ITERATING THE ALGEBRAIC ÉTALE-BRAUER SET

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1 ITERATING THE ALGEBRAIC ÉTALE-BRAUER SET F BALESTRIERI Abstract In this paper, we iterate the algebraic étale-brauer set for any nice variety X over a number field k with πét 1 (X) finite and we show that the iterated set coincides with the original algebraic étale-brauer set This provides some evidence towards the conjectures by Colliot-Thélène on the arithmetic of rational points on nice geometrically rationally connected varieties over k and by Skorobogatov on the arithmetic of rational points on K3 surfaces over k; moreover, it gives a partial answer to an algebraic analogue of a question by Poonen about iterating the descent set 1 Introduction 11 Notation In this paper, k C will be a number field, k C a fixed algebraic closure of k, A k the ring of adeles of k, Ω k the set of places of k, and k v the completion of k at v Ω k For any variety X over k, we endow X(A k ) with the adelic topology and v Ω k X(k v ) with the product topology; when X is proper, X(A k ) = v Ω k X(k v ) and the product and adelic topologies are equivalent A variety which is smooth, projective, and geometrically integral over k will be called a nice variety over k Let {X ω } ω be a family of smooth, geometrically integral varieties over k If X(A k ) X(k) for all X {X ω } ω, we say that {X ω } ω satisfies the Hasse principle (HP), while if X(k) and X(k) = X(A k ) for all X {X ω } ω, we say that {X ω } ω satisfies strong approximation (SA) When the varieties in the family {X ω } ω are moreover proper, strong approximation is equivalent to weak approximation (WA), ie to the property that X(k) and X(k) = v Ω k X(k v ) for all X {X ω } ω ; in general, however, we just have the chain of implications (SA) (WA) (HP) For any smooth variety X over k, the Brauer group of X is Br X := Hét 2 (X, G m) and the Brauer-Manin set of X is X(A k ) Br := (x v) X(A k ) : inv v (α(x v )) = 0, α Br X v Ω k where the inv v : Br k v Q/Z are the local invariant maps coming from class field theory The algebraic Brauer group of X is Br 1 (X) := ker(br X Br X), where X := X Spec k Spec k and where Br X Br X is the canonical map induced by the natural morphism X X We define the algebraic Brauer-Manin set X(A k ) Br 1 by restricting the intersection in the definition of the Brauer-Manin set to the elements in Br 1 X One can show that both X(A k ) Br and X(A k ) Br 1 are closed in X(A k ) and that X(k) X(A k ) Br X(A k ) Br 1 (see eg [Sko01, 52]) Let L k := {G : G is a linear algebraic k-group}/, where G 1 G 2 if and only if G 1 and G 2 are k-isomorphic as k-groups We will abuse notation and write G L k also to mean a representative of the k-isomorphism class of G For any A, B L k, we let Ext(A, B) = {G L k : G is an extension of A by B, for some A A and B B}/ For any S L k, the S-descent set is X(A k ) S := G S [f:y X] H 1 ét (X,G) [τ] H 1 ét (k,g) f τ (Y τ (A k )); MSC2010: Primary 11G35; Secondary 14G05, 14G25, 20G30 Keywords: Rational points, weak approximation, Brauer-Manin obstruction, étale-brauer obstruction, universal torsors, algebraic groups 1

2 when S =, we define X(A k ) := X(A k ), while when S = L k, the L k -descent set is just called descent set For any S L k, the set X(A k ) S is closed in X(A k ) and contains the adelic closure of X(k) (see [CDX16, Prop 64]) Let F k := {G L k : G is finite}/ The étale-brauer set of X is X(A k )ét Br := f τ (Y τ (A k ) Br ) F F k [f:y X] Hét 1 (X,F ) [τ] Hét 1 (k,f ) Similarly, we can define the algebraic étale-brauer set X(A k )ét Br 1 by replacing Br with Br 1 in the definition above Both X(A k )ét Br and X(A k )ét Br 1 are closed in X(A k ) (see the discussion after [CDX16, Prop 66]), and they both contain the adelic closure of X(k) Finally, for any S, S L k and any {, Br, Br 1, ét Br, ét Br 1, S }, we define Iter S (X /k, ) := f τ (Y τ (A k ) ) G S [f:y X] Hét 1 (X,G) [τ] Hét 1 (k,g) 12 Motivation The aim of this paper is to give some evidence and partial answers to various conjectures and open questions about the arithmetic behaviour of rational points on certain classes of varieties over k More specifically, the conjectures that we are interested in are the following Conjecture 11 (Colliot-Thélène, [CT03, p174]) Let X be a nice geometrically rationally connected variety over k Then X(k) = X(A k ) Br In other words, the Brauer-Manin obstruction is the only one for strong (equivalently, weak) approximation (Recall that X is geometrically rationally connected if any two general points x 1, x 2 X can be joined by a chain of k-rational curves; examples of geometrically rationally connected varieties include geometrically unirational varieties and Fano varieties) Conjecture 12 (Skorobogatov) Let X be a nice K3 surface over k Then X(k) = X(A k ) Br In other words, the Brauer-Manin obstruction is the only one for strong (equivalently, weak) approximation Conjecture 13 Let X be a nice Enriques surface over k Then X(k) = X(A k )ét Br In other words, the étale-brauer obstruction is the only one for strong (equivalently, weak) approximation Remark 14 In general, K3 surfaces over k do not satisfy X(k) = X(A k ); see [HVA13] for an example over k = Q violating the Hasse principle Similarly, in [BBM + 16], the authors have constructed an Enriques surface X over k = Q such that X(A k ) Br but X(k) = ; this implies that, for Enriques surfaces, X(k) = X(A k ) Br does not hold in general Another source of motivation for this paper is the following: for any nice variety X over k, the étale- Brauer set X(A k )ét Br is currently the smallest general obstruction set known Unfortunately, the étale-brauer set is not small enough to explain all the failures of the Hasse principle: see eg [Poo10] for a counterexample We thus want a way to construct obstruction sets smaller than X(A k )ét Br A possible strategy is to mimick the construction of the étale-brauer set itself: for any nice variety X over k, the results in [Dem09b] and [Sko09] imply that X(A k )ét Br = X(A k ) L k; if S L k contains the trivial group, then the obstruction set Iter S (X /k, L k ) is certainly potentially smaller than X(A k )ét Br It turns out, however, that for certain choices of S the set Iter S (X /k, L k ) is the same as the original étale-brauer set: this is the case, for example, when S = F k (cf [Sko09, Thm 11]) It is natural to ask about the case when S is maximal, ie when S = L k ; in this case, we can think of Iter Lk (X /k, L k ) as an iteration of the descent set Question 15 (Poonen) Let X be a nice variety over k Is Iter Lk (X /k, L k ) = X(A k )ét Br? Remark 16 In [CDX16, Thm 75], the authors show that Y (A k ) L k = Y (A k )ét Br for any smooth, quasi-projective, geometrically connected variety Y over k, thus removing the properness condition 2

3 from the earlier results in [Dem09b] and [Sko09] As a consequence, we have that Iter Lk (X /k, L k ) = Iter Lk (X /k, ét Br) for any nice variety X over k We focus on a question similar to Question 15: we want to iterate the algebraic étale-brauer set X(A k )ét Br 1 To make sense of this, we first need an analogue of the result X(A k )ét Br = X(A k ) L k for X(A k )ét Br 1 Such an analogue is given by [Bal16, Thm 58]: if X is a nice variety over k, then where T k := {G L k : G is a torus}/ X(A k )ét Br 1 = X(A k ) Ext(F k,t k ), Question 17 Let X be a nice variety over k Is Iter Ext(Fk,T k )(X /k, ét Br 1 ) = X(A k )ét Br 1? Remark 18 By putting together the results in [CDX16] and [Bal16], we have Y (A k ) Ext(F k,t k ) = Y (A k )ét Br 1 for any smooth, quasi-projective, geometrically connected variety Y over k From this, we can easily deduce that Iter Ext(Fk,T k )(X /k, Ext(F k, T k )) = Iter Ext(Fk,T k )(X /k, ét Br 1 ) for any nice variety X over k 13 Main result Motivated by the above conjectures and questions, our main theorem is the following Theorem 19 (Main Theorem) Let X be a nice variety over k such that πét 1 (X) is finite Then Iter Ext(Fk,T k )(X /k, ét Br 1 ) = X(A k )ét Br 1 Some comments: Let X be a nice geometrically rationally connected variety over k Then πét 1 (X) = 0, meaning that the hypotheses of Theorem 19 are satisfied If moreover Br X = 0, as is the case when dim X = 2 or Hét 3 (X, Z l(1)) tors = 0 for all primes l (cf [CTS13, Lemma 13]), then X(A k )ét Br 1 = X(A k ) Br 1 = X(A k ) Br Hence, in this case, Theorem 19 tells us that Iter Tk (X /k, ét Br 1 ) = X(A k ) Br, thus giving some evidence for Conjecture 11 Let X be a nice K3 surface over k Then πét 1 (X) = 0, and thus the hypotheses of Theorem 19 are satisfied When, moreover, Br X = 0 (see eg the comment after [SZ12, Prop 51]), then Theorem 19 yields Iter Tk (X /k, ét Br 1 ) = X(A k ) Br, thus giving some evidence for Conjecture 12 Let X be a nice Enriques surface over k Then πét 1 (X) = Z/2Z, and so the hypotheses of Theorem 19 are satisfied When X(A k )ét Br = X(A k )ét Br 1 (eg in [VAV11]), then Iter Ext(Fk,T k )(X /k, ét Br 1 ) = X(A k )ét Br, which is some evidence for Conjecture 13 Theorem 19 also applies to some higher-dimensional analogues of Enriques surfaces (cf [BNWS11, 2]) Theorem 19 gives a positive answer to Question 17, assuming the finiteness of πét 1 (X); it would be interesting to see whether this condition can be weakened or removed (a possible weakening could be that of considering nice varieties X over k such that Pic Y is finitely generated as a Z-module for any finite cover Y X) In the literature, there are several conditional proofs of the existence of nice varieties X over k with πét 1 (X) = 0, X(k) =, and X(A k) Br : see [SW95] for an example conditional on Lang s conjectures, [Poo01] for one conditional on the existence of a complete intersection satisfying certain properties, and [Sme14, Thm 41] for one conditional on the abc conjecture Theorem 19 would apply to these examples 2 Some properties of universal torsors Let X be a variety over k with k[x] = k and with Pic X finitely generated as a Z-module Let M k := {G L k : G is of multiplicative type}/ and let S M k An S-torsor Y X is a universal torsor for X if its type λ Y : Ŝ Pic X is an isomorphism (for the definition of the type 3

4 of a torsor, see eg [Sko01, Cor 239]); here Ŝ := Hom GrpSch k (S, G m,k ) denotes the Cartier dual As explained in [Sko01, 23], universal torsors do not always exist over k, as a universal torsor over k might not descend to k The following proposition gives sufficient conditions for the existence of universal torsors Proposition 21 ([Sko01, Cor 613(1)]) Let X be a variety over k such that k[x] = k, Pic X is finitely generated as a Z-module, and X(A k ) Br1 Then there exists a universal torsor W X with an adelic point When universal torsors exist, they have some desirable properties The following easy lemmas (which can be deduced from results in eg [Sko01, 23], and with the additional help of the Hochschild-Serre spectral sequence H ṕ et (k, Hq ét (W, G m)) H p+q ét (W, G m ) for the second lemma) give some of these properties Lemma 22 Let X be a nice variety over k with Pic X torsion-free Suppose that there exists a universal torsor W X under S (a torus) Then W is geometrically connected, k[w ] = k, and Pic W = 0 Lemma 23 Let W be a smooth, geometrically integral variety over k such that k[w ] = k and Pic W = 0 Then Br 1 (W ) = Br k In particular, W (A k ) Br 1 = W (A k ) Finally, universal torsors satisfy the following universal property: let X be a variety over k such that Pic X is finitely generated as a Z-module and k[x] = k ; given a universal torsor W X and any other torsor Y X under some M M k, there is a [σ] Hét 1 (k, M) such that there exists a map W Y σ of X-torsors (see the discussion after [Sko01, Defn 233]) Remark 24 Using universal torsors, one can easily prove results such as the following: if X is a nice variety over k with Pic X finitely generated as a Z-module, then Iter Mk (X(A k ) Br 1 ) = X(A k ) Br 1 (compare this with [CDX16, Cor 42]) Lemma 25 Let W be a smooth, geometrically integral variety over k with k[w ] = k and πét 1 (W ) = 0 Let F F k and let f : U W be a torsor under F Then [τ] Hét 1 (k,f ) f τ (U τ (A k )) = W (A k ) Proof Recall from eg [Sko01, 53] that, since W is geometrically integral and smooth, f τ (U τ (A k )) = (x v) W (A k ) : ev [U] ((x v )) im H ét 1 (k, F ) Hét 1 (k v, F ), v Ω k [τ] where ev [U] ((x v )) is the evalutation of [f : U W ] H 1 ét (W, F ) at (x v) and the map H 1 ét (k, F ) v Ω k H 1 ét (k v, F ) is the canonical restriction map Since W is geometrically integral, we have the exact sequence of fundamental groups (omitting base-points) 1 πét 1 (W ) πét 1 (W ) Gal(k/k) 1 From the hypothesis that πét 1 (W ) = 0, we deduce that πét 1 (W ) = Gal(k/k) Since F F k, using the Grothendieck-Galois theory we have that Hét 1 (W, F ) = H1 (πét 1 (W ), F (k)), where the action of πét 1 (W ) on F (k) is via Gal(k/k) By using [Ser01, 58(a)], we deduce that H1 ét (k, F ) = H1 ét (W, F ), which implies that the F -torsor U W comes from some F -torsor V Spec k Then, for each place v Ω k, the evaluation map ev [U] (x v ) is defined by the following commutative diagram of pullbacks: V Spec k v U = V W V F F F Spec k v x v W p Spec k, 4

5 ie ev [U] (x v ) = [V Spec k v Spec k v ] Note that the composition p x v : Spec k v Spec k is the canonical morphism induced by the inclusion k k v Since the pullback of a pullback is again a pullback, we have that the F -torsor V Spec k v Spec k v is the pullback of the F -torsor V Spec k along the canonical morphism p x v : Spec k v Spec k, meaning that [V Spec k v Spec k v ] = res v ([V ]) In other words, ev [U] (x v ) = res v ([V ]) By considering all the places v Ω k, we get that ev [U] ((x v )) is the image of [V ] under the canonical restriction map Hét 1 (k, F ) v Ω k Hét 1 (k v, F ), as required Lemma 26 Let W be a smooth, geometrically integral variety over k such that W (A k ) = W (A k ) F k Let (x v ) W (A k ) and let f : R W be an F -torsor, for some F F k Then there exists an F F k, a 1-cocycle σ Z 1 (k, F ), a W -torsor R W under F, a morphism p : F F σ, and a morphism of W -torsors R R σ such that the following diagram commutes R F R σ W F σ and such that R is geometrically connected and (x v ) lifts to a point in R (A k ) Proof By assumption, W (A k ) = W (A k ) F k Modifying the proof of [Sto07, Prop 517] by using [CDX16, Prop 63] yields W (A k ) = f σ (R σ (A k ) Fk ) [σ] Hét 1 (k,f ) Since (x v ) W (A k ), there exists some σ Z 1 (k, F ) such that (x v ) lifts to some (y v ) R σ (A k ) F k Let Ω k,c denote the set of complex places of k and let R σ (A nc k ) denote the restricted product v (Ω k \Ω Rσ k,c) (k v ), where the restriction is the usual adelic one Let R σ = R 1 Rn be the decomposition of R σ into its k-connected components By [Sto07, Prop 511] (also, see [HS13, Remark 9114]), we have that R σ (A nc k )F k = R 1 (A nc k )F k Rn (A nc k )F k Now, since (x v ) lifts to some point (y v ) R σ (A k ) F k, we can assume that y v R (k v ) for all non-complex places v, where R is a k-connected component of R σ ; hence, R (A k ) Since R is connected and R (A k ), by [Sto07, Lemma 55] we can deduce that R is geometrically connected, and thus that R W is a torsor under the stabiliser F F σ of R In particular, R W is surjective, so we can change (y v ) at the complex places if necessary to obtain an adelic point (y v ) R (A k ) above (x v ) 3 Proof of the main theorem Lemma 31 Let X be a nice variety over k with πét 1 (X) finite Then Pic X is finitely generated as a Z-module Proof Let r N, and consider the Kummer sequence 0 µ r,k G m,k t t r G m,k 0 Passing to cohomology and identifying (non-canonically) µ r,k with Z/rZ, we obtain an isomorphism Hét 1 (X, Z/rZ) = (Pic X)[r], where we have used the fact that H 0 (X, G m ) = k[x] = k is divisible Further, Hét 1 (X, Z/rZ) = Hom(πét 1 (X), Z/rZ) (cf [Fu11, Prop 5720]), and hence Hom(πét 1 (X), Z/rZ) = (Pic X)[r] 5

6 Since πét 1 (X) is finite, say with πét 1 (X) = d, it follows that (Pic X)[r] = (Pic0 X)[r] = 0 for any r N with gcd(r, d) = 1 Since X is proper, Pic 0 X is an abelian variety over k; if Pic 0 X 0, 2 dim Pic0 X is then Pic 0 X[r] = (Z/rZ) non-trivial for all r N, a contradiction to the fact that (Pic 0 X)[r] = 0 when gcd(r, d) = 1 Hence, Pic 0 X = 0, which implies that Pic X = NS X is finitely generated as a Z-module Proposition 32 Let Y be a smooth and geometrically connected variety over k with πét 1 (Y ) = 0 Let W Y be a torsor under some connected linear algebraic group T over k Then πét 1 (W ) is abelian Proof Since k is an algebraically closed field of characteristic 0 and since the étale fundamental group does not change under base-change over extensions K/k of algebraically closed fields (cf [Sza09, Second proof of Cor 576 and Rmk 578] together with [Gro71, XII] and [Org03]), there is a Lefschetz principle and we can work over C instead of k Let LFT C and AN C denote, respectively, the category of schemes locally of finite type over C and the category of complex analytic spaces The analytification functor ( ) an : LFT C AN C (cf [Gro71, XII]) induces an equivalence of categories from the category of finite étale covers of X LFT C to the category of finite étale covers of X an AN C (cf [Gro71, XII, Thm 51 Théorème d existence de Riemann ]); by [Gro71, XII, Cor 52], if X LFT C is connected, then (omitting base-points) πét 1 (X) = π top 1 (X an ) The fibration obtained by applying ( ) an to the T C -torsor W C Y C induces the homotopy (exact) sequence π top 1 ((T C ) an ) π top 1 ((W C ) an ) π top 1 ((Y C ) an ) π top 0 ((T C ) an ), where π top 1 ((T C ) an ) is abelian (since (T C ) an is a topological group) and where π top 0 ((T C ) an ) = 0 as (T C ) an is connected (cf [Gro71, XII, Prop 24]) Since taking the profinite completion is right-exact (cf [RZ00, Prop 325]), we obtain the exact sequence πét 1 (T C ) πét 1 (W C ) πét 1 (Y C ) 0, where πét 1 (T C) is abelian (as the profinite completion of an abelian group is abelian); since, by assumption (and by the Lefschetz principle) πét 1 (Y C) = 0, from the above sequence we deduce that πét 1 (W C) (and thus πét 1 (W )) is a quotient of an abelian group and hence abelian, as required Lemma 33 Let W be a geometrically integral variety over k with k[w ] = k and Pic W torsionfree Then π1 ab (W ) = 0, where πab 1 denotes the abelianised ètale fundamental group Proof From eg [Sko01, pp 35-36], we have that π ab 1 (W ) = lim n Hom(H 1 (W, µ n ), k ) For any n N, the Kummer sequence yields the short exact sequence of Gal(k/k)-modules (cf [Sko01, p 36]) 0 k[w ] /k[w ],n H 1 (W, µ n ) Pic W [n] 0 and, since k[w ] = k is divisible (implying that k[w ] /k[w ],n = 0) and Pic W [n] = 0, we get that H 1 (W, µ n ) = 0 for each n and thus that π1 ab (W ) = 0, as required Proof of Theorem 19 The inclusion Iter Ext(Fk,T k )(X /k, ét Br 1 ) X(A k )ét Br 1 holds by construction, so we just need to prove the opposite inclusion We may assume that X(A k )ét Br1, since otherwise the conclusion of the theorem is trivially true as Iter Ext(Fk,T k )(X(A k )ét Br 1 ) X(A k )ét Br 1 Let (x v ) X(A k )ét Br 1 We need to prove that, for any G Ext(F k, T k ) and for any [Z X] Hét 1 (X, G), there exists some [ξ] Hét 1 (k, G) such that, for any F F k and for any [U Z ξ ] Hét 1 (Zξ, F ), 6

7 there exists some [ψ] H 1 ét (k, F ) such that (x v ) lifts to a point in U ψ (A k ) Br 1 Step 1 Let Z X be a torsor under G, for some G Ext(F k, T k ), say with G fitting into a short exact sequence 1 T G F 1, with T T k and F F k Let Y := Z/T and decompose the G-torsor Z X into the T -torsor Z Y and the F -torsor Y X By [Dem09a, Lemme 227] (see also [CDX16, Lemma 71]) there exists some [σ] Hét 1 (k, F ), some F 1 F k, some F 1 -torsor Y 1 X, and a X-torsor morphism Y 1 Y σ such that Y 1 is geometrically integral, and (x v ) lifts to a point in Y 1 (A k ) Br 1 Moreover, since X is smooth and projective and Y 1 X is étale, it follows that Y is smooth, projective, and geometrically integral over k By [Dem09a, Prop 229] (see also [CDX16, Prop 74]), we have that [σ] Hét 1 (k, F ) lifts to some [τ] H1 ét (k, G), meaning that the diagram Y 1 F1 Z τ T τ Y σ F σ X G τ is commutative Step 2 Since πét 1 (X) is finite, so is πét 1 (Y 1) We now construct a geometrically connected torsor Y 2 Y 1 under some F 2 F k such that πét 1 (Y 2) = 0 Let U Y 1 be a torsor under some B F k with πét 1 (U ) = 0 and U (geometrically) connected We claim that, up to twisting Y 1 by some element in H 1 (k, F 1 ), there exists some F 2 F k and some F 2 -torsor Y 2 Y 1 such that the B -torsor U Y 1 is obtained from the F 2 -torsor Y 2 Y 1 by base-changing k to k; in particular, such a Y 2 is geometrically connected and satisfies πét 1 (Y 2) = 0 Indeed, by [HS02, Prop 22 and 31] and [HS12, Thm 21 and Rmk 22(1)], the torsor U Y 1 has a k-form over Y 1 if Y 1 (A k ) F k But the latter is true, up to twisting Y 1, by [Sto07, Prop 517] Hence, U Y 1 has a k-form, say Y 2 Y 1 under some F 2 F k satisfying πét 1 (Y 2) = 0, as claimed We now claim that, without loss of generality, (x v ) lifts to a point in Y 2 (A k ) Br 1 Indeed, by [Sko09, Prop 23] there exists a B-torsor V X under some B F k and a surjective X-torsor morphism h : E Y 1 under ker(b F 1 ); moreover, when considered as a Y 1 -torsor via h, E admits a surjective Y 1 -torsor morphism to Y 2 By a modification of [Sko09, Lemma 22] (we replace the assumption (x v ) X(A k ) desc with (x v ) X(A k )ét Br 1 and then use [Sto07, Prop 517] to check that the proof holds under this new assumption), there exists some γ H 1 (k, ker(b F 1 )) and some point (M v ) E γ (A k ) Br 1 which lifts (x v ) Let γ be the image of γ in H 1 (k, F 2 ) under the image of ker(b F 1 ) F 2 Then E γ Y 1 factors through Y γ 2 Y 1, implying that we can use the functoriality of Br 1 to push (M v ) to a point in Y γ 2 (A k) Br 1 above (x v ) Hence, without loss of generality (up to twisting everything as above if necessary), we can assume that (x v ) lifts to a point in Y 2 (A k ) Br 1 Let R := Y 2 Y σ Z τ Y 2 be the pullback of Z τ Y σ along Y 2 Y 1 Y σ ; this is naturally a T τ -torsor By Lemma 31, Pic Y 2 is finitely generated as a Z-module and (Pic Y 2 ) tors = 0; since Y 2 (A k ) Br1, by Proposition 21 there is a universal torsor W 2 Y 2 under a torus T 2 T k with W 2 (A k ) Since the type λ W2 : T 2 Pic Y 2 is an isomorphism, from the exact sequence of Colliot-Thélène and Sansuc (cf [CTS87, (211)]) 0 k[w 2 ] /k T 2 λ T2 Pic Y 2 Pic W 2 0, 7

8 we deduce that Pic W 2 = 0 and k[w 2 ] = k By the universal property of universal torsors, there is also a morphism of Y 2 -torsors W 2 R µ, for some [µ] Hét 1 (k, T τ ) Let µ be the image of µ under the map Z 1 (k, T τ ) Z 1 (k, G τ ) Then (Z τ ) µ = (Z τ ) µ Let t τ : Z 1 (k, G τ ) Z 1 (k, G) be the bijection as in [Ser94, I53, Prop 35bis], and let ν := t τ ( µ) Then (Z τ ) µ = Z ν, (G τ ) µ = G ν, and (T τ ) µ = T ν Since (x v ) lifts to a point in Y 2 (A k ) Br 1 and since by [Sko99, Thm 3] we have that Y 2 (A k ) Br 1 = Y 2 (A k ) M k, there is some [λ] Hét 1 (k, T 2) such that (x v ) lifts to a point in W2 λ(a k) Let λ be the image of λ under the map Z 1 (k, T 2 ) Z 1 (k, T ν ) induced by the type λ R µ : T ν Pic Y 2 ; then we get a morphism of Y 2 -torsors W λ (R µ ) λ Let ω be the image of λ under the morphism Hét 1 (k, T ν ) Hét 1 (k, Gν ) Then (Z ν ) λ = (Z ν ) ω Let t ν : Z 1 (k, G ν ) Z 1 (k, G) be the bijection as in [Ser94, I53, Prop 35bis], and let ξ := t ν (ω) Then (Z ν ) ω = Z ξ and (G ν ) ω = G ξ Summarising, we have the commutative diagram W λ 2 (R µ ) λ Z ξ T λ 2 T ξ T ξ G ξ Y 2 Y 1 Y σ F σ X Step 3 Since πét 1 (Y 2) = 0, by Proposition 32 we have that πét 1 (W 2 λ) is abelian; hence, since W 2 λ is geometrically connected, k[w2 λ] = k, and Pic W2 λ = 0, by Lemma 33 we deduce that πét 1 (W 2 λ) = 0 Let F F k and let [U Z ξ ] Hét 1 (Zξ, F ) Consider the fibred product V := W2 λ Z ξ U; this is naturally an F -torsor over W2 λ Since πét 1 (W 2 λ ) = 0, by Lemma 25 and Lemma 26 we get the existence of some [ρ] Hét 1 (k, F ), some F F k, an F -torsor V 1 W2 λ, and a W 2 λ-torsor morphism V 1 V ρ with V 1 geometrically connected and with (x v ) lifting to a point in V 1 (A k ) But πét 1 (W 2 λ) = 0 implies that F is trivial and that V 1 is isomorphic to W2 λ Hence, by using W2 λ(a k) = W2 λ(a k) Br 1, we can easily see that the fact that (x v ) lifts to a point (w v ) W2 λ(a k) Br 1 implies that (x v ) lifts to a point (u v ) V 1 (A k ) Br 1, which can then be pushed by functoriality of Br 1 to a point (u v ) U ψ (A k ) Br 1 above (x v ), as required Acknowledgments This work was completed as part of the author s DPhil and was supported by the EPSRC scholarship EP/L505031/1 The author would like to warmly thank the anonymous referee for their many useful comments, which greatly improved this manuscript, and for providing help with Step 2 in the proof of the main theorem; her supervisor, Victor Flynn, for his encouragement and advice; Alexander Betts and Yang Cao, for many helpful remarks and comments References [Bal16] F Balestrieri, Obstruction sets and extensions of groups, Acta Arith 173 (2016), [BBM + 16] F Balestrieri, J Berg, M Manes, J Park, and B Viray, Insufficiency of the Brauer-Manin obstruction for Enriques surfaces, Directions in Number Theory: Proceedings of the 2014 WIN3 Workshop, AWM Springer Series, vol 3, 2016 [BNWS11] S Boissière, M Nieper-Wisskirchen, and A Sarti, Higher dimensional Enriques varieties and automorphisms of generalized Kummer varieties, J Math Pures Appl (9) 95(5) (2011), [CDX16] Y Cao, C Demarche, and F Xu, Comparing descent obstruction and Brauer-Manin obstruction for open varieties, 2016 Preprint available at [CT03] J-L Colliot-Thélène, Points rationnels sur les fibrations, Higher dimensional varieties and rational points (Budapest, 2001), Bolyai Soc Math Stud 12 (2003), [CTS13] J-L Colliot-Thélène and A N Skorobogatov, Good reduction of the brauermanin obstruction, Trans Amer Math Soc 365 (2013),

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