Manin obstruction to strong approximation for homogeneous spaces

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32 Mikhail Borovoi and Cyril Demarchehave a canonical epimorphism ϕ: G → G ′ and a canonical smooth ϕ-equivariantmorphism ψ : X → Y . We have G ′ lin = G lin /G u , hence G ′u = 1. We haveG ′ abvar = G abvar , hence X(G ′ abvar ) is finite.Assume that the pair (Y, G ′ ) has Property (P S ). We prove that the pair (X, G)has this property. Let x ∈ X(A) be a point orthogonal <strong>to</strong> Br 1 (X, G). Set y :=ψ(x) ∈ Y (A). By func<strong>to</strong>riality, y is orthogonal <strong>to</strong> Br 1 (Y, G ′ ). Let U f Xbe as in(P S ), and let U X , U X ′ be the special neighbourhoods of x defined by U f X . Notethat U X ′ = U X ′ f ×U X,∞ , where U X ′ f is an open subset of X(A f ). Indeed, G scu (k v )is connected for all v ∈ Ω ∞ (see [34], Theorem 5.2.3).Set U fY := ψ(U f X ) ⊂ Y (Af ), U Y := ψ(U X ) ⊂ Y (A) and U Y′ := ψ(U X ′ ) ⊂Y (A). Since G u is connected, by Lemma 4.2 U fY is open in Y (Af ) and U Y andU Y ′ are open in Y (A). Set U Y,v := ψ(U X,v ). By [7], Lemma A.2, for each v ∈ Ω ∞the set U Y,v is the connected component of y v in Y (k v ). Set U Y,∞ := ∏ v∈Ω ∞U Y,v .We have U Y = U fY × U Y,∞. We see that U Y is the special neighbourhood of ydefined by U fY. From the split exact sequence1 → G u → G scu → G ′ sc → 1 ,we see that the map G scu (k S ) → G ′ sc (k S ) is surjective. Note that G ′ scu = G ′ sc . Itfollows thatU ′Y = U Y .G scu (k S ) = U Y .G ′ sc (kS ) = U Y .G ′ scu (kS ).Since the pair (Y, G ′ ) has Property (P S ), there exists a k-point y 0 ∈ Y (k) ∩ U Y ′ .Let X y0 denote the fibre of X over y 0 . It is a homogeneous space of the unipotentgroup G u . By Lemma 4.7, X y0 (k) ≠ ∅ and X y0 has the strong approximationproperty away from Ω ∞ : the set X y0 (k) is dense in X y0 (A f ). Consider the setV f := X y0 (A f ) ∩ U ′ fX . By Corollary 4.6, for any v ∈ Ω ∞ the set X y0 (k v ) isconnected, and by Lemma 4.5, X y0 (k v ) is one orbit under G u (k v ). Set V :=V f × X y0 (k ∞ ).Let v ∈ Ω ∞ . We show that X y0 (k v ) ⊂ U X,v . Since y 0 ∈ U ′Y= ψ(U X ′ ), thereexists a point x v ∈ U X,v such that y 0 = ψ(x v ). Clearly x v ∈ X y0 (k v ). SinceX y0 (k v ) is one orbit under G u (k v ), we see that X y0 (k v ) = x v .G u (k v ) ⊂ U X,v ,because G u (k v ) is a connected group. Thus X y0 (k ∞ ) ⊂ U X,∞ and V ⊂ U ′fX ×U X,∞ = U X ′ , hence V ⊂ X y 0(A) ∩ U X ′ .Since y 0 ∈ ψ(U X ′ ), the set V is non-empty. Since by Lemma 4.7 X y 0(k) is densein X y0 (A f ), there is a point x 0 ∈ X y0 (k) ∩ V . Clearly x 0 ∈ X(k) ∩ U X ′ . Thus thepair (X, G) has Property (P S ). We see that in the proof of Theorem 1.4 we mayassume that G u = 1.7.2. Second reduction.By [7], Proposition 3.1 we may regard X as a homogeneous space of anotherconnected group G ′ such that G ′u = {1}, G ′ ss is semisimple simply connected,
<strong>Manin</strong> <strong>obstruction</strong> <strong>to</strong> strong approximation for homogeneous spaces 33and the stabilizers of the geometric points of X in G ′ are linear and connected.We have G ′ sc = G sc , hence G ′ scu = G scu (because G scu = G sc and G ′ scu = G ′ sc ).It follows from the construction in the proof of Proposition 3.1 of [7] that there isa surjective homomorphism G abvar → G ′ abvar . Since by assumption X(G abvar ) isfinite, we obtain from [7], Lemma A.3 that X(G ′ abvar ) is finite.Let us prove that if a point x ∈ X(A) is orthogonal <strong>to</strong> Br 1 (X, G), then it isorthogonal <strong>to</strong> Br 1 (X, G ′ ). More precisely, we prove that Br 1 (X, G ′ ) is a subgroupof Br 1 (X, G).By construction (see [7], proof of Proposition 3.1), there is an exact sequenceof connected algebraic groups1 → S → G ′ q−→ G1 → 1,where G 1 is the quotient of G by the central subgroup Z(G)∩H and S is a k-<strong>to</strong>rus.Consider the following natural commutative diagram Xπ ′ π 1π1 S qG ′ G 1 1where the maps π, π ′ and π 1 are the natural quotient maps. From this diagram,we deduce the following one, where the second line is exact (see the <strong>to</strong>p row ofdiagram (11)):Br(X) π ∗π ∗ 10 = Pic(S) Br(G 1 )p ∗pGπ ′∗,q ∗ Br(G ′ )Br(G) .Therefore, the injectivity of the map q ∗ : Br(G 1 ) → Br(G ′ ) implies that the naturalinclusion Br 1,x0 (X, G 1 ) ⊂ Br 1,x0 (X, G ′ ) is an equality. And by func<strong>to</strong>rialityBr 1,x0 (X, G 1 ) is a subgroup of Br 1,x0 (X, G).Thus Br 1,x0 (X, G ′ ) = Br 1,x0 (X, G 1 ) is a subgroup of Br 1,x0 (X, G). It followsthat if a point x ∈ X(A) is orthogonal <strong>to</strong> Br 1 (X, G), then it is orthogonal <strong>to</strong>Br 1 (X, G ′ ).Thus if Theorem 1.4 holds for the pair (X, G ′ ), then it holds for (X, G). Wesee that we may assume in the proof of Theorem 1.4 that G lin is reductive, G ss is
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Manin obstruction to strong approximation for homogeneous spaces

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