Manin obstruction to strong approximation for homogeneous spaces

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46 Mikhail Borovoi and Cyril DemarcheWe set w := w ′ .p ∈ W (A), then m W,F (w) = 0 and t(w) = x.7.22. We can now resume the proof of Theorem 7.3. We use Construction 7.18.Let U f X ⊂ X(Af ) be an open neighbourhood of x f . Let U X ⊂ X(A) be thecorresponding special neighbourhood of x. SetU f W := t−1 (U f X ) ⊂ W (Af ).For v ∈ Ω ∞ let U W,v be the connected component of w v in W (k v ), then by[7], Lemma A.2, we have t(U W,v ) = U X,v . Set U W,∞ := ∏ v∈Ω ∞U W,v . SetU W := U f W × U W,∞, then U W is the special open neighbourhood of w defined byU f W , and t(U W ) ⊂ U X .The pair (W, F ) of F satisfies the hypotheses of Proposition 7.4 (see Lemma7.19), so by that proposition, there is a point w 1 ∈ W (k) ∩ U W .F sc (k S ). Notethat F sc = G sc . Set x 1 := t(w 1 ), then x 1 ∈ X(k) ∩ U X .G sc (k S ). Thus the pair(X, G) has Property (P S ).This completes the proofs of Theorem 7.3 and proves the nontrivial inclusionof Theorem 1.4, that is, that any element of (X(A) • ) Br1(X,G) lies in the closureof the set X(k).G scu (k Sf ). The argument in the proof of the trivial inclusion ofTheorem 6.1 also proves the trivial inclusion of Theorem 1.4, that is, that eachelement of this closure is orthogonal <strong>to</strong> Br 1 (X, G). This completes the proof ofMain Theorem 1.4.8 The algebraic <strong>Manin</strong> <strong>obstruction</strong>In this section we prove Theorem 1.7 about the algebraic <strong>Manin</strong> <strong>obstruction</strong> (“algebraic”means coming from Br 1 (X)). We prove this result without using theresult of Colliot-Thélène and Xu (Theorem 6.1 or [11], Theorem 3.7(b)).8.1. Before proving Theorem 1.7, we need <strong>to</strong> prove a special case – an analogueof Theorem 6.1. In [6], the first-named author defined, for any connected groupH over a field k of characteristic 0, a Galois module π 1 (H), an abelian groupHab 1 (k, H) and a canonical abelianization mapab 1 : H 1 (k, H) → H 1 ab(k, H)(see also [8] in any characteristic). These π 1 (H), H 1 ab (k, H) and ab1 are func<strong>to</strong>rialin H.Now let k be a number field. Set Γ := Gal(k/k), Γ v := Gal(k v /k v ). We regardΓ v as a subgroup of Γ.For v ∈ Ω f we defined in [6], Proposition 4.1(i), a canonical isomorphismλ v : H 1 ab(k v , H) ∼ → (π 1 (H) Γv ) <strong>to</strong>rs ,
<strong>Manin</strong> <strong>obstruction</strong> <strong>to</strong> strong approximation for homogeneous spaces 47where π 1 (H) Γv denotes the groups of coinvariants of Γ v in π 1 (H), and ( ) <strong>to</strong>rsdenotes the <strong>to</strong>rsion subgroup. Here we set λ ′ v := λ v .For v ∈ Ω ∞ we defined in [6], Proposition 4.2, a canonical isomorphismλ v : H 1 ab(k v , H) ∼ → H −1 (Γ v , π 1 (H)).Here we define a homomorphism λ ′ v as the compositionλ ′ v : H 1 ab(k v , H)λ v−−→ H −1 (Γ v , π 1 (H)) ↩→ (π 1 (H) Γv ) <strong>to</strong>rs .For any v ∈ Ω we define the Kottwitz map β v as the compositionβ v : H 1 (k v , H) −−→ ab1Hab(k 1 v , H)λ ′ v−−→ (π 1 (H) Γv ) <strong>to</strong>rs .This map β v is func<strong>to</strong>rial in H. Note that for v ∈ Ω f the maps ab 1 : H 1 (k v , H) →H 1 ab (k v, H) and β v are bijections. Thus for v ∈ Ω f we have a canonical andfunc<strong>to</strong>rial in H bijection β v : H 1 (k v , H) ∼ → (π 1 (H) Γv ) <strong>to</strong>rs .For any v ∈ Ω we define a map µ v as the compositionµ v : H 1 (k v , H)β v−−→ (π 1 (H) Γv ) <strong>to</strong>rscor v−−→ (π 1 (H) Γ ) <strong>to</strong>rs , (39)where cor v is the obvious corestriction map.We write ⊕ v H1 (k v , H) for the set of families (ξ v ) v∈Ω such that ξ valmost all v. We define a map= 1 forµ := ∑ v∈Ωµ v : ⊕ v∈ΩH 1 (k v , H) → (π 1 (H) Γ ) <strong>to</strong>rs .Proposition 8.2 (Kottwitz [30], Proposition 2.6, see also [6], Theorem 5.15).The kernel of the map µ is equal <strong>to</strong> the image of the localization map H 1 (k, H) →⊕v H1 (k v , H).Proposition 8.3. Let G be a simply connected k-group over a number field k,and let H ⊂ G be a connected geometrically character-free subgroup (i.e. H <strong>to</strong>r =1). Set X := H\G. Let S be a finite set of places of k containing at least onenonarchimedean place. Then any orbit of G(A S ) in X(A S ) contains a k-point.Proof. Write M := π 1 (H). First we prove that the mapµ S = ∑ ∏µ v : ker[H 1 (k v , H) → H 1 (k v , G)] → (M Γ ) <strong>to</strong>rsv∈Sv∈Sis surjective. Since H is geometrically character-free, the group M = π 1 (H) isfinite, and therefore (M Γ ) <strong>to</strong>rs = M Γ and (M Γv ) <strong>to</strong>rs = M Γv . In this case themap cor v is the canonical map M Γv → M Γ , which is clearly surjective. Let
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Manin obstruction to strong approximation for homogeneous spaces

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