Manin obstruction to strong approximation for homogeneous spaces

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24 Mikhail Borovoi and Cyril DemarcheIn the case x = x 0 = π(y 0 ) we obtain(π ∗ b)(y 0 .g) = b(π(y 0 ).g) = b(π(y 0 )) + π G (m ∗ b)(g) = π G (m ∗ b)(g), (27)since b(π(y 0 )) = b(x 0 ) = 0. Consequently, (26) and (27) give the expected formula,that is:b(x.g) = b(x) + (π ∗ b)(y 0 .g) .Corollary 3.4. Let X := H\G be a right homogeneous space of a connected k-group G over a field k of characteristic 0, where H ⊂ G is a connected lineark-subgroup. Then for all b ∈ Br 1,x0 (X, G), x ∈ X(k), g ∈ G(k) we haveb(x.g) = b(x) + π ∗ b(g),where π : G → X is the quotient map and x 0 = π(e).Proof. We take Y = G, y 0 = e ∈ G(k), and x 0 = π(y 0 ) = π(e) in Lemma 3.3, thenBr 1,x0 (X, Y ) = Br 1,x0 (X, G) and π ∗ b(y 0 .g) = π ∗ b(g).Corollary 3.5. Let k be a number field. Let X := H\G be a homogeneous spaceof a connected k-group G, where H ⊂ G is a connected linear k-subgroup. Let 〈, 〉denote the <strong>Manin</strong> pairingBr(X) × X(A) → Q/Z.Let b ∈ Br 1,x0 (X, G), x ∈ X(A), g ∈ G(A). Then〈b, x.g〉 = 〈b, x〉 + 〈π ∗ (b), g〉,where π : G → X is the quotient map and x 0 = π(e).Corollary 3.6. Let k, G, H and X be as in Corollary 3.5. Let ϕ: G ′ → G bea homomorphism of k-groups, where G ′ is a simply connected k-group. Let b ∈Br 1,x0 (X, G), x ∈ X(A), g ′ ∈ G ′ (A). ThenProof. By Corollary 3.5 we haveBy func<strong>to</strong>riality we have〈b, x.ϕ(g ′ )〉 = 〈b, x〉.〈b, x.ϕ(g ′ )〉 = 〈b, x〉 + 〈π ∗ (b), ϕ(g ′ )〉.〈π ∗ (b), ϕ(g ′ )〉 = 〈ϕ ∗ π ∗ (b), g ′ 〉.Since b ∈ Br 1,x0 (X, G), we have π ∗ b ∈ Br 1,e (G) and ϕ ∗ π ∗ b ∈ Br 1,e (G ′ ) = 0. Thusϕ ∗ π ∗ b = 0, hence 〈π ∗ (b), ϕ(g ′ )〉 = 0, and the corollary follows.
<strong>Manin</strong> <strong>obstruction</strong> <strong>to</strong> strong approximation for homogeneous spaces 254 Some lemmasFor an abelian group A we write A D := Hom(A, Q/Z).Lemma 4.1. Let P be a quasi-trivial k-<strong>to</strong>rus over a number field k. Then thecanonical map λ: P (A) → Br a (P ) D induced by the <strong>Manin</strong> pairing is surjective.Proof. We have Br a (P ) = H 2 (k, ̂P ), see [38], Lemma 6.9(ii). By [38], (8.11.2), themapλ: P (A) → Br a (P ) D = H 2 (k, ̂P ) Dis given by the canonical pairingP (A) × H 2 (k, ̂P ) → Q/Z.Consider the map µ from the Tate-Poi<strong>to</strong>u exact sequence(P (A) • ) ∧ µ−−→ H 2 (k, ̂P ) D → H 1 (k, P ), (28)see [25], Theorem 5.6 or [14], Theorem 6.3. By (P (A) • ) ∧ we mean the completionof P (A) • for the <strong>to</strong>pology of open subgroups of finite index. Then the map µ isinduced by λ. Since P is a quasi-trivial <strong>to</strong>rus, we have H 1 (k, P ) = 0, and we seefrom (28) that the map µ is surjective. But by [22], Lemma 4, im µ = im λ. Thusλ is surjective.Lemma 4.2. Let X be a right homogeneous space (not necessarily principal) ofa connected k-group G over a number field k. Let N ⊂ G be a connected normalk-subgroup. Set Y := X/N, and let π : X → Y be the canonical map. Then theinduced map X(A) → Y (A) is open.Note that the geometric quotient X/N exists in the category of k-varieties by[5], Lemma 3.1.Proof. If v is a nonarchimedean place of k, we denote by O v the ring of integers ofk v , and by κ v the residue field of O v . For an O v -scheme Z v we set ˜Z v := Z v × Ov κ v .Since the morphism π is smooth, the map X(k v ) → Y (k v ) is open for any placev of k.Let S be a finite set of places of k containing all the archimedean places. WriteO S for the ring of elements of k that are integral outside S. Taking S sufficientlylarge, we can assume that G and N extend <strong>to</strong> smooth group schemes G and Nover Spec(O S ), and that X and Y extend <strong>to</strong> homogeneous spaces X of G andY of G /N over Spec(O S ) such that Y = X /N . In particular, the reductionÑ v := N × O S κ v is connected for v /∈ S.Let v /∈ S and let y v ∈ Y (O v ). Set X yv := X × Y Spec(O v ), the morphismsbeing given by π and by y v : Spec (O v ) → Y . It is an O v -scheme. Then itsreduction ˜X yv is a homogeneous space of the connected κ v -group Ñv over the
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Manin obstruction to strong approximation for homogeneous spaces

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