(P)=H^1(K,\widehat{P})=0$ since P̂$\widehat{P}$ is a permutation module.)The exactness of the sequence 3.5.1 is known for T$T$ ([7, Corollary 9.9] for K$K$ of type (a), and [13, Theorem 4.9] for K$K$ of type (b)), by a similar diagram chasing, we can prove the result for G′$G^\prime$, thus completing the proof. The essential condition we need in this diagram chasing is that we should have an injection БS(G′)D↪БS(T)D$\Be _S(G^\prime )^D\hookrightarrow \Be _S(T)^D$, or equivalently a surjection БS(T)↠БS(G′)$\Be _S(T)\twoheadrightarrow \Be _S(G^\prime )$. This is indeed true because we have an exact sequence Bra(T)→Bra(G′)→Bra(Gsc)$\operatorname{Br}_{{\rm a}}(T)\rightarrow \operatorname{Br}_{{\rm a}}(G^\prime )\rightarrow \operatorname{Br}_{{\rm a}}(G^{{\rm sc}})$ and Br(Gsc)=Br(K)$\operatorname{Br}(G^{{\rm sc}})=\operatorname{Br}(K)$ since Gsc$G^{{\rm sc}}$ a simply connected group over a field of characteristic 0 (cf. Corollary in Section  0 of [9]).Now for a connected linear group G$G$ (not necessarily reductive), we use again the resolution 1→Gu→G→Gred→1$1\rightarrow G^{{\rm u}}\rightarrow G\rightarrow G^{{\rm red}}\rightarrow 1$ and the induced commutative diagramtaking into account the vanishing of H1(Kv,Gu)$H^1(K_v,G^{{\rm u}})$. The injectivity of the middle line follows from the fact that Bra(Gu)=Bra(AKn)=(Bra(AK1))n=0$\operatorname{Br}_{{\rm a}}(G^{{\rm u}})=\operatorname{Br}_{{\rm a}}(\bm {\mathrm{A}}_K^n)=(\operatorname{Br}_{{\rm a}}(\bm {\mathrm{A}}_K^1))^n=0$ (cf. Lemma 6.6 of [15], Theorem 4.5.1(viii) of [8]). Then a diagram chasing gives the desired result.□$\Box$3.6CorollaryTheorem 3.1With the same notation as above, there is an exact sequence1→G(K)¯→G(KΩ)→Бω(G)D→Б(G)D→1.$$\begin{equation*} 1\rightarrow \overline{G(K)}\rightarrow G(K_\Omega )\rightarrow \Be _\omega (G)^D\rightarrow \Be (G)^D\rightarrow 1. \end{equation*}$$ProofWe extend the result with respect to a finite set S$S$ of places to all the Ω$\Omega$ by following the same proof used in Corollary 9.9 of [7]. Let A(G):=coker(G(K)¯→G(KΩ))$A(G):=\operatorname{coker}(\overline{G(K)}\rightarrow G(K_\Omega ))$ and AS(G):=coker(G(K)¯→∏v∈S(G(Kv)))$A_S(G):=\operatorname{coker}(\overline{G(K)}\rightarrow \prod _{v\in S}(G(K_v)))$ measuring the failure of weak approximation. Let 1→Q→R→G→1$1\rightarrow Q\rightarrow R\rightarrow G\rightarrow 1$ be a flasque resolution of G$G$ (cf. Proposition–Definition 3.1 of [5]). By Proposition 9.1 of [7] (for K$K$ of type (a)) and Theorem 3.7 with its remark in [6] (for K$K$ of type (b)), we have AS(G)=coker(H1(K,Q)→∏v∈SH1(Kv,Q))$A_S(G)=\operatorname{coker}(H^1(K,Q)\rightarrow \prod _{v\in S}H^1(K_v,Q))$ and H1(Kv,Q)=0$H^1(K_v,Q)=0$ for all v$v$ outside the finite set S0$S_0$ of places of bad reduction of Q$Q$. Therefore, we have AS(G)=A(G)$A_S(G)=A(G)$ for S0⊆S$S_0\subseteq S$. Let S0⊆S⊆S′$S_0\subseteq S\subseteq S^\prime$, and then we have a commutative diagramwhere the left and the right vertical maps are isomorphisms. Thus the middle map is also an isomorphism and БS(G)=Бω(G)$\Be _S(G)=\Be _{\omega }(G)$.□$\Box$WEAK APPROXIMATION FOR HOMOGENEOUS SPACESNow we look at weak approximation for homogeneous spaces over such fields K$K$. The usual Brauer–Manin obstruction we used in the last section is not enough in this case, as shown by the following construction of examples:4.1PropositionLet Q$Q$ be a flasque torus such thatH1(K,Q)→⊕v∈ΩH1(Kv,Q)→Шω1(Q̂)D$$\begin{equation*} H^1(K,Q)\rightarrow \oplus _{v\in \Omega }H^1(K_v,Q)\rightarrow \Sha ^1_{\omega }(\widehat{Q})^D \end{equation*}$$is not exact. Such a torus exists (see Corollary 9.16 of [7] for a construction.) Embed Q$Q$ into some SLn$\operatorname{SL}_n$ and let Z:=SLn/Q$Z:=\operatorname{SL}_n/Q$. Then .ProofWe consider the following commutative diagram with exact rows:The vanishing of Bra(SLn)$\operatorname{Br}_{{\rm a}}(\operatorname{SL}_n)$ on the bottom‐left corner comes from the Corollary of [9] (section 0). We prove the result by diagram chasing. Since the right column is not exact, there exists a∈∏v∈ΩH1(Kv,Q)$a\in \prod _{v\in \Omega }H^1(K_v,Q)$ such that f6(a)$f_6(a)$ vanishes but a∉f3(H1(K,Q))$a\notin f_3(H^1(K,Q))$. Since f4$f_4$ is surjective, we can find b∈Z(KΩ)$b\in Z(K_\Omega )$ such that f4(b)=a$f_4(b)=a$. By the commutativity of the bottom square, f5(b)$f_5(b)$ vanishes. We prove that b∉Z(K)¯$b\notin \overline{Z(K)}$ by contradiction. We suppose b∈Z(K)¯$b\in \overline{Z(K)}$. The torus Q$Q$ being flasque implies that ∏v∈ΩH1(Kv,Q)$\prod _{v\in \Omega }H^1(K_v,Q)$ is finite (cf. [7, Proposition 9.1] for K$K$ of type (a); Theorem 3.7 with its remark in [6] for K$K$ of type (b)), thus the preimage f4−1(a)$f_4^{-1}(a)$ is open (f4$f_4$ is continuous, cf. [4]), containing b∈Z(K)¯$b\in \overline{Z(K)}$, so we should be able to find c∈Z(K)$c\in Z(K)$ lying in f4−1(a)$f_4^{-1}(a)$. Then f3(f1(c))=a$f_3(f_1(c))=a$ by the commutativity of the top square, contradicting a∉Imf3$a\notin \operatorname{Im}f_3$. Therefore, we found .□$\Box$Therefore, we need to find other obstructions. In the rest of this section, we consider homogeneous spaces Z$Z$ under a connected linear group G$G$ with geometric stabilizer H¯$\bar{H}$ such that Gss$G^{{\rm ss}}$ is simply connected and H¯torf$\bar{H}^{{\rm torf}}$ is abelian. This is the assumption already used in [11] and [3], and is satisfied by every homogeneous space under a connected linear K$K$‐group with connected stabilizers (cf. Lemma 5.2 in [3]).We can find Z←W→W′$Z\leftarrow W\rightarrow W^\prime$ such thatW$W$ is a K$K$‐homogeneous space under G×T$G\times T$,T$T$ is a quasitrivial torus into which Htorf$H^{{\rm torf}}$ injects, where Htorf$H^{{\rm torf}}$ is the canonical K$K$‐form of H¯torf$\bar{H}^{{\rm torf}}$ associated to Z$Z$,W→Z$W\rightarrow Z$ is a T$T$‐torsor,W′$W^\prime$ is the quotient variety Z/Gss$Z/G^{{\rm ss}}$, which is also a homogeneous space of Gtor×T$G^{{\rm tor}}\times T$ with geometric stabilizer H¯torf$\bar{H}^{{\rm torf}}$ and the fibers of W→W′$W\rightarrow W^\prime$ are homogeneous spaces of Gssu$G^{{\rm ssu}}$ with geometric stabilizers H¯ssu$\bar{H}^{{\rm ssu}}$.Indeed, we can embed Htor$H^{{\rm tor}}$ in a quasi‐trivial torus T$T$ and consider the diagonal morphism H→G×T$H\rightarrow G\times T$ induced by the inclusion H↪G$H\hookrightarrow G$ and the composition H→Htor→T$H\rightarrow H^{{\rm tor}}\rightarrow T$. Then we define W=(G×T)/H$W=(G\times T)/H$, and W→Z$W\rightarrow Z$ is induced by the projection to the first coordinate. Define W′$W^{\prime }$ to be the quotient variety W/Gssu$W/G^{{\rm ssu}}$ and we get what we want.4.2PropositionThe fiber WP$W_P$ above a K$K$‐point P∈W′(K)$P\in W^\prime (K)$ satisfies weak approximation.ProofSince Hss$H^{{\rm ss}}$ is semi‐simple, we consider its simply connected covering 1→F→Hsc→Hss→1$1\rightarrow F\rightarrow H^{{\rm sc}}\rightarrow H^{{\rm ss}}\rightarrow 1$ where F$F$ is finite and Hsc$H^{{\rm sc}}$ is simply connected.We first prove that this covering induces isomorphisms H1(K,Hss)≃H2(K,F)$H^1(K,H^{{\rm ss}})\simeq H^2(K,F)$ and H1(Kv,Hss)≃H2(Kv,F)$H^1(K_v,H^{{\rm ss}})\simeq H^2(K_v,F)$ for all v∈Ω$v\in \Omega$. For K$K$ and Kv$K_v$, the two conditions in [6, Theorem 2.1] are satisfied, and thus we have a bijection H1(K,Had)→H2(K,μ)$H^1(K,H^{\rm ad})\rightarrow H^2(K,\mu )$ coming from the central isogeny1→μ→Hsc→Had→1$$\begin{equation*} 1\rightarrow \mu \rightarrow H^{{\rm sc}}\rightarrow H^{\rm ad}\rightarrow 1 \end{equation*}$$associated to the center μ$\mu$ of Hsc$H^{{\rm sc}}$. Since F$F$ is contained in the center μ$\mu$, we have the exact sequence1→μ/F→Hss→Had→1$$\begin{equation*} 1\rightarrow \mu /F\rightarrow H^{{\rm ss}}\rightarrow H^{\rm ad}\rightarrow 1 \end{equation*}$$and the commutative diagram with exact rowsThen the four‐lemma gives the surjectivity of H1(K,Hss)→H2(K,F)$H^1(K,H^{{\rm ss}})\rightarrow H^2(K,F)$. The injectivity comes from the vanishing of H1(K,Hsc)$H^1(K,H^{{\rm sc}})$. The proof is the same when K$K$ is replaced by Kv$K_v$.Then, we prove thatH2(K,F)→⊕v∈SH2(Kv,F)$$\begin{equation*} H^2(K,F)\rightarrow \oplus _{v\in S}H^2(K_v,F) \end{equation*}$$is surjective, where S$S$ is a finite set of places. This is proved by chasing the following commutative diagram:The row in the middle is exact. (See [12] Theorem 2.7 applying d=0$d=0$ for K$K$ of type (a). For K$K$ of type (b), use [13, Corollary 2.3] and the exact sequence H2(U,F)→⨁v∈X∖UH2(Kv,F)→Hc3(U,F)$H^2(U,F)\rightarrow \bigoplus _{v\in X\backslash U}H^2(K_v,F)\rightarrow H^3_c(U, F)$ with X$X$ and U$U$ defined in [13], and then take direct limit over U$U$.) The surjectivity of ⊕v∉SH2(Kv,F)→H0(K,F̂)D$\oplus _{v\notin S}H^2(K_v,F)\rightarrow H^0(K,\widehat{F})^D$ comes from the injectivity taking the duals H0(K,F̂)↪⊕v∉SH0(Kv,F̂)$H^0(K,\widehat{F})\hookrightarrow \oplus _{v\notin S} H^0(K_v,\widehat{F})$. This yields the surjectivity of H2(K,F)→∏v∈SH2(Kv,F)$H^2(K,F)\rightarrow \prod _{v\in S}H^2(K_v,F)$, which is H1(K,Hss)→⊕v∈SH1(Kv,Hss)$H^1(K,H^{{\rm ss}})\rightarrow \oplus _{v\in S}H^1(K_v,H ^{{\rm ss}})$.Finally, by [15, Lemma 1.13], we have H1(K,Hss)=H1(K,Hssu)$H^1(K,H^{{\rm ss}})=H^1(K,H^{{\rm ssu}})$ and H1(Kv,Hss)=H1(Kv,Hssu)$H^1(K_v,H^{{\rm ss}})=H^1(K_v,H^{{\rm ssu}})$. With the same argument as in [15, Proposition 3.2], we can prove that Gssu$G^{{\rm ssu}}$ satisfies weak approximation (cf. [2, Proposition 4.1] for the vanishing of H1(K,Hu)$H^1(K,H^u)$ and H1(Kv,Hu)$H^1(K_v,H^u)$. The unipotent radical Hu$H^{{\rm u}}$ satisfies weak approximation). Therefore, we consider the commutative diagram with exact rows (the set WP(K)$W_P(K)$ is non‐empty by [11, Proposition 3.1]):A diagram chasing then gives the desired result.□$\Box$4.3PropositionWe have .ProofConsider , and any open subset U$U$ containing (Pv)$(P_v)$, we want to find Q∈U∩W(K)$Q\in U\cap W(K)$ using fibration methods.Denote by (Pv′)$(P_v^\prime )$ the image of (Pv)$(P_v)$ under the induced map g:W(KΩ)→W′(KΩ)$g: W(K_\Omega )\rightarrow W^\prime (K_\Omega )$. By the functorality of the Brauer–Manin pairing, (Pv′)$(P_v^\prime )$ lies in . Since W→W′$W\rightarrow W^ \prime$ is smooth, and the Kv$K_v$ are henselian, we have open maps W(Kv)→W′(Kv)$W(K_v)\rightarrow W^\prime (K_v)$, and thus g$g$ is open. In particular, V=g(U)⊆W′(KΩ)$V=g(U)\subseteq W^\prime (K_\Omega )$ is also open. Since W′$W^{\prime }$ is a torus (cf. [11, Theorem 3.2]), we have . Therefore, there exists P′∈V∩W′(K)$P^\prime \in V\cap W^\prime (K)$. Let (Qv)∈g−1(P′)∩U⊆W(KΩ)$(Q_v)\in g^{-1}(P^{\prime })\cap U\subseteq W(K_\Omega )$, and it can be seen as in WP′(KΩ)$ W_{P^\prime }(K_\Omega )$ too, as shown in the following diagram.This diagram also shows that WP′(KΩ)⊆W(KΩ)$W_{P^\prime }(K_\Omega )\subseteq W(K_\Omega )$, and U|WP′(KΩ):=U∩WP′(KΩ)$U_{|_{ W_{P^\prime }(K_\Omega )}}:=U\bigcap W_{P^\prime }(K_\Omega )$ is an open neighborhood of (Qv)$(Q_v)$ in WP′(KΩ)$W_{P^\prime }(K_\Omega )$. Since WP′$W_{P^\prime }$ satisfies weak approximation by Proposition 4.2, there exists Q∈WP′(K)∩U|WP′(KΩ)$Q\in W_{P^\prime }(K)\cap U_{|_{ W_{P^\prime }(K_\Omega )}}$, and thus in W(K)∩U$W(K)\cap U$, which proves that .□$\Box$As in [11], we are naturally led to consider the following definition.4.4DefinitionFor an arbitrary K$K$‐variety Z$Z$, we defineUsing the torsor W→Z$W\rightarrow Z$ defined above, we have . In fact, this is an equality.4.5PropositionLet the notation be as above. Then .ProofSince for all f$f$, we haveso . To prove Proposition 4.5, it is equivalent to prove that is closed. It suffices to prove that is closed for every torus f:W→TZ$f: W\xrightarrow T{Z}$ with T$T$ quasitrivial. Since f$f$ is smooth and the Kv$K_v$ are henselian, the induced map f:W(KΩ)→Z(KΩ)$f:W(K_\Omega )\rightarrow Z(K_\Omega )$ is open. Now we will prove that does not meet , and combined with the surjectivity of f$f$, we will get  closed.Fixing w∈W(K)$w\in W(K)$ which is a K$K$‐point of W$W$, we have the canonical isomorphism Bra(W)≃Br1,w(W)$\operatorname{Br}_{{\rm a}}(W)\simeq \operatorname{Br}_{1,w}(W)$. Define the map iw:T→W$i_w: T\rightarrow W$ as t↦t.w$t\mapsto t.w$, and we have an induced map iw∗:Br1,w(W)→Br1,e(T)$i_w^*: \operatorname{Br}_{1,w}(W)\rightarrow \operatorname{Br}_{1,e}(T)$. Let α$\alpha$ be an element in Бω(W)$\Be _\omega (W)$ and we still denote by α$\alpha$ its image in Br1,w(W)$\operatorname{Br}_{1,w}(W)$. Let (xv),(yv)∈W(KΩ)$(x_v),(y_v)\in W(K_\Omega )$ above (Pv)$(P_v)$. We will prove that ∑v∈Ω<xv,α>=∑v∈Ω<yv,α>$\sum _{v\in \Omega }&lt;x_v,\alpha &gt;=\sum _{v\in \Omega }&lt;y_v,\alpha &gt;$. Let m:T×W→W$m: T\times W\rightarrow W$ denotes the action of T$T$ on W$W$. Since WPv$W_{P_v}$ is a Kv$K_v$‐torsor under T$T$, there exists tv∈T(Kv)$t_v\in T(K_v)$ such that xv=tv.yv$x_v=t_v.y_v$. By the functorality of the Brauer–Manin paring, we have <xv,α>=<(tv,yv),m∗α>$&lt;x_v,\alpha &gt;=&lt;(t_v,y_v),m^* \alpha &gt;$. By [15, Lemma 6.6], we have Br1,(w,e)(W×T)=Br1,w(W)×Br1,e(T)$\operatorname{Br}_{1,(w,e)}(W\times T)=\operatorname{Br}_{1,w}(W)\times \operatorname{Br}_{1,e}(T)$, and the formula (24) in [1] gives m∗α=pT∗iw∗(α)+pW∗α$m^*\alpha =p_T^*i_w^*(\alpha )+p_W^*\alpha$ where pT$p_T$ and pW$p_W$ are the two natural projections. Then <xv,α>=<tv,iw∗(α)>+<yv,α>$&lt;x_v,\alpha &gt;=&lt;t_v,i_w^*(\alpha )&gt;+&lt;y_v,\alpha &gt;$. But P$P$ is a quasi‐trivial torus, we have Бω(P)≃Б(P)$\Be _\omega (P)\simeq \Be (P)$ (cf. Proposition 2.6 and section  8.1 of [7]). Therefore, by the exact sequence in Theorem 3.1, the map P(KΩ)→Бω(P)D$P(K_\Omega )\rightarrow \Be _\omega (P)^D$ is 0, that is, ∑v∈Ω<tv,iw∗(α)>=0$\sum _{v\in \Omega }&lt;t_v,i_w^*(\alpha )&gt;=0$ and we get what we want.□$\Box$4.6RemarkActually the above proof also shows thatWith this description, we can prove as done in [11, Theorem 6.4] that4.6.1But this does not necessarily give an obstruction to weak approximation since we do not know if Z(KΩ)tor$Z(K_\Omega )^{{\rm tor}}$ is closed. We do not know if (4.6.1) is an equality. One could wonder whether a ‘purely descent’ description of  exists.ACKNOWLEDGEMENTThe author would like to thank Cyril Demarche for his enormous help as a PhD supervisor and a great number of useful discussions, without which it would not be possible to see this article coming out.JOURNAL INFORMATIONThe Bulletin of the London Mathematical Society is wholly owned and managed by the London Mathematical Society, a not‐for‐profit Charity registered with the UK Charity Commission. All surplus income from its publishing programme is used to support mathematicians and mathematics research in the form of research grants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics.REFERENCESM. Borovoi and C. Demarche, Manin obstruction to strong approximation for homogeneous spaces, Comment. Math. Helv. 88 (2013), no. 1, 1–54.A. Borel, Algebraic groups and discontinuous subgroups, Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, 1966.M. Borovoi, The Brauer‐Manin obstructions for homogeneous spaces with connected or abelian stabilizer, Journal für die reine und angewandte Mathematik (Crelles Journal) 1996 (1996), no. 473, 181–194.K. Česnavičius, Topology on cohomology of local fields, Forum Math. Sigma 3 (2015), E16. https://doi.org/10.1017/fms.2015.18.J.‐L. Colliot, Résolutions flasques des groupes linéaires connexes, J. reine angew. Math. 618 (2008), 77–133.J.‐L. Colliot‐Thélène, P. Gille, and R. Parimala, Arithmetic of linear algebraic groups over 2‐dimensional geometric fields, Duke Math. J. 121 (2004), no. 2, 285–342.J.‐L. Colliot‐Thélène and D. Harari, Dualité et principe local‐global pour les tores sur une courbe au‐dessus de C((t))$\mathbb {C}((t))$, Proc. Lond. Math. Soc. 110 (Apr 2015), no. 6, 1475–1516.J.‐L. Colliot‐Thélene and A. N. Skorobogatov, The Brauer–Grothendieck group, vol. 71, Springer Nature, Switzerland AG, 2021.S. Gille, On the brauer group of a semisimple algebraic group, Adv. Math. 220 (2009), no. 3, 913–925.D. Harbater, J. Hartmann, and D. Krashen, Applications of patching to quadratic forms and central simple algebras, Invent. Math. 178 (2009), no. 2, 231–263.D. Izquierdo and G. L. Arteche, Local‐global principles for homogeneous spaces over some two‐dimensional geometric global fields, Journal für die reine und angewandte Mathematik (Crelles Journal), 2021.D. Izquierdo, Théorèmes de dualité pour les corps de fonctions sur des corps locaux supérieurs, Math. Z. 284 (2016), no. 1, 615–642.D. Izquierdo, Dualité et principe local‐global pour les anneaux locaux henséliens de dimension 2, with an appendix by Joël Riou. Algebr. Geom 6 (2019), no. 2, 148–176.B. Poonen, Rational points on varieties, vol. 186, American Mathematical Soc., Providence, RI, 2017.J.‐J. Sansuc, Groupe de brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, Journal für die reine und angewandte Mathematik 327 (1981), 12–80. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Weak approximation for homogeneous spaces over some two‐dimensional geometric global fields

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Wiley
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0024-6093
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10.1112/blms.12728
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Abstract

INTRODUCTIONGiven an algebraic variety Z$Z$ over a number field K$K$, we say that Z$Z$ satisfies weak approximation if Z(K)$Z(K)$ is dense in ∏v∈ΩZ(Kv)$\prod _{v\in {\Omega }}Z(K_v)$ with respect to the product of the v$v$‐adic topologies, where Kv$K_v$ denotes the v$v$‐adic completion and Ω$\Omega$ denotes the set of places of K$K$. Weak approximation is satisfied for simply connected groups, and when it is not satisfied, we want to characterize the closure Z(K)¯$\overline{Z(K)}$ of the set of rational points in ∏v∈ΩZ(Kv)$\prod _{v\in {\Omega }}Z(K_v)$ as some subset of elements satisfying certain compatibility conditions, for example, as the Brauer–Manin set Z(KΩ)Br$Z(K_{\Omega })^{\operatorname{Br}}$ (cf. [14, Section 8.2]), and this gives an obstruction to weak approximation.Going beyond number fields, recently there has been an increasing interest in studying analogous questions over some two‐dimensional geometric global fields. As in [11], we consider a field K$K$ of one of the following two types:(a)the function field of a smooth projective curve C$C$ over C((t))$\mathbb {C}((t))$;(b)the fraction field of a local, henselian, two‐dimensional, excellent domain A$A$ with algebraically closed residue field of characteristic 0 (for example, any finite extension of C((x,y))$\mathbb {C}((x,y))$, the field of Laurent series in two variables over the field of complex numbers).In case (a), one can consider the set Ω$\Omega$ of valuations coming from the closed points of the curve C$C$. Colliot‐Thélène/Harari (cf. [7]) proved the following exact sequence describing the obstruction to weak approximation for the case of tori T$T$:1.0.1where Б(Z)$\Be (Z)$ (resp. Бω(Z)$\Be _\omega (Z)$) is defined to be the subgroup of Br1(Z)/Br(K)$\operatorname{Br}_1(Z)/\operatorname{Br}(K)$ containing elements vanishing in Br1(Zv)/Br(Kv)$\operatorname{Br}_1(Z_v)/\operatorname{Br}(K_v)$ for all places (resp. almost all places) v∈C(1)$v\in C^{(1)}$, and AD$A^{D}$ denotes the group of continuous homomorphisms from A$A$ to Q/Z(−1)$\mathbb {Q}/\mathbb {Z}(-1)$.In case (b), one can take Ω$\Omega$ to be the set of valuations coming from prime ideals of height one in A$A$. Izquierdo (cf. [13]) proved the exact sequence 1.0.1 in such situations for tori T$T$.Using some dévissage arguments as in [7], we can first generalize this result to a connected linear group G$G$ over K$K$.1.1TheoremTheorem 3.1We keep the notation as above. For a connected linear group G$G$ over K$K$ of the type (a) or (b), there is an exact sequence1.1.1In particular, the obstruction to weak approximation is controlled by the Brauer set . However, such an exact sequence does not generalize to homogeneous spaces, and we do find counterexamples:1.2TheoremProposition 4.1Let Q$Q$ be a flasque torus such thatH1(K,Q)→⊕v∈ΩH1(Kv,Q)→Шω1(Q̂)D$$\begin{equation*} H^1(K,Q)\rightarrow \oplus _{v\in \Omega }H^1(K_v,Q)\rightarrow \Sha ^1_{\omega }(\widehat{Q})^D \end{equation*}$$is not exact (for example, the one constructed in [7, Corollary 9.16]). Embed Q$Q$ into some SLn$\operatorname{SL}_n$ and let Z:=SLn/Q$Z:=\operatorname{SL}_n/Q$. Then .In order to better understand the obstruction to weak approximation for homogeneous spaces over fields of the type (a) or (b), we should somehow combine the Brauer–Manin obstruction with the descent obstruction, another natural tool used in the study of such questions, as done by Izquierdo and Lucchini Arteche in [11] for the study of obstruction to rational points. We present the following result which applies in particular to homogeneous spaces under connected stabilizers:1.3TheoremProposition 4.5For G$G$ a connected linear group, we consider a homogeneous space Z$Z$ under G$G$ with geometric stabilizer H¯$\bar{H}$ such that Gss$G^{{\rm ss}}$ is simply connected and H¯torf$\bar{H}^{{\rm torf}}$ is abelian. Definewhere f$f$ runs over torsors W→Z$W\rightarrow Z$ under quasi‐trivial tori T$T$. Then .NOTATION AND PRELIMINARIESThe notation will be fixed in this section and used throughout this article. The setting is pretty much the same as in [11] where they treated the problem of Hasse principle for such varieties.CohomologyThe cohomology groups we consider are always in terms of Galois cohomology or étale cohomology.FieldsThroughout this article, we consider a field K$K$ of one of the following two types:(a)the function field of a smooth projective curve C$C$ over C((t))$\mathbb {C}((t))$;(b)the field of fractions of a two‐dimensional, excellent, henselian, local domain A$A$ with algebraically closed residue field of characteristic zero. An example of such a field is any finite extension of the field of fractions C((X,Y))$\mathbb {C}((X, Y))$ of the formal power series ring C[[X,Y]]$\mathbb {C}[[X,Y]]$.Both these two types of fields share a number of properties that hold also for totally imaginary number fields (cf. [10, Theorem 5.5], Theorem 1.2 and Theorem 1.4 of [6]):(i)the cohomological dimension is two,(ii)index and exponent of central simple algebras coincide,(iii)for any semisimple simply connected group G$G$ over K$K$, we have H1(K,G)=1$H^1(K,G)=1$,(iv)there is a natural set Ω$\Omega$ of rank one discrete valuations (that is, with values in Z$\mathbb {Z}$), with respect to which one can take completions. For type (a), let Ω$\Omega$ be the set of valuations coming from C(1)$C^{(1)}$ the set of closed points of the curve C$C$. For type (b), let Ω$\Omega$ be the set of valuations coming from prime ideals of height one in A$A$.Weak approximation is satisfied for semisimple simply connected groups over such fields (cf. [6, Theorem 4.7] for type (b), and [7, section 10.1 ] for type (a)).Sheaves and abelian groupsFor i>0$i&gt;0$, we denote by μn⊗i$\mu _n^{\otimes i}$ the i$i$‐fold tensor product of the étale sheaf μn$\mu _n$ of n$n$‐th roots of unity with itself. We set μn0=Z/nZ$\mu _n^0=\mathbb {Z}/n\mathbb {Z}$ and for i<0$i&lt;0$, and we define μn⊗i=Hom(μn⊗(−i),Z/nZ)$\mu _n^{\otimes i}=\operatorname{Hom}(\mu _n^{\otimes (-i)},\mathbb {Z}/n\mathbb {Z})$. Over the fields K$K$ we consider, since K$K$ contains an algebraically closed field and thus all the roots of unity, we have a (non‐canonical) isomorphism μn≃Z/nZ$\mu _n\simeq \mathbb {Z}/n\mathbb {Z}$. Denote by Q/Z(i)$\mathbb {Q}/\mathbb {Z}(i)$ the direct limit of the sheaves μm⊗i$\mu _m^{\otimes i}$ for all m>0$m&gt;0$. By choosing a compatible system of primitive n$n$‐th roots of unity for every n$n$ (for example, ξn=exp(2πi/n)$\xi _n=\exp (2\pi i/n)$), we can identify Q/Z(i)$\mathbb {Q}/\mathbb {Z}(i)$ with Q/Z$\mathbb {Q}/\mathbb {Z}$.For an abelian group A$A$ (that we are always supposed to be equipped with the discrete topology if there is no other topology defined on it), we denote by AD$A^D$ the group of continuous homomorphisms from A$A$ to Q/Z(−1)$\mathbb {Q}/\mathbb {Z}(-1)$. The functor A↦AD$A\mapsto A^D$ is an anti‐equivalence of categories between torsion abelian groups and profinite groups.Weak approximation and Brauer–Manin obstructionGiven a set Ω$\Omega$ of places of K$K$, let Z(KΩ):=∏v∈ΩKZ(Kv)$Z(K_\Omega ):=\prod _{v\in \Omega _K}Z(K_v)$ be the topological product where each Z(Kv)$Z(K_v)$ is equipped with the v$v$‐adic topology, and Kv$K_v$ denotes the completion at v$v$. We say that a K$K$‐variety Z$Z$ satisfies weak approximation with respect to Ω$\Omega$ if the set of rational points Z(K)$Z(K)$, considered through the diagonal map as a subset of Z(KΩ)$Z(K_\Omega )$, is a dense subset. Weak approximation is a birational invariant of smooth geometrically integral varieties, which is a consequence of the implicit function theorem for Kv$K_v$ (cf. Theorem 9.5.1 of [8] and the argument in [8, Proposition 12.2.3]).There are varieties for which weak approximation fails, thus we try to introduce obstructions that give a more precise description of the closure of Z(K)$Z(K)$ inside Z(KΩ)$Z(K_\Omega )$, thus explaining such failures.For a variety Z$Z$, the cohomological Brauer group Br(Z)$\operatorname{Br}(Z)$ is defined to be H2(Z,Gm)$H^2(Z,\mathbb {G}_m)$. Define also the following:Br0(Z):=Im(Br(K)→Br(Z))$_0(Z):=\operatorname{Im}(\operatorname{Br}(K)\rightarrow \operatorname{Br}(Z))$.Br1(Z):=ker(Br(Z)→Br(ZK¯))$_1(Z):=\ker (\operatorname{Br}(Z)\rightarrow \ Br(Z_{\bar{K}}))$.Bra(Z):=Br1(Z)/Br0(Z)$_{{\rm a}}(Z):=\operatorname{Br}_1(Z)/\operatorname{Br}_0(Z)$.Let w∈Z(K)$w\in Z(K)$ be a K$K$‐point of Z$Z$, we set Brw(Z):=Ker(w∗:Br(Z)→Br(K))$\operatorname{Br}_w(Z):=\operatorname{Ker}(w^*:\operatorname{Br}(Z)\rightarrow \operatorname{Br}(K))$ and Br1,w:=Ker(w∗:Br1(Z)→Br(K))$\operatorname{Br}_{1,w}:=\operatorname{Ker}(w^*:\operatorname{Br}_1(Z)\rightarrow \operatorname{Br}(K))$. We have a canonical isomorphism Br1,w(Z)≃Bra(Z)$\operatorname{Br}_{1,w}(Z)\simeq \operatorname{Br}_{{\rm a}}(Z)$.Б(Z):=ker(Bra(Z)→∏v∈ΩBra(ZKv))$\Be (Z):=\ker (\operatorname{Br}_{{\rm a}}(Z)\rightarrow \prod _{v\in \Omega }\operatorname{Br}_{{\rm a}}(Z_{K_{v}}))$.БS(Z):=ker(Bra(Z)→∏v∈Ω∖SBra(ZKv))$\Be _S(Z):=\ker (\operatorname{Br}_{{\rm a}}(Z)\rightarrow \prod _{v\in \Omega \backslash S}\operatorname{Br}_{{\rm a}}(Z_{K_{v}}))$ where S$S$ is a finite subset of Ω$\Omega$.Бω(Z)$\Be _\omega (Z)$: subgroup of Bra(Z)$\operatorname{Br}_{{\rm a}}(Z)$ containing the elements vanishing in Bra(ZKv)$\operatorname{Br}_{{\rm a}}(Z_{K_v})$ for almost all places v∈Ω$v\in \Omega$.Similar to the exact sequence from Class Field Theory for number fields, there is also an exact sequence (cf. Proposition 2.1.(v) of [7] and Theorem 1.6 of [13])Br(K)→⨁v∈ΩBr(Kv)→Q/Z→0$$\begin{equation*} \operatorname{Br}(K)\rightarrow \bigoplus _{v\in \Omega }\operatorname{Br}(K_v) \rightarrow \mathbb {Q}/\mathbb {Z}\rightarrow 0 \end{equation*}$$and a Brauer–Manin pairingZ(KΩ)×Бω(Z)→Q/Z((Pv),α)↦∑v∈Ω<Pv,α>$$\begin{align*} Z(K_\Omega )\times \Be _\omega (Z)&\rightarrow \mathbb {Q}/\mathbb {Z}\\ ((P_v),\alpha )& \mapsto \sum _{v\in \Omega }&lt;P_v,\alpha &gt; \end{align*}$$with the following property: Z(K)$Z(K)$ is contained in the subset of Z(KΩ)$Z(K_\Omega )$ defined as {(Pv)∈Z(KΩ):((Pv),α)=0forallα∈Бω(Z)}$\lbrace (P_v)\in Z(K_\Omega ):((P_v),\alpha )=0\text{ for all }\alpha \in \Be _{\omega }(Z)\rbrace$: the set of points that are orthogonal to Бω(Z)$\Be _{\omega }(Z)$. When Z(K)$Z(K)$ is non‐empty and , we say that the Brauer–Manin obstruction with respect to Бω$\Be _\omega$ is the only one to weak approximation.Tate–Shafarevich groupsFor a Galois module M$M$ over the field K$K$ and an integer i⩾0$i\geqslant 0$, we define the following Tate–Shafarevich groups:Шi(K,M):=ker(Hi(K,M)→∏v∈ΩHi(Kv,M))$\Sha ^i(K,M):=\ker (H^i(K,M)\rightarrow \prod _{v\in \Omega }H^i(K_v,M))$.ШSi(K,M):=ker(Hi(K,M)→∏v∈Ω∖SHi(Kv,M))$\Sha _S^i(K,M):=\ker (H^i(K,M)\rightarrow \prod _{v\in \Omega \backslash S}H^i(K_v,M))$ where S$S$ is a finite subset of Ω$\Omega$.Шωi(K,M):$\Sha _{\omega }^i(K,M):$ subgroup of Hi(K,M)$H^i(K,M)$ containing the elements vanishing in Hi(Kv,M)$H^i(K_v,M)$ for almost all places v∈Ω$v\in \Omega$.Algebraic groups and homogeneous spacesFor a linear algebraic K$K$‐group G$G$, the following notation will be used:D(G)$D(G)$: the derived subgroup of G$G$,G∘$G^{\circ }$: the neutral connected component of G$G$,Gf:=G/G∘$G^{{\rm f}}:=G/G^{\circ}$ the group of connected components of G$G$, which is a finite group,Gu:$G^{{\rm u}}:$ the unipotent radical of G∘$G^{\circ}$,Gred:=G∘/Gu$G^{{\rm red}}:=G^{\circ} /G^{\rm u}$ which is a reductive group,Gss:=D(Gred)$G^{{\rm ss}}:=D(G^{{\rm red}})$ which is a semisimple group,Gtor:=Gred/Gss$G^{{\rm tor}}:=G^{{\rm red}}/G^{{\rm ss}}$ which is a torus,Gssu:=ker(G∘→Gtor)$G^{{\rm ssu}}:=\ker (G^{\circ }\rightarrow G^{{\rm tor}})$ which is an extension of Gss$G^{{\rm ss}}$ by Gu$G^{\rm u}$,Gtorf:=G/Gssu$G^{{\rm torf}}:=G/G^{{\rm ssu}}$ which is an extension of Gf$G^{\rm f}$ by Gtor$G^{{\rm tor}}$,Ĝ$\widehat{G}$: the Galois module of the geometric characters of G$G$.A unipotent group over K$K$ a field of characteristic 0 is isomorphic to an affine space AKn$\bm {\mathrm{A}}_K^n$, thus K$K$‐rational and satisfies weak approximation.A torus T$T$ is said to be quasi‐trivial if T̂$\widehat{T}$ is an induced Gal(K¯/K)$\operatorname{Gal}(\bar{K}/K)$‐module.WEAK APPROXIMATION FOR CONNECTED LINEAR GROUPSThe aim of this section is to prove the following theorem which concerns the Brauer–Manin obstruction to weak approximation for connected linear groups:3.1TheoremLet K$K$ be a field of the type (a) or (b) and let G$G$ be a connected linear group over K$K$. Then there is an exact sequence1→G(K)¯→G(KΩ)→Бω(G)D→Б(G)D→1.$$\begin{equation*} 1\rightarrow \overline{G(K)}\rightarrow G(K_\Omega )\rightarrow \Be _\omega (G)^D\rightarrow \Be (G)^D\rightarrow 1. \end{equation*}$$We complete the proof by a series of lemmas.3.2LemmaSuppose that Theorem 3.1 holds for Gn$G^n$ where n$n$ is a positive integer, then it also holds for G$G$.ProofIt follows from the fact that all of the operations in the exact sequence above take products to products. (For example, the closure of a product is the product of closures.)□$\Box$An exact sequence1→F→H×KP→G→1,$$\begin{equation*} 1\rightarrow F\rightarrow H\times _K P\rightarrow G\rightarrow 1, \end{equation*}$$where H$H$ is semi‐simple simply connected, P$P$ is a quasitrivial K$K$‐torus and F$F$ is finite and central is called a special covering of the reductive K$K$‐group G$G$. For any reductive K$K$‐group G$G$, there exists an integer n>0$n&gt;0$ such that Gn$G^n$ admit a special covering. By the lemma above, we can suppose without loss of generality that G$G$ admits a special covering itself.3.3LemmaLet S$S$ be a finite set of places in Ω$\Omega$. The following sequence is exact1→G(K)¯→∏v∈SG(Kv)→БS(G)D,$$\begin{equation*} 1\rightarrow \overline{G(K)}\rightarrow \prod _{v\in S}G(K_v)\rightarrow \Be _S(G)^D, \end{equation*}$$where G$G$ is a reductive connected linear group.ProofThe proof is exactly the same as in [7, Lemma 9.6], using the special covering and the fact that H$H$ and P$P$ both satisfy weak approximation (so does their product) and H1(K,H)=H1(K,P)=H1(K,H×KP)=0$H^1(K,H)=H^1(K,P)=H^1(K,H\times _K P)=0$. We have a commutative diagram where the rows are exact and the columns are complexes.The exactness of the last row comes from the fact that H1(K,F̂)≃Ker(Bra(G)→Bra(H×KP))$H^1(K,\widehat{F})\simeq \operatorname{Ker}(\operatorname{Br}_{{\rm a}}(G)\rightarrow \operatorname{Br}_{{\rm a}}(H\times _K P))$ and Bra(H×KP)≃H2(K,P̂)$\operatorname{Br}_{{\rm a}}(H\times _K P)\simeq H^2(K,\widehat{P})$ (Corollary 7.4 and Lemma 6.9 of [15]), with the latter isomorphism inducing an isomorphism БS(H×P)≃ШS2(P̂)$\Be _S(H\times P)\simeq \Sha ^2_S(\widehat{P})$ by the same argument as in [7, Section 8.1]. The commutativity of the right‐bottom square can be deduced from [15, Lemma 8.11].□$\Box$3.4LemmaLet S$S$ be a finite set of places in Ω$\Omega$. The following sequence is exact:1→G(K)¯→∏v∈SG(Kv)→БS(G)D,$$\begin{equation*} 1\rightarrow \overline{G(K)}\rightarrow \prod _{v\in S}G(K_v)\rightarrow \Be _S(G)^D, \end{equation*}$$where G$G$ is a connected linear group over K$K$.ProofWe use the resolution 1→Gu→G→Gred→1$1\rightarrow G^{{\rm u}}\rightarrow G\rightarrow G^{{\rm red}}\rightarrow 1$ and the induced diagramtaking into account the vanishing of H1(K,Gu)$H^1(K,G^{{\rm u}})$. The unipotent radical Gu$G^{{\rm u}}$ satisfies weak approximation (see Section 2). The exactness of the rightmost column is known from the previous lemma, then a diagram chasing gives the result.□$\Box$3.5LemmaLet S$S$ be a finite set of places in Ω$\Omega$. The sequence3.5.1∏v∈SG(Kv)→БS(G)D→Б(G)D$$\begin{equation} \prod _{v\in S}G(K_v)\rightarrow \Be _S(G)^D\rightarrow \Be (G)^D\end{equation}$$is exact, where G$G$ is a connected linear group over K$K$.ProofAs in the previous lemmas, we can first treat the case where G$G$ is reductive. Suppose now that G$G$ is reductive, we use a coflasque resolution1→P→G′→G→1,$$\begin{equation*} 1\rightarrow P\rightarrow G^\prime \rightarrow G\rightarrow 1, \end{equation*}$$where P$P$ is a quasitrivial torus and G′$G^\prime$ fits into the exact sequence1→Gsc→G′→T→1,$$\begin{equation*} 1\rightarrow G^{{\rm sc}}\rightarrow G^\prime \rightarrow T\rightarrow 1, \end{equation*}$$where T$T$ is a coflasque torus and Gsc$G^{{\rm sc}}$ is a semisimple simply connected group. Suppose that the exactness of the sequence in question is known for G′$G^\prime$ (replacing G$G$). Then we can prove the exactness for G$G$ by chasing in the following diagram:The leftmost arrow is an isomorphism by [7, Proposition 2.6]. The last two rows are exact, and this can be proved by chasing the following diagram, as done in the proof of [3, Lemma 4.4].(The exactness of the rows can be deduced from [15, Proposition 6.10] and the fact that Pic(P)=H1(K,P̂)=0$\operatorname{Pic}(P)=H^1(K,\widehat{P})=0$ since P̂$\widehat{P}$ is a permutation module.)The exactness of the sequence 3.5.1 is known for T$T$ ([7, Corollary 9.9] for K$K$ of type (a), and [13, Theorem 4.9] for K$K$ of type (b)), by a similar diagram chasing, we can prove the result for G′$G^\prime$, thus completing the proof. The essential condition we need in this diagram chasing is that we should have an injection БS(G′)D↪БS(T)D$\Be _S(G^\prime )^D\hookrightarrow \Be _S(T)^D$, or equivalently a surjection БS(T)↠БS(G′)$\Be _S(T)\twoheadrightarrow \Be _S(G^\prime )$. This is indeed true because we have an exact sequence Bra(T)→Bra(G′)→Bra(Gsc)$\operatorname{Br}_{{\rm a}}(T)\rightarrow \operatorname{Br}_{{\rm a}}(G^\prime )\rightarrow \operatorname{Br}_{{\rm a}}(G^{{\rm sc}})$ and Br(Gsc)=Br(K)$\operatorname{Br}(G^{{\rm sc}})=\operatorname{Br}(K)$ since Gsc$G^{{\rm sc}}$ a simply connected group over a field of characteristic 0 (cf. Corollary in Section  0 of [9]).Now for a connected linear group G$G$ (not necessarily reductive), we use again the resolution 1→Gu→G→Gred→1$1\rightarrow G^{{\rm u}}\rightarrow G\rightarrow G^{{\rm red}}\rightarrow 1$ and the induced commutative diagramtaking into account the vanishing of H1(Kv,Gu)$H^1(K_v,G^{{\rm u}})$. The injectivity of the middle line follows from the fact that Bra(Gu)=Bra(AKn)=(Bra(AK1))n=0$\operatorname{Br}_{{\rm a}}(G^{{\rm u}})=\operatorname{Br}_{{\rm a}}(\bm {\mathrm{A}}_K^n)=(\operatorname{Br}_{{\rm a}}(\bm {\mathrm{A}}_K^1))^n=0$ (cf. Lemma 6.6 of [15], Theorem 4.5.1(viii) of [8]). Then a diagram chasing gives the desired result.□$\Box$3.6CorollaryTheorem 3.1With the same notation as above, there is an exact sequence1→G(K)¯→G(KΩ)→Бω(G)D→Б(G)D→1.$$\begin{equation*} 1\rightarrow \overline{G(K)}\rightarrow G(K_\Omega )\rightarrow \Be _\omega (G)^D\rightarrow \Be (G)^D\rightarrow 1. \end{equation*}$$ProofWe extend the result with respect to a finite set S$S$ of places to all the Ω$\Omega$ by following the same proof used in Corollary 9.9 of [7]. Let A(G):=coker(G(K)¯→G(KΩ))$A(G):=\operatorname{coker}(\overline{G(K)}\rightarrow G(K_\Omega ))$ and AS(G):=coker(G(K)¯→∏v∈S(G(Kv)))$A_S(G):=\operatorname{coker}(\overline{G(K)}\rightarrow \prod _{v\in S}(G(K_v)))$ measuring the failure of weak approximation. Let 1→Q→R→G→1$1\rightarrow Q\rightarrow R\rightarrow G\rightarrow 1$ be a flasque resolution of G$G$ (cf. Proposition–Definition 3.1 of [5]). By Proposition 9.1 of [7] (for K$K$ of type (a)) and Theorem 3.7 with its remark in [6] (for K$K$ of type (b)), we have AS(G)=coker(H1(K,Q)→∏v∈SH1(Kv,Q))$A_S(G)=\operatorname{coker}(H^1(K,Q)\rightarrow \prod _{v\in S}H^1(K_v,Q))$ and H1(Kv,Q)=0$H^1(K_v,Q)=0$ for all v$v$ outside the finite set S0$S_0$ of places of bad reduction of Q$Q$. Therefore, we have AS(G)=A(G)$A_S(G)=A(G)$ for S0⊆S$S_0\subseteq S$. Let S0⊆S⊆S′$S_0\subseteq S\subseteq S^\prime$, and then we have a commutative diagramwhere the left and the right vertical maps are isomorphisms. Thus the middle map is also an isomorphism and БS(G)=Бω(G)$\Be _S(G)=\Be _{\omega }(G)$.□$\Box$WEAK APPROXIMATION FOR HOMOGENEOUS SPACESNow we look at weak approximation for homogeneous spaces over such fields K$K$. The usual Brauer–Manin obstruction we used in the last section is not enough in this case, as shown by the following construction of examples:4.1PropositionLet Q$Q$ be a flasque torus such thatH1(K,Q)→⊕v∈ΩH1(Kv,Q)→Шω1(Q̂)D$$\begin{equation*} H^1(K,Q)\rightarrow \oplus _{v\in \Omega }H^1(K_v,Q)\rightarrow \Sha ^1_{\omega }(\widehat{Q})^D \end{equation*}$$is not exact. Such a torus exists (see Corollary 9.16 of [7] for a construction.) Embed Q$Q$ into some SLn$\operatorname{SL}_n$ and let Z:=SLn/Q$Z:=\operatorname{SL}_n/Q$. Then .ProofWe consider the following commutative diagram with exact rows:The vanishing of Bra(SLn)$\operatorname{Br}_{{\rm a}}(\operatorname{SL}_n)$ on the bottom‐left corner comes from the Corollary of [9] (section 0). We prove the result by diagram chasing. Since the right column is not exact, there exists a∈∏v∈ΩH1(Kv,Q)$a\in \prod _{v\in \Omega }H^1(K_v,Q)$ such that f6(a)$f_6(a)$ vanishes but a∉f3(H1(K,Q))$a\notin f_3(H^1(K,Q))$. Since f4$f_4$ is surjective, we can find b∈Z(KΩ)$b\in Z(K_\Omega )$ such that f4(b)=a$f_4(b)=a$. By the commutativity of the bottom square, f5(b)$f_5(b)$ vanishes. We prove that b∉Z(K)¯$b\notin \overline{Z(K)}$ by contradiction. We suppose b∈Z(K)¯$b\in \overline{Z(K)}$. The torus Q$Q$ being flasque implies that ∏v∈ΩH1(Kv,Q)$\prod _{v\in \Omega }H^1(K_v,Q)$ is finite (cf. [7, Proposition 9.1] for K$K$ of type (a); Theorem 3.7 with its remark in [6] for K$K$ of type (b)), thus the preimage f4−1(a)$f_4^{-1}(a)$ is open (f4$f_4$ is continuous, cf. [4]), containing b∈Z(K)¯$b\in \overline{Z(K)}$, so we should be able to find c∈Z(K)$c\in Z(K)$ lying in f4−1(a)$f_4^{-1}(a)$. Then f3(f1(c))=a$f_3(f_1(c))=a$ by the commutativity of the top square, contradicting a∉Imf3$a\notin \operatorname{Im}f_3$. Therefore, we found .□$\Box$Therefore, we need to find other obstructions. In the rest of this section, we consider homogeneous spaces Z$Z$ under a connected linear group G$G$ with geometric stabilizer H¯$\bar{H}$ such that Gss$G^{{\rm ss}}$ is simply connected and H¯torf$\bar{H}^{{\rm torf}}$ is abelian. This is the assumption already used in [11] and [3], and is satisfied by every homogeneous space under a connected linear K$K$‐group with connected stabilizers (cf. Lemma 5.2 in [3]).We can find Z←W→W′$Z\leftarrow W\rightarrow W^\prime$ such thatW$W$ is a K$K$‐homogeneous space under G×T$G\times T$,T$T$ is a quasitrivial torus into which Htorf$H^{{\rm torf}}$ injects, where Htorf$H^{{\rm torf}}$ is the canonical K$K$‐form of H¯torf$\bar{H}^{{\rm torf}}$ associated to Z$Z$,W→Z$W\rightarrow Z$ is a T$T$‐torsor,W′$W^\prime$ is the quotient variety Z/Gss$Z/G^{{\rm ss}}$, which is also a homogeneous space of Gtor×T$G^{{\rm tor}}\times T$ with geometric stabilizer H¯torf$\bar{H}^{{\rm torf}}$ and the fibers of W→W′$W\rightarrow W^\prime$ are homogeneous spaces of Gssu$G^{{\rm ssu}}$ with geometric stabilizers H¯ssu$\bar{H}^{{\rm ssu}}$.Indeed, we can embed Htor$H^{{\rm tor}}$ in a quasi‐trivial torus T$T$ and consider the diagonal morphism H→G×T$H\rightarrow G\times T$ induced by the inclusion H↪G$H\hookrightarrow G$ and the composition H→Htor→T$H\rightarrow H^{{\rm tor}}\rightarrow T$. Then we define W=(G×T)/H$W=(G\times T)/H$, and W→Z$W\rightarrow Z$ is induced by the projection to the first coordinate. Define W′$W^{\prime }$ to be the quotient variety W/Gssu$W/G^{{\rm ssu}}$ and we get what we want.4.2PropositionThe fiber WP$W_P$ above a K$K$‐point P∈W′(K)$P\in W^\prime (K)$ satisfies weak approximation.ProofSince Hss$H^{{\rm ss}}$ is semi‐simple, we consider its simply connected covering 1→F→Hsc→Hss→1$1\rightarrow F\rightarrow H^{{\rm sc}}\rightarrow H^{{\rm ss}}\rightarrow 1$ where F$F$ is finite and Hsc$H^{{\rm sc}}$ is simply connected.We first prove that this covering induces isomorphisms H1(K,Hss)≃H2(K,F)$H^1(K,H^{{\rm ss}})\simeq H^2(K,F)$ and H1(Kv,Hss)≃H2(Kv,F)$H^1(K_v,H^{{\rm ss}})\simeq H^2(K_v,F)$ for all v∈Ω$v\in \Omega$. For K$K$ and Kv$K_v$, the two conditions in [6, Theorem 2.1] are satisfied, and thus we have a bijection H1(K,Had)→H2(K,μ)$H^1(K,H^{\rm ad})\rightarrow H^2(K,\mu )$ coming from the central isogeny1→μ→Hsc→Had→1$$\begin{equation*} 1\rightarrow \mu \rightarrow H^{{\rm sc}}\rightarrow H^{\rm ad}\rightarrow 1 \end{equation*}$$associated to the center μ$\mu$ of Hsc$H^{{\rm sc}}$. Since F$F$ is contained in the center μ$\mu$, we have the exact sequence1→μ/F→Hss→Had→1$$\begin{equation*} 1\rightarrow \mu /F\rightarrow H^{{\rm ss}}\rightarrow H^{\rm ad}\rightarrow 1 \end{equation*}$$and the commutative diagram with exact rowsThen the four‐lemma gives the surjectivity of H1(K,Hss)→H2(K,F)$H^1(K,H^{{\rm ss}})\rightarrow H^2(K,F)$. The injectivity comes from the vanishing of H1(K,Hsc)$H^1(K,H^{{\rm sc}})$. The proof is the same when K$K$ is replaced by Kv$K_v$.Then, we prove thatH2(K,F)→⊕v∈SH2(Kv,F)$$\begin{equation*} H^2(K,F)\rightarrow \oplus _{v\in S}H^2(K_v,F) \end{equation*}$$is surjective, where S$S$ is a finite set of places. This is proved by chasing the following commutative diagram:The row in the middle is exact. (See [12] Theorem 2.7 applying d=0$d=0$ for K$K$ of type (a). For K$K$ of type (b), use [13, Corollary 2.3] and the exact sequence H2(U,F)→⨁v∈X∖UH2(Kv,F)→Hc3(U,F)$H^2(U,F)\rightarrow \bigoplus _{v\in X\backslash U}H^2(K_v,F)\rightarrow H^3_c(U, F)$ with X$X$ and U$U$ defined in [13], and then take direct limit over U$U$.) The surjectivity of ⊕v∉SH2(Kv,F)→H0(K,F̂)D$\oplus _{v\notin S}H^2(K_v,F)\rightarrow H^0(K,\widehat{F})^D$ comes from the injectivity taking the duals H0(K,F̂)↪⊕v∉SH0(Kv,F̂)$H^0(K,\widehat{F})\hookrightarrow \oplus _{v\notin S} H^0(K_v,\widehat{F})$. This yields the surjectivity of H2(K,F)→∏v∈SH2(Kv,F)$H^2(K,F)\rightarrow \prod _{v\in S}H^2(K_v,F)$, which is H1(K,Hss)→⊕v∈SH1(Kv,Hss)$H^1(K,H^{{\rm ss}})\rightarrow \oplus _{v\in S}H^1(K_v,H ^{{\rm ss}})$.Finally, by [15, Lemma 1.13], we have H1(K,Hss)=H1(K,Hssu)$H^1(K,H^{{\rm ss}})=H^1(K,H^{{\rm ssu}})$ and H1(Kv,Hss)=H1(Kv,Hssu)$H^1(K_v,H^{{\rm ss}})=H^1(K_v,H^{{\rm ssu}})$. With the same argument as in [15, Proposition 3.2], we can prove that Gssu$G^{{\rm ssu}}$ satisfies weak approximation (cf. [2, Proposition 4.1] for the vanishing of H1(K,Hu)$H^1(K,H^u)$ and H1(Kv,Hu)$H^1(K_v,H^u)$. The unipotent radical Hu$H^{{\rm u}}$ satisfies weak approximation). Therefore, we consider the commutative diagram with exact rows (the set WP(K)$W_P(K)$ is non‐empty by [11, Proposition 3.1]):A diagram chasing then gives the desired result.□$\Box$4.3PropositionWe have .ProofConsider , and any open subset U$U$ containing (Pv)$(P_v)$, we want to find Q∈U∩W(K)$Q\in U\cap W(K)$ using fibration methods.Denote by (Pv′)$(P_v^\prime )$ the image of (Pv)$(P_v)$ under the induced map g:W(KΩ)→W′(KΩ)$g: W(K_\Omega )\rightarrow W^\prime (K_\Omega )$. By the functorality of the Brauer–Manin pairing, (Pv′)$(P_v^\prime )$ lies in . Since W→W′$W\rightarrow W^ \prime$ is smooth, and the Kv$K_v$ are henselian, we have open maps W(Kv)→W′(Kv)$W(K_v)\rightarrow W^\prime (K_v)$, and thus g$g$ is open. In particular, V=g(U)⊆W′(KΩ)$V=g(U)\subseteq W^\prime (K_\Omega )$ is also open. Since W′$W^{\prime }$ is a torus (cf. [11, Theorem 3.2]), we have . Therefore, there exists P′∈V∩W′(K)$P^\prime \in V\cap W^\prime (K)$. Let (Qv)∈g−1(P′)∩U⊆W(KΩ)$(Q_v)\in g^{-1}(P^{\prime })\cap U\subseteq W(K_\Omega )$, and it can be seen as in WP′(KΩ)$ W_{P^\prime }(K_\Omega )$ too, as shown in the following diagram.This diagram also shows that WP′(KΩ)⊆W(KΩ)$W_{P^\prime }(K_\Omega )\subseteq W(K_\Omega )$, and U|WP′(KΩ):=U∩WP′(KΩ)$U_{|_{ W_{P^\prime }(K_\Omega )}}:=U\bigcap W_{P^\prime }(K_\Omega )$ is an open neighborhood of (Qv)$(Q_v)$ in WP′(KΩ)$W_{P^\prime }(K_\Omega )$. Since WP′$W_{P^\prime }$ satisfies weak approximation by Proposition 4.2, there exists Q∈WP′(K)∩U|WP′(KΩ)$Q\in W_{P^\prime }(K)\cap U_{|_{ W_{P^\prime }(K_\Omega )}}$, and thus in W(K)∩U$W(K)\cap U$, which proves that .□$\Box$As in [11], we are naturally led to consider the following definition.4.4DefinitionFor an arbitrary K$K$‐variety Z$Z$, we defineUsing the torsor W→Z$W\rightarrow Z$ defined above, we have . In fact, this is an equality.4.5PropositionLet the notation be as above. Then .ProofSince for all f$f$, we haveso . To prove Proposition 4.5, it is equivalent to prove that is closed. It suffices to prove that is closed for every torus f:W→TZ$f: W\xrightarrow T{Z}$ with T$T$ quasitrivial. Since f$f$ is smooth and the Kv$K_v$ are henselian, the induced map f:W(KΩ)→Z(KΩ)$f:W(K_\Omega )\rightarrow Z(K_\Omega )$ is open. Now we will prove that does not meet , and combined with the surjectivity of f$f$, we will get  closed.Fixing w∈W(K)$w\in W(K)$ which is a K$K$‐point of W$W$, we have the canonical isomorphism Bra(W)≃Br1,w(W)$\operatorname{Br}_{{\rm a}}(W)\simeq \operatorname{Br}_{1,w}(W)$. Define the map iw:T→W$i_w: T\rightarrow W$ as t↦t.w$t\mapsto t.w$, and we have an induced map iw∗:Br1,w(W)→Br1,e(T)$i_w^*: \operatorname{Br}_{1,w}(W)\rightarrow \operatorname{Br}_{1,e}(T)$. Let α$\alpha$ be an element in Бω(W)$\Be _\omega (W)$ and we still denote by α$\alpha$ its image in Br1,w(W)$\operatorname{Br}_{1,w}(W)$. Let (xv),(yv)∈W(KΩ)$(x_v),(y_v)\in W(K_\Omega )$ above (Pv)$(P_v)$. We will prove that ∑v∈Ω<xv,α>=∑v∈Ω<yv,α>$\sum _{v\in \Omega }&lt;x_v,\alpha &gt;=\sum _{v\in \Omega }&lt;y_v,\alpha &gt;$. Let m:T×W→W$m: T\times W\rightarrow W$ denotes the action of T$T$ on W$W$. Since WPv$W_{P_v}$ is a Kv$K_v$‐torsor under T$T$, there exists tv∈T(Kv)$t_v\in T(K_v)$ such that xv=tv.yv$x_v=t_v.y_v$. By the functorality of the Brauer–Manin paring, we have <xv,α>=<(tv,yv),m∗α>$&lt;x_v,\alpha &gt;=&lt;(t_v,y_v),m^* \alpha &gt;$. By [15, Lemma 6.6], we have Br1,(w,e)(W×T)=Br1,w(W)×Br1,e(T)$\operatorname{Br}_{1,(w,e)}(W\times T)=\operatorname{Br}_{1,w}(W)\times \operatorname{Br}_{1,e}(T)$, and the formula (24) in [1] gives m∗α=pT∗iw∗(α)+pW∗α$m^*\alpha =p_T^*i_w^*(\alpha )+p_W^*\alpha$ where pT$p_T$ and pW$p_W$ are the two natural projections. Then <xv,α>=<tv,iw∗(α)>+<yv,α>$&lt;x_v,\alpha &gt;=&lt;t_v,i_w^*(\alpha )&gt;+&lt;y_v,\alpha &gt;$. But P$P$ is a quasi‐trivial torus, we have Бω(P)≃Б(P)$\Be _\omega (P)\simeq \Be (P)$ (cf. Proposition 2.6 and section  8.1 of [7]). Therefore, by the exact sequence in Theorem 3.1, the map P(KΩ)→Бω(P)D$P(K_\Omega )\rightarrow \Be _\omega (P)^D$ is 0, that is, ∑v∈Ω<tv,iw∗(α)>=0$\sum _{v\in \Omega }&lt;t_v,i_w^*(\alpha )&gt;=0$ and we get what we want.□$\Box$4.6RemarkActually the above proof also shows thatWith this description, we can prove as done in [11, Theorem 6.4] that4.6.1But this does not necessarily give an obstruction to weak approximation since we do not know if Z(KΩ)tor$Z(K_\Omega )^{{\rm tor}}$ is closed. We do not know if (4.6.1) is an equality. 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