aa r X i v : . [ m a t h . N T ] M a r BAD REDUCTION OF THE BRAUER–MANIN OBSTRUCTION
MARTIN BRIGHT
Abstract.
We relate the Brauer group of a smooth variety over a p -adicfield to the geometry of the special fibre of a regular model, using the puritytheorem in ´etale cohomology. As an illustration, we describe how the Brauergroup of a smooth del Pezzo surface is determined by the singularity type ofits reduction. We then relate the evaluation of an element of the Brauer groupto the existence of points on certain torsors over the special fibre; we use thisto describe situations when the evaluation is constant, and situations when theevaluation is surjective. In the latter case, we describe how this surjectivitycan be used to prove vanishing of the Brauer–Manin obstruction on varietiesover number fields. Introduction
Background.
An important tool for studying rational points on surfaces andhigher-dimensional varieties is the
Brauer–Manin obstruction . This is an obstruc-tion to the existence of rational points, first constructed by Manin [27], based onthe Brauer group of the variety in question. For background information on Brauergroups of schemes, see the three articles by Grothendieck [18, 19, 20] or Chapter IVof Milne’s book [28]. Let X be a smooth, geometrically irreducible variety over anumber field K . For each place v of K , there is a pairing(1) Br X × X ( K v ) → Br K v given by evaluation: ( A , P )
7→ A ( P ). Let A K denote the ring of ad`eles of K .Applying the local invariant map and summing over all places of K , we obtain apairing Br X × X ( A K ) → Q / Z defined by ( A , ( P v )) P v inv v A ( P v ). Denote by X ( A K ) Br the right kernel ofthis pairing, that is, X ( A K ) Br = (cid:8) ( P v ) ∈ X ( A K ) (cid:12)(cid:12) X v inv v A ( P v ) = 0 for all A ∈ Br X (cid:9) . Manin’s observation was that the set of rational points X ( K ) must be contained,under the diagonal inclusion in X ( A K ), in X ( A K ) Br . In particular, if X ( A K ) Br is empty, then X ( K ) is also empty, and we say that there is a Brauer–Maninobstruction to the existence of K -rational points on X . In some situations, theBrauer group Br X and the set X ( A K ) Br are effectively computable, making theBrauer–Manin obstruction a valuable tool for the explicit study of rational points.Central to understanding the Brauer–Manin obstruction is the purely local prob-lem of understanding the pairing (1). One way to view the local structure of avariety at a finite place v is to look at the geometry of a regular model over the Mathematics Subject Classification. ring of integers at v . In this article, we describe how the geometry of a regularmodel is reflected both in the Brauer group of the variety X v = X × K K v and inthe evaluation pairing. As an application, we study two cases of bad reduction ofdel Pezzo surfaces: that where the special fibre is a singular del Pezzo surface, andthat where the special fibre is a cone.There are several results in the literature which we will recover and generalise.Colliot-Th´el`ene and Skorobogatov [7] studied the case of good reduction. In par-ticular, they gave various conditions under which the evaluation map is constant ata place of good reduction. In a remark, they outlined an extension of some of theirresults to the case of bad reduction; Section 5.1 of this article describes very similarresults. On the other hand, several authors have used surjectivity of the evaluationmap associated to an algebra to show that it cannot give an obstruction. For ex-ample, Colliot-Th´el`ene, Kanevsky and Sansuc [4] used this approach for diagonalcubic surfaces, and the present author has applied a similar technique to diagonalquartic surfaces [1]. In Sections 5.2 and 5.3, we will show how these proofs can beviewed as special cases of a very general technique.1.2. Overview.
Our main tool is the purity theorem for the Brauer group, dis-cussed by Grothendieck in Section 6 of [20], which describes what happens to theBrauer group of a regular scheme when a regular subscheme is removed. Section 2is devoted to recalling the absolute purity theorem in ´etale cohomology and de-riving several versions of purity for the Brauer group. In Section 3, we apply thepurity results to relate the Brauer group of a variety to that of a regular model; asan application, in Section 4 we calculate the Brauer group of a del Pezzo surfacehaving the mildest sort of bad reduction, that is, where the reduction is a singulardel Pezzo surface. After an unramified base change, the Brauer group is entirelydetermined by the singularity type of the bad reduction. We use this result toexhibit a del Pezzo surface of degree 1 having an element of order 5 in its Brauergroup.In Section 5, we turn our attention from the Brauer group itself to the evalua-tion pairing. Restricting to a particular element of Br X gives an evaluation map X ( K v ) → Br K v . We begin by giving some criteria for the evaluation map to beconstant. Then, in the opposite direction, we investigate when the image of theevaluation map is as large as possible; using the Weil conjectures, we prove that thishappens whenever the residue field is sufficiently large. In particular, any differ-ence between the evaluation map on points and that on zero-cycles can only happenwhen the residue field is small. This allows us to recover a result of Colliot-Th´el`eneand Saito characterising those elements of the Brauer group which vanish on allzero-cycles of degree one. In Section 6, we describe in more detail the example ofa smooth variety reducing to a cone. In Section 7, we return to global fields anddescribe how the surjectivity results of Section 5 may be used to prove that certainvarieties have no Brauer–Manin obstruction to the existence of rational points. Asan application, we give criteria for vanishing of the Brauer–Manin obstruction whena variety reduces to a cone at a bad place.1.3. Notation. If A is an Abelian group, then A [ n ] and A/n denote the kerneland cokernel, respectively, of the multiplication-by- n map on A ; if ℓ is prime, then A ( ℓ ) denotes the ℓ -power torsion subgroup of A , and A ( ℓ ′ ) denotes the prime-to- ℓ torsion subgroup. AD REDUCTION OF THE BRAUER–MANIN OBSTRUCTION 3 If X is a scheme, then the Brauer group of X , denoted Br X , is defined to bethe ´etale cohomology group H ( X, G m ).We will often identify the group Z /n with the n -torsion subgroup of Q / Z , andis doing so identify H ( X, Z /n ) with the n -torsion subgroup of H ( X, Q / Z ).2. Variations on the purity theorem
In [20, Section 6], Grothendieck describes the purity theorem for the Brauergroup: that is, the relationship between Br X and Br( X \ Z ) when Z is a regularsubscheme of a regular scheme X . In this section, we derive some variants ofthat result. We refer to Section I.16 of Milne’s lecture notes [29] for notation andbackground. The absolute cohomological purity theorem, proved by Gabber [15,30], asserts the following: Theorem 2.1 (Absolute cohomological purity) . Let X be a regular scheme, andlet Z be a regular closed subscheme of X , everywhere of codimension c . Write i for the inclusion Z → X . Let n be coprime to the residue characteristics of X ; let Λ denote the constant sheaf Z /n on X . Then we have R r i ! Λ = 0 for r = 2 c , andthere is a canonical isomorphism R c i ! Λ ∼ = Λ( − c ) of sheaves on Z . A standard argument then shows that, for any finitely generated locally freesheaf F of Λ-modules on X , we have R r i ! F = 0 for r = 2 c , and an isomorphismR c i ! F ∼ = i ∗ F ( − c ). As described by Milne [29, p. 112], the Purity Theorem givesrise to isomorphisms H rZ ( X, F ) ∼ = H r − c ( Z, F ( − c )) for all r ≥ c , and H rZ ( X, F ) =0 for r < c . Using these isomorphisms, the long exact sequence for the pair ( X, Z )shows that there are isomorphisms H r ( X, F ) ∼ = H r ( X \ Z, F ) for r ≤ c −
2, and along exact sequence of Z /n -modules (the Gysin sequence ):(2) 0 → H c − ( X, F ) → H c − ( X \ Z, F ) → H ( Z, F ( − c )) →→ H c ( X, F ) → H c ( X \ Z, F ) → H ( Z, F ( − c )) → · · · . The Gysin sequence is functorial in the pair (
X, Z ). Remark . In what follows, we do not need the full force of the purity theorem:we will only make use of R r i ! µ n for r ≤
3. For r ≤
2, the calculation of thesesheaves is standard (see for example [10, Cycle, Proposition 2.1.4]); so it is only thevanishing of R i ! µ n which matters to us.It will be helpful to deduce several corollaries where Z is not necessarily regu-lar. (Note that Grothendieck’s statements of Theorem 6.1, and therefore of Corol-lary 6.2, in [20] are missing some necessary hypotheses, as pointed out in [12].) Corollary 2.3 (Semi-purity in codimension ≥ . Suppose that X is a regularexcellent scheme, and Z any closed subscheme of codimension c ≥ . Then thenatural map H ( X, F ) → H ( X \ Z, F ) is an isomorphism; and the natural map H ( X, F ) → H ( X \ Z, F ) is injective, and an isomorphism if c ≥ .Proof. The statement is unaffected by replacing Z with the reduced subscheme Z red , so we may assume that Z is reduced. Now we use descending induction onthe codimension c of Z . If c = dim X , then dim Z = 0, so Z is regular and we canapply the Purity Theorem directly. Otherwise, let S denote the non-regular locusof Z , together with any components of Z of codimension strictly greater than c ; MARTIN BRIGHT since Z is reduced and excellent, S is a closed subscheme of codimension at least 1in Z , so of codimension at least c + 1 in X . By induction, we have isomorphismsH ( X, F ) ∼ = H ( X \ S, F ) and H ( X, F ) ∼ = H ( X \ S, F ) . Now Z \ S is a regular closed subscheme of X \ S , and so a further application ofthe Purity Theorem gives the desired result. (cid:3) Corollary 2.4 (Semi-purity in codimension 1) . Let X be a regular excellent scheme,and Z any reduced, closed subscheme everywhere of codimension . Suppose thatthe non-regular locus S of Z is of codimension c in Z . Write X ◦ for X \ S , and Z ◦ for Z \ S . Then there is an exact sequence → H ( X, F ) → H ( X \ Z, F ) → H ( Z, F ( − →→ H ( X, F ) → H ( X \ Z, F ) → H ( Z ◦ , F ( − →→ H ( X ◦ , F ) → H ( X \ Z, F ) . If c ≥ , then we may replace H ( X ◦ , F ) with H ( X, F ) .Proof. This comes from applying the Purity Theorem to Z ◦ ⊂ X ◦ to give an exactsequence as in (2). Now X ◦ \ Z ◦ is the same as X \ Z ; and Corollary 2.3 givesisomorphisms H p ( X ◦ , F ) ∼ = H p ( X, F ) for p = 1 ,
2, and also for p = 3 in the case c ≥ (cid:3) Corollary 2.5.
Under the conditions of Corollary 2.4, there is an exact sequence → Br X [ n ] → Br( X \ Z )[ n ] → H ( Z ◦ , Z /n ) → H ( X ◦ , µ n ) → H ( X \ Z, µ n ) for any n coprime to the residue characteristics of X . If c ≥ , then we may replace H ( X ◦ , µ n ) with H ( X, µ n ) .Proof. Applying Corollary 2.4 with F = µ n produces the exact sequenceH ( X, µ n ) → H ( X \ Z, µ n ) → H ( Z ◦ , Z /n ) → H ( X ◦ , µ n ) → H ( X \ Z, µ n ) . Now the Kummer sequence gives a commutative diagram with exact rows0 −−−−→
Pic
X/n −−−−→ H ( X, µ n ) −−−−→ Br X [ n ] −−−−→ α y β y γ y −−−−→ Pic( X \ Z ) /n −−−−→ H ( X \ Z, µ n ) −−−−→ Br( X \ Z )[ n ] −−−−→ . Since X is regular, the restriction map Pic X → Pic( X \ Z ) is surjective, and so α issurjective; therefore coker β ∼ = coker γ . Moreover, it follows from [19, Corollary 1.8]that Br X → Br( X \ Z ) is injective, and therefore γ is injective. Putting these factstogether gives the claimed result. (cid:3) Corollary 2.6 (Semi-purity for the Brauer group) . Under the conditions of Corol-lary 2.4, there is an exact sequence → Br X ( ℓ ) → Br(
X/Z )( ℓ ) → H ( Z ◦ , Q ℓ / Z ℓ ) →→ H ( X ◦ , G m )( ℓ ) → H ( X \ Z, G m )( ℓ ) for any prime ℓ not dividing the residue characteristics of X . If c ≥ , then wemay replace H ( X ◦ , G m ) with H ( X, G m ) . AD REDUCTION OF THE BRAUER–MANIN OBSTRUCTION 5
Proof.
We apply Corollary 2.5 with n = ℓ r for increasing r , and then take the directlimit to obtain an exact sequence0 → Br X ( ℓ ) → Br(
X/Z )( ℓ ) → H ( Z ◦ , Q ℓ / Z ℓ ) →→ H ( X ◦ , µ ℓ ∞ ) → H ( X \ Z, µ ℓ ∞ ) . On the other hand, the Kummer sequence gives rise to an exact sequence0 → Br X ◦ ⊗ Q ℓ / Z ℓ → H ( X ◦ , µ ℓ ∞ ) → H ( X ◦ , G m )( ℓ ) → X \ Z . Since X ◦ is regular, Br X ◦ is torsion by [19, Proposition 1.4] and therefore Br X ◦ ⊗ Q ℓ / Z ℓ is trivial, givingan isomorphism H ( X ◦ , µ ℓ ∞ ) ∼ = H ( X ◦ , G m )( ℓ ). Similarly, we obtain a compatibleisomorphism H ( X \ Z, µ ℓ ∞ ) ∼ = H ( X \ Z, G m )( ℓ ), and hence the claimed result. (cid:3) The arithmetic setting
Throughout this section, we fix the following notation. k is a finite extension of Q p , with ring of integers O and residue field F . X is a regular scheme, faithfully flat,separated and of finite type over O , such that the generic fibre X = X × O Spec k is a smooth, geometrically irreducible variety over k . The special fibre is denotedby X = X × O Spec F . The non-singular locus of X is X ◦ , and the complement in X of the singular locus of X is denoted X ◦ . Because X ◦ is a non-singular varietyover F , its irreducible components are also its connected components; call them Z , . . . , Z r .We will sometimes want to pass to the maximal unramified extension of k . Let k nr denote the maximal unramified extension of k , and O nr its ring of integers; theresidue field is ¯ F , an algebraic closure of F . Denote by X nr the base change of X to O nr , and by ¯ X ◦ the base change of X ◦ to ¯ F .Under the assumptions above, we may apply Corollary 2.6 to the pair X ⊂ X and take the product over all primes ℓ = p to obtain the following exact sequence:(3) 0 → Br X ( p ′ ) → Br X ( p ′ ) Q i ∂ i −−−→ Y i H ( Z i , Q / Z )( p ′ ) →→ H ( X ◦ , G m )( p ′ ) → H ( X, G m )( p ′ ) . In the special case X = Spec O , we have Br O = 0 and we simply recover theprime-to- p part of the residue map Br k ∼ = H ( F , Q / Z ) of class field theory. Remark . Colliot-Th´el`ene and Skorobogatov [7] made use of the first three termsof (3), but instead derived it from a complex defined by Kato using K-theory.
Remark . If X is smooth and proper over O , with geometrically irreduciblefibres, then the exact sequence (3) gives a split short exact sequence0 → Br X ( p ′ ) → Br X ( p ′ ) ∂ −→ H ( X , Q / Z )( p ′ ) → . This may be seen as follows, as pointed out by Olivier Wittenberg. Let c be anelement of H ( X , Q / Z ) of order coprime to p , which we think of as an elementof H ( X , Z /n ) for suitable n . Proper base change shows that the natural mapH ( X , Z /n ) → H ( X , Z /n ) is an isomorphism; composing its inverse with therestriction map H ( X , Z /n ) → H ( X, Z /n ) we obtain a map φ : H ( X , Z /n ) → H ( X, Z /n ). Now let π be a uniformising element in k , and denote also by π itsclass in H ( k, µ n ) ∼ = k × / ( k × ) n . Then the cup product φ ( c ) ∪ π lies in H ( X, µ n ), MARTIN BRIGHT and can be pushed forward to Br X . We claim that this construction gives a sectionof ∂ . To see this, let K denote the function field of X and ˆ K its completion under thevaluation corresponding to X , and let K be the function field of X . Then thereis a natural inclusion H ( X , Q / Z ) ⊂ H ( K , Q / Z ), and the result follows from thecorresponding fact for Br ˆ K → H ( K , Q / Z ), which goes back to Witt [36] (see also[33, Exercise XII.3.3] for the statement without the assumption that the residuefield be perfect).The sequence (3) involves the cohomology of X ◦ , which in general may not beeasy to understand. However, when X is proper over O and not too singular, wecan use the proper base change theorem to relate the cohomology groups of X ◦ tothose of X . In doing so, we see a strong relationship between the cohomology of X and the Brauer group of X . Remark . For the following proposition, we assume that the special fibre X isnon-singular in codimension 1. Since X is regular and X is an effective Cartierdivisor in X , so X is Cohen–Macaulay. By Serre’s criterion, we deduce that X isnormal and therefore geometrically integral. In particular, the special fibre of ourmodel can have only one component. Proposition 3.4.
Suppose that X is proper over O , that X is non-singular incodimension , and that the natural map Pic
X →
Pic X is surjective. Let n be coprime to p . Then there are natural isomorphisms Br X [ n ] → Br X [ n ] and H ( X ◦ , Z /n ) → H ( X , Z /n ) , and so the sequence of Corollary 2.5 becomes → Br X [ n ] → Br X [ n ] ∂ −→ H ( X ◦ , Z /n ) → H ( X , µ n ) → H ( X, µ n ) . Proof.
The proper base change theorem shows that the natural maps H i ( X , µ n ) → H i ( X , µ n ) are isomorphisms. Now the Kummer sequence gives a commutativediagram with exact rows as follows.0 −−−−→ Pic X /n −−−−→ H ( X , µ n ) −−−−→ Br X [ n ] −−−−→ y y ∼ = y −−−−→ Pic X /n −−−−→ H ( X , µ n ) −−−−→ Br X [ n ] −−−−→ X →
Pic X is surjective, the left-hand vertical mapis also surjective. By the Snake Lemma, the right-hand vertical map is an isomor-phism.Now, since X is non-singular in codimension 1, Corollary 2.3 shows that therestriction map H ( X , µ n ) → H ( X ◦ , µ n ) is an isomorphism; combining this withthe isomorphism from proper base change gives the desired result. (cid:3) Remark . The hypothesis that Pic
X →
Pic X be surjective is satisfied whenever X satisfies H ( X , O X ) = 0: see [17, Corollaire 1 to Proposition 3]. Corollary 3.6.
Assume further that Br X is torsion. Then the following sequenceis exact: → Br X ( p ′ ) → Br X ( p ′ ) → H ( X ◦ , Q / Z )( p ′ ) →→ H ( X , G m )( p ′ ) → H ( X, G m )( p ′ ) . AD REDUCTION OF THE BRAUER–MANIN OBSTRUCTION 7
Proof.
As in the proof of Corollary 2.6, the assumption that Br X is torsion givesisomorphisms H ( X , µ ℓ ∞ ) ∼ = H ( X , G m )( ℓ ) and H ( X, µ ℓ ∞ ) ∼ = H ( X, G m )( ℓ ) foreach prime ℓ = p . Putting these together gives the stated sequence. (cid:3) Application: del Pezzo surfaces
In this section we apply Corollary 3.6 to the situation where X is a smooth delPezzo surface over k , and the reduction X is a singular del Pezzo surface over F ,in the sense of Coray and Tsfasman [8]. (For example, a singular del Pezzo surfaceof degree 3 is a cubic surface having only rational double points as singularities.)The purpose of this section is to demonstrate how the geometry of the reductionof a variety controls the Brauer group of the variety. For background informationon Brauer groups of smooth del Pezzo surfaces, including a list of all the possibleBrauer groups, see [9]. In particular, recall that the Brauer group of any rationalvariety over an algebraically closed field is trivial.Our main theorem is the following statement concerning Br X nr . Theorem 4.1.
Let X be a regular scheme, proper and flat over O , such that thegeneric fibre X is a smooth del Pezzo surface over k and the special fibre X is asingular del Pezzo surface over F . Then the sequence → Br ¯ X ( p ′ ) → Br X nr ( p ′ ) → H ( ¯ X ◦ , Q / Z )( p ′ ) → is exact. Before proving the theorem, we extract some corollaries showing how the Brauergroup of X nr is determined by the reduction type. The Brauer groups of singulardel Pezzo surfaces were calculated in [3]. It turns out that, if Y is a singular delPezzo surface over an algebraically closed field, then there is a (non-canonical)isomorphism Br Y ∼ = Pic( Y ◦ ) tors . In particular, in our situation, the two groupsBr ¯ X and H ( ¯ X ◦ , Q / Z ) appearing in Theorem 4.1 are always isomorphic. Corollary 4.2.
Suppose that p > . Let X be a smooth del Pezzo surface of degree over k admitting a flat, regular, proper model over O , the special fibre of which isa singular del Pezzo surface of degree . Then Br X nr is isomorphic to ( Z / if thespecial fibre has singularity type A + A or A , and Br X nr is trivial otherwise.Proof. The calculations of [3] show that Br ¯ X and H ( ¯ X ◦ , Q / Z ) are both trivial,and so Br X nr is trivial by Theorem 4.1, except in the two special cases stated. Inthe two special cases, Theorem 4.1 gives an exact sequence0 → Z / → Br X nr ( p ′ ) → Z / → . Swinnerton-Dyer [35] has listed the possible Brauer groups of del Pezzo surfaces ofdegree 4, and the only one which fits into this sequence is ( Z / . (cid:3) Corollary 4.3.
Suppose that p > . Let X be a smooth del Pezzo surface of degree over k admitting a flat, regular, proper model over O , the special fibre of whichis a singular del Pezzo surface of degree . Then: • Br X nr is trivial unless the special fibre has singularity type A + A , A + A , A or A ; • if the special fibre has singularity type A + A , A + A or A , then Br X nr is isomorphic to ( Z / ; MARTIN BRIGHT • if the special fibre has singularity type A , then Br X nr is isomorphic to ( Z / .Proof. As for the previous corollary, the isomorphic groups Br ¯ X and H ( ¯ X ◦ , Q / Z )are given in [3]; the list of possibilities for Br X nr given by Swinnerton-Dyer [35]then shows that, in each case, there is only one which fits with Theorem 4.1. (cid:3) One application of Theorem 4.1 and the above corollaries is that it gives anapproach to writing down explicit del Pezzo surfaces with particular Brauer groups,at least over local fields. In the past, this problem has been approached from thepoint of view of constructing surfaces with a particular Galois action on the Picardgroup by various geometric techniques: see, for example, [13] and [14]. As a moreextended application of Theorem 4.1, we now exhibit a smooth del Pezzo surfaceof degree 1 over Q having a non-constant element of order 5 in its Brauer group.Although it is well known that such surfaces can exist, the author is not aware ofany explicit examples in the literature. Example . Let X be the subscheme of weighted projective space P (1 , , , x, y, z, w , over the ring Z , defined by the equation(4) w = z + (2 x + 5 x y − x y − xy + 2 y ) z + ( − x + 4 x y − x y − x y − xy + 8 y ) . A straightforward calculation verifies that the generic fibre X is smooth over Q and that the special fibre X has two singular points at ( x, y, z, w ) = ( ± , , , X , they are both of multiplicity 1, andso X is regular. Explicitly resolving the singularities (which takes two blow-ups)shows that they are both of type A , with the exceptional curves all individuallydefined over F . The calculation of [3] then shows that Br ¯ X and the torsionsubgroup of Pic ¯ X ◦ both have order 5. Because X × ¯ Q has trivial Brauer group,the Hochschild–Serre spectral sequence shows that Br X nr is isomorphic to theGalois cohomology group H ( ¯ Q / ( Q ) nr , Pic( X × ¯ Q )). Since the Galois actionfactors through a subgroup of the Weyl group of E , the only primes which mightdivide the order of Br X nr are 2, 3, 5 and 7. So the prime-to-11 part of Br X nr isthe whole of it. Theorem 4.1 gives the exact sequence(5) 0 → Z / → Br X nr → Z / → . Now the list of cases given by Corn [9, Theorem 4.1] shows that the only possibilityis that the sequence (5) splits (as a sequence of Abelian groups), giving Br X nr ∼ =( Z / .Over Q , we cannot directly apply the same method, since the equivalent ofLemma 4.6 does not hold. However, to see that at least one non-trivial element ofBr X nr descends to Q , it is enough to show that Br X is non-trivial and then applyCorollary 3.6. Let Γ denote the absolute Galois group of F p . The Hochschild–Serrespectral sequence for the Galois covering ¯ X → X gives an exact sequence0 → Br X → (Br ¯ X ) Γ β −→ H ( F p , Pic ¯ X ) . Let n be an integer coprime to 11. According to [6, Proposition 3.3], the map β restricted to the n -torsion in (Br ¯ X ) Γ is the boundary map associated to the 4-termexact sequence0 → Pic ¯ X n −→ Pic ¯ X → H ( ¯ X , µ n ) → Br ¯ X [ n ] → . AD REDUCTION OF THE BRAUER–MANIN OBSTRUCTION 9
As a consequence, any element of (Br ¯ X [ n ]) Γ that lifts to a Γ-invariant element ofH ( ¯ X , µ n ) lies in the kernel of β .Let f : Y → ¯ X be a minimal resolution. The Leray spectral sequence gives anexact sequenceH ( ¯ X , R f ∗ µ n ) → H ( ¯ X , µ n ) → H ( Y, µ n ) → H ( ¯ X , R f ∗ µ n ) . Since the fibre of f above each singular point of ¯ X is a tree of P s, we haveR f ∗ µ n = 0 and therefore H ( ¯ X , µ n ) injects into H ( Y, µ n ). As Y is a rationalvariety, the Brauer group of Y is trivial and so the Kummer sequence gives anisomorphism (Pic Y ) /n → H ( Y, µ n ). But Pic Y is generated by the canonicalclass and the classes of the eight exceptional curves, which are all defined over F ;thus the Galois action on Pic Y is trivial, and we deduce that the Galois actionon H ( ¯ X , µ n ) is also trivial. The Kummer sequence further shows that the Galoisaction on Br ¯ X [ n ] is trivial. Applying this with n = 5 shows that β is the zeromap, and the natural map Br X → (Br ¯ X ) Γ = Br ¯ X is an isomorphism. Thereforewe have Br X ∼ = Z / X . Remark . I thank Ronald van Luijk for providing me with the equation of anelliptic surface over P Q having two singular fibres of type I , which arose in theproof of [31, Proposition 5.3]. The equation (4) is simply the reduction of thatequation modulo 11. Viewed as the equation of an elliptic surface in Weierstrassform, equation (4) defines a surface over F having two singular fibres of type I .Viewing the equation instead as defining a surface in P (1 , , ,
3) is equivalent toblowing down the zero section of the elliptic surface and gives a singular del Pezzosurface of degree 1 with two A singularities.Let us now prove Theorem 4.1. We will apply Corollary 3.6 to X nr ; the resultwill be established if we can show that H ( ¯ X , G m )( ℓ ) is trivial for all primes ℓ = p .That calculation is accomplished by the following lemma. Lemma 4.6.
Let Y be a singular del Pezzo surface over an algebraically closedfield. Then, for any n prime to the characteristic, we have H ( Y, G m )[ n ] = 0 .Proof. Consider the resolution of singularities f : ˜ Y → Y , where ˜ Y is the corre-sponding generalised del Pezzo surface (see [8]). Because the sheaves R q f ∗ G m , for q >
0, are supported at the finitely many singular points of Y , so H p ( Y, R q f ∗ G m )vanishes for p, q >
0. Also, f is proper and birational, giving f ∗ G m = G m . Usingthese facts, we extract from the Leray spectral sequence for f an exact sequenceBr ˜ Y → H ( Y, R f ∗ G m ) → H ( Y, G m ) → H ( ˜ Y , G m ) . Because ˜ Y is a smooth, rational surface, we have Br ˜ Y = 0. Taking n -torsion inthe above sequence, we have0 → H ( Y, R f ∗ G m )[ n ] → H ( Y, G m )[ n ] → H ( ˜ Y , G m )[ n ] . Now, Pic ˜ Y is torsion-free, giving H ( ˜ Y , µ n ) = 0; by Poincar´e duality, we deduceH ( ˜ Y , µ n ) = 0 as well. The Kummer sequence then shows H ( ˜ Y , G m )[ n ] = 0. Toconclude, it will be enough to prove that R f ∗ G m [ n ] = 0, and we can check this onstalks as follows.The Kummer sequence on ˜ Y gives a short exact sequence of sheaves on Y :0 → (R f ∗ G m ) /n → R f ∗ µ n → (R f ∗ G m )[ n ] → . Let y be a singular point of Y . Taking stalks at y , which is exact and so preserveskernels and cokernels, gives the top exact row of the following commutative diagram;the bottom row comes from the Kummer sequence on the fibre ˜ Y y , which is a unionof exceptional curves.0 −−−−→ (R f ∗ G m ) y /n −−−−→ (R f ∗ µ n ) y −−−−→ (R f ∗ G m ) y [ n ] −−−−→ y y y −−−−→ Pic ˜ Y y /n −−−−→ H ( ˜ Y y , µ n ) −−−−→ Br ˜ Y y [ n ] −−−−→ f ∗ G m ) y is the Picard group of ˜ Y × Y O sh Y,y , the base change of ˜ Y to thestrictly Henselian local ring at y ; so the left-hand vertical map in the diagramis surjective by [26, Lemma 14.3]. By the Snake Lemma, the right-hand verticalarrow is an isomorphism. But, because ˜ Y y is a curve, we have Br ˜ Y y = 0 by [20,Corollary 1.2]; note that this corollary does indeed apply to reducible curves. Itfollows that (R f ∗ G m )[ n ] vanishes, and we deduce H ( Y, G m )[ n ] = 0 as claimed. (cid:3) Evaluating elements of Brauer groups
We return to the general situation of Section 3. Given an element
A ∈ Br X ,we obtain an evaluation map A : X ( k ) → Br k by thinking of a k -point P as amorphism Spec k → X and defining A ( P ) = P ∗ A . In this section, we show howthe boundary maps ∂ of the Gysin sequence allow us to describe the evaluationmap in terms of the special fibre of the given model.5.1. Preliminaries.
The natural map X ( O ) → X ( k ) is not necessarily surjective;in other words, there may be k -points of X which do not extend to O -points of ourmodel. We will not be able to say anything about such points. As X is regular, nopoint of X ( O ) reduces to a singular point of X ; that is, we have X ( O ) = X ◦ ( O ).Denote by X i the open subscheme of X ◦ obtained by removing all componentsof the special fibre apart from Z i . Then X ( O ) = S i X i ( O ). The results in thissection describe in more detail the relationship between ∂ i A and the evaluationmap A : X i ( O ) → Br k . Since this may be studied separately on each component ofthe special fibre, we may replace X by some X i and therefore assume from now onthat the non-singular locus of the special fibre X ◦ has only one connectedcomponent . In this case, the exact sequence (3) gives(6) 0 → Br X ( p ′ ) → Br X ( p ′ ) ∂ −→ H ( X ◦ , Q / Z )( p ′ ) . We will further assume that X ( O ) is non-empty , since most of our statementswill be vacuous otherwise. Note that this condition implies that X ◦ is geometricallyconnected. We will sometimes identify X ( O ) with its image in X ( k ). Proposition 5.1.
Let P be a k -point of X lying in the image of X ( O ) and reducingto a point P of X ◦ . Then the following diagram commutes: Br X ( p ′ ) ∂ −−−−→ H ( X ◦ , Q / Z )( p ′ ) P ∗ y P ∗ y Br k ( p ′ ) ∼ = −−−−→ H ( F , Q / Z )( p ′ ) where the bottom map is the residue map of class field theory. AD REDUCTION OF THE BRAUER–MANIN OBSTRUCTION 11
Proof.
By assumption, P extends to a morphism Spec O → X . The diagram nowfollows from the functoriality of the Gysin sequence applied to both X and O . (cid:3) The following corollaries are immediate.
Corollary 5.2.
Let A be an element of Br X ( p ′ ) . Then A ( P ) depends only on P ;that is, the evaluation map A : X ( O ) → Br k factors through reduction mod p . Corollary 5.3.
For any
A ∈ Br X ( p ′ ) , the evaluation map A : X ( O ) → Br k isdetermined by the class of ∂ A in H ( X ◦ , Q / Z ) . Corollaries 5.2 and 5.3 may be summarised by saying that the following diagramcommutes, where the top row is the evaluation pairing for the Brauer group andthe bottom row is the pairing defined by taking the isomorphism class of the fibreof a torsor at a point.Br X ( p ′ ) × X ( O ) −→ Br k ( p ′ ) ∂ y y y H ( X ◦ , Q / Z )( p ′ ) × X ◦ ( F ) −→ H ( F , Q / Z )( p ′ ) Corollary 5.4. If X is any model of a smooth k -variety X , A an element of Br X ( p ′ ) , and P a k -point of X reducing to a smooth point P of the special fibre,then A ( P ) depends only on the reduction P .Proof. Apply Proposition 5.1 to a regular Zariski neighbourhood of P in X . (cid:3) In order to say more about how the evaluation map associated to an algebra A depends on the class of ∂ A , it will be useful to pass to the maximal unramifiedextension of k . We keep the notation of Section 3. Because X nr is ´etale over X andtherefore regular, we can consider the exact sequence (6) with X replaced by X nr .Since we are assuming that X ◦ ( F ) is non-empty, it follows that ¯ X ◦ is connected. Lemma 5.5.
Let
Br( X nr / X ) denote the kernel of the natural map Br X → Br X nr ,and similarly let Br( X nr /X ) denote the kernel of the natural map Br X → Br X nr .Then there is a commutative diagram, with exact rows and columns, as follows. (7) 0 −−−−→ Br( X nr / X )( p ′ ) −−−−→ Br( X nr /X )( p ′ ) ∂ −−−−→ H ( F , Q / Z )( p ′ ) y y y −−−−→ Br X ( p ′ ) −−−−→ Br X ( p ′ ) ∂ −−−−→ H ( X ◦ , Q / Z )( p ′ ) y y y −−−−→ Br X nr ( p ′ ) −−−−→ Br X nr ( p ′ ) −−−−→ H ( ¯ X ◦ , Q / Z )( p ′ ) The top row is split, giving a canonical isomorphism
Br( X nr /X )( p ′ ) ∼ = Br( X nr / X )( p ′ ) ⊕ Br k ( p ′ ) . Proof.
The second and third rows simply come from the functoriality of our exactsequence (6). Now the Hochschild–Serre spectral sequence applied to ¯ X ◦ → X ◦ gives an exact sequence0 → H ( F , Q / Z ) → H ( X ◦ , Q / Z ) → H ( ¯ X ◦ , Q / Z ) , identifying H ( F , Q / Z ) with the subgroup of H ( X ◦ , Q / Z ) consisting of the isomor-phism classes of constant torsors. Therefore the top row of our diagram consistsof the kernels of the three vertical base change maps, and so the whole diagramcommutes.Now Br k is certainly contained in Br( X nr /X ), and class field theory shows thatthe restricted map ∂ : Br k ( p ′ ) → H ( F , Q / Z )( p ′ ) is an isomorphism; its inversegives a canonical section H ( F , Q / Z )( p ′ ) → Br( X nr /X )( p ′ ) whose image is Br k ( p ′ ).This shows that the top row is a split exact sequence, and identifies Br( X nr /X )( p ′ )as a direct sum as claimed. (cid:3) Denote by ¯ ∂ : Br X ( p ′ ) → H ( ¯ X ◦ , Q / Z )( p ′ ) the composite homomorphism fromthe diagram (7). Proposition 5.6.
Let A be an element of Br X ( p ′ ) , and suppose that ¯ ∂ ( A ) is zero.Then the evaluation map A : X ( O ) → Br k is constant.Proof. Lemma 5.5 shows that assuming ¯ ∂ ( A ) = 0 implies that ∂ ( A ) is the class of aconstant torsor. By Proposition 5.1, the evaluation map is constant on X ( O ). (cid:3) Corollary 5.7.
Let n be coprime to p . If H ( ¯ X ◦ , Z /n ) is trivial then, for every A ∈ Br X [ n ] , the evaluation map A : X ( O ) → Br k is constant.Remark . Applying this to the del Pezzo surfaces of Section 4, we obtain thefollowing: • Let X be a del Pezzo surface of degree 4 satisfying the conditions of Corol-lary 4.2. If the singularity type of the special fibre X is neither 2 A + A nor 4 A then, for any A ∈ Br X , the evaluation map associated to A isconstant. • Let X be a del Pezzo surface of degree 3 satisfying the conditions of Corol-lary 4.3. If the singularity type of the special fibre X is neither A + A ,2 A + A , 4 A nor 3 A then, for any A ∈ Br X , the evaluation map asso-ciated to A is constant.Proposition 5.6 is unsurprising if A is either in Br k , or in Br X (in the lattercase, the evaluation map factors through Br O , which is trivial). The followingproposition states that those classes generate ker ¯ ∂ . Proposition 5.9.
The kernel of ¯ ∂ is Br X ( p ′ ) ⊕ Br k ( p ′ ) ⊂ Br X ( p ′ ) .Proof. ¯ ∂ is defined to be composition of two homomorphisms, so we can describeits kernel using the kernel-cokernel exact sequence. Combined with the diagram ofLemma 5.5, this gives an exact sequence0 → Br X ( p ′ ) → ker ¯ ∂ ∂ −→ H ( F , Q / Z )( p ′ ) . Since ∂ maps Br k ( p ′ ) ⊆ ker ¯ ∂ isomorphically to H ( F , Q / Z )( p ′ ), this sequencesplits. (cid:3) Surjectivity of evaluation maps.
It is natural to ask to what extent theconverse of Proposition 5.6 is true: if the evaluation of an algebra A is constanton X ( O ), under what circumstances can we deduce that ¯ ∂ ( A ) = 0? Certainly, theconverse does not always hold: for example, if X ◦ ( F ) has only one point, then theevaluation map will be constant no matter what ¯ ∂ ( A ) is. However, we shall see thatthis phenomenon can only happen for small residue fields F , in the following sense: AD REDUCTION OF THE BRAUER–MANIN OBSTRUCTION 13 whenever F is sufficiently large, the evaluation map associated to an Azumayaalgebra takes all possible values on points of X ( O ). Here “takes all possible values”means that, up to the class of a constant algebra, the image of the evaluation mapis all of Br k [ m ], where m is the order of ¯ ∂ ( A ); and the meaning of “sufficientlylarge” depends on m and geometric invariants of X ◦ .In order to simplify the statements of the remaining results in this section, wewould like to ignore constant algebras. A convenient way to do that is given by thefollowing definition. Definition 5.10.
An element
A ∈ Br X is normalised if there exists a point P ∈X ( O ) such that A ( P ) = 0.Note that this condition is not vacuous, since by assumption X ( O ) is non-empty.Any choice of P ∈ X ( O ) gives a retraction of the inclusion Br k → Br X ; lettingBr P X denote the subgroup of those A ∈ Br X satisfying A ( P ) = 0, we obtain anisomorphism Br X ∼ = Br k ⊕ Br P X . Thus any element of Br X may be expressedas an element of Br k plus a normalised element of Br X . Lemma 5.11.
Let A be a normalised element of Br X ( p ′ ) , and suppose that ¯ ∂ A hasorder n in H ( ¯ X ◦ , Q / Z ) . Then ∂ A has order n in H ( X ◦ , Q / Z ) , and the evaluationmap A : X ( O ) → Br k takes values in Br k [ n ] .Proof. By hypothesis, ¯ ∂ ( n A ) is the trivial class in H ( ¯ X ◦ , Q / Z ); as in the proof ofProposition 5.6, we deduce that ∂ ( n A ) lies in the image of H ( F , Q / Z ). But n A is normalised, and so Proposition 5.1 shows that ∂ ( n A ) is trivial. Therefore ∂ A is of order dividing n in H ( X ◦ , Q / Z ). Since ¯ ∂ A is of order exactly n , we deducethat ∂ A is also of order exactly n . The final statement follows immediately fromProposition 5.1. (cid:3) Let us now notice that whether the evaluation map takes any particular valueis controlled by the existence of points on a certain variety over F . Recall thatany class in H ( X ◦ , Z /n ) is represented by a torsor Y → X ◦ under Z /n . Givenan element α ∈ Br k [ n ], we obtain a class ∂α ∈ H ( F , Z /n ); twisting Y → X ◦ byGalois descent gives a new torsor Y ∂α → X ◦ , geometrically isomorphic to Y . Theclass of Y ∂α in H ( X ◦ , Z /n ) is simply the sum of the classes of Y and ∂α . (See [34],Section 2.2 and particularly p. 22.) Lemma 5.12.
Let A be an element of Br X ( p ′ ) , and suppose that ∂ A ∈ H ( X ◦ , Q / Z ) has order n . Let Y → X ◦ be a torsor representing the class of ∂ A in H ( X ◦ , Z /n ) .Let α be a class in Br k [ n ] , and let P ∈ X ( O ) be a point reducing to P ∈ X ◦ ( F ) .Then A ( P ) is equal to − α if and only if P lies in the image of the twisted tor-sor Y ∂α ( F ) → X ◦ ( F ) . In particular, − α is in the image of the evaluation map A : X ( O ) → Br k if and only if Y ∂α ( F ) is non-empty.Proof. By Proposition 5.1, we have A ( P ) = − α if and only if the class in H ( F , Z /n )of the fibre Y P is ∂α . Twisting by ∂α , this is true if and only if the class of the fibre( Y ∂α ) P is trivial. This happens precisely when that fibre contains an F -rationalpoint, that is, when P lies in the image of Y ∂α ( F ) → X ◦ ( F ). (cid:3) Lemma 5.12 shows that, to prove that the evaluation map associated to anAzumaya algebra takes a particular value, it is enough to prove existence of rationalpoints on a certain torsor over X ◦ . A method to guarantee the existence of pointson varieties over finite fields comes from the Weil conjectures. Fact 5.13.
Let Y be a geometrically irreducible variety over a finite field F , anddenote by ¯ Y the base change of Y to the algebraic closure of F . Then there is abound B ( ¯ Y ) , depending only on cohomological invariants of ¯ Y , such that, whenever | F | > B ( ¯ Y ) , then Y ( F ) is non-empty. This fact follows from the generalised form of the Weil conjectures proved byDeligne [11] together with the Lefschetz trace formula. More specifically, the bounddepends on the Betti numbers of ¯ Y in compactly supported ℓ -adic cohomology. Remark . The usefulness of this fact is that the bound B ( ¯ Y ) depends not onthe precise geometry of ¯ Y , but only on certain geometric invariants. The existenceof such a bound is the famous result of Lang and Weil [25], who use much moreelementary methods than those of Deligne. They show that, for a projective varietyin P n , a bound can be chosen that depends only on n and the dimension anddegree of the variety. For many of our applications, and in particular for all thecorollaries to Theorem 5.16 below, this is enough. The only reason for appealingto Deligne instead of Lang–Weil is that many natural applications of the results inthis section arise in the context of a family of varieties, and Deligne’s statementsoften immediately show how the bound B ( Y ) varies as Y moves in a family. Forexample, Betti numbers are constant in a smooth, proper family of varieties, andso we see immediately that the same bound will work for every member of such afamily.In order to apply Fact 5.13 to torsors, we need to understand when a torsor isgeometrically irreducible. Let S be a scheme, and G a finite Abelian group. Recallthat the cohomology group H ( S, G ) classifies S -torsors under G ; these are the sameas Galois covers of S with group G , and so are also classified by homomorphismsfrom π ( S ) to G . This correspondence gives an isomorphism of groups H ( S, G ) ∼ =Hom( π ( S ) , G ). Lemma 5.15.
Suppose that S is normal and connected. Then, under the isomor-phism H ( S, G ) ∼ = Hom( π ( S ) , G ) , the connected torsors correspond to the surjec-tive homomorphisms. In particular, if G is the cyclic group Z /n , then the connectedtorsors are those representing classes of exact order n in H ( S, G ) .Proof. Let K denote the function field of S . Because S is normal, taking genericfibres induces a one-to-one correspondence from finite ´etale covers of S to ´etalealgebras over K , preserving connectedness (see [21, Expos´e I, Section 10]); so itsuffices to prove the theorem when S is the spectrum of a field. In this case, let T → K be a torsor under G , and recall the definition of the associated element φ ∈ Hom(Gal( ¯
K/K ) , G ): pick a geometric point x of T ; then, for σ ∈ Gal( ¯
K/K ),we define φ ( σ ) to be the unique g ∈ G such that σx = gx . From this description itis easy to see that φ is surjective if and only if the Galois action on the geometricpoints of T is transitive, that is, if and only if T is connected. (cid:3) Armed with this, we can prove a partial converse to Proposition 5.6. The fol-lowing theorem is a special case of Theorem 5.24 below, but we state and prove itseparately both because it motivates the statement and proof of Theorem 5.24 andbecause it has corollaries of independent interest.
Theorem 5.16.
Fix a class c ∈ H ( ¯ X ◦ , Q / Z ) , of order n coprime to p . Thenthere is a constant B , depending only on c , such that, whenever | F | > B and for all AD REDUCTION OF THE BRAUER–MANIN OBSTRUCTION 15 normalised
A ∈ Br X such that ¯ ∂ A = c , the evaluation map A : X ( O ) → Br k [ n ] issurjective. If ¯ Y → ¯ X ◦ is a torsor representing the class of c in H ( ¯ X ◦ , Z /n ) , thenwe can take B to be B ( ¯ Y ) .Proof. By Lemma 5.11, ∂ A has order n in H ( X ◦ , Q / Z ), and so we may think of ∂ A as lying in H ( X ◦ , Z /n ). Let Y → X ◦ be a torsor representing the class of ∂ A in H ( X ◦ , Z /n ). Then ¯ Y → ¯ X ◦ is a torsor representing c in H ( ¯ X ◦ , Z /n ); byhypothesis, c is of order n , and so Lemma 5.15 shows that ¯ Y is connected. Since¯ Y is also smooth, ¯ Y is irreducible. If | F | > B ( ¯ Y ), then Fact 5.13 shows that Y hasan F -point, as does every Galois twist of Y . By Lemma 5.12 the evaluation map A : X i ( O ) → Br k [ n ] is surjective. (cid:3) Corollary 5.17.
Fix an integer n coprime to p . Then there is a constant B ,depending only on ¯ X ◦ and n , such that, whenever | F | > B and for all normalised A ∈ Br X such that ¯ ∂ A has order n , the evaluation map A : X ( O ) → Br k [ n ] issurjective.Proof. The group H ( ¯ X ◦ , Z /n ) is finite [28, VI, Corollary 5.5]. So we may applyTheorem 5.16 to each class c ∈ H ( ¯ X ◦ , Z /n ) and take B to be as large as necessaryfor the conclusion to hold in all cases. (cid:3) Corollary 5.18.
Let A be a normalised element of Br X ( p ′ ) , and suppose that ¯ ∂ A is of order n in H ( ¯ X ◦ , Q / Z ) . If ℓ is an unramified extension of k of sufficientlylarge degree, with ring of integers O ℓ , then the evaluation map A : X ( O ℓ ) → Br k [ n ] is surjective.Proof. Replacing X by its base change to O ℓ changes neither ¯ X ◦ nor ¯ ∂ A . SoTheorem 5.16 shows that the conclusion will hold as long as the size of the residuefield of ℓ is greater than B . (cid:3) We can use Corollary 5.18 to obtain further corollaries about zero-cycles on X ;first let us briefly recall some notation. A zero-cycle on a variety X is a finite formallinear combination P n i P i of closed points P i . The group of zero-cycles on X isdenoted Z ( X ). Given A ∈ Br X , we define A ( P n i P i ) = P n i cores k ( P i ) /k A ( P i ),giving a pairing Br X × Z ( X ) → Br k . The degree of the zero-cycle P n i P i is P n i [ k ( P i ) : k ]. Let Z ( X ) denote the group of zero-cycles of degree 0. If A ∈ Br k ,then we have A ( z ) = 0 for any z ∈ Z ( X ); so we obtain a pairing (Br X/ Br k ) × Z ( X ) → Br k . For each of these pairings, we may try to describe the left kernel.In our situation, we need to talk about the group of zero-cycles on X whichextend to a section of X , and so specialise to a point of the special fibre. Denotethis group by Z ( X, X ). Equivalently, Z ( X, X ) is the image of the (injective)natural map Z ( X ) → Z ( X ). Corollary 5.19.
Let A be an element of Br X ( p ′ ) , and suppose that ¯ ∂ ( A ) has order n in H ( ¯ X ◦ , Q / Z ) . Then the image of the evaluation map A : Z ( X, X ) → Br k is Br k [ n ] .Proof. By assumption, X ( O ) is non-empty; let P ∈ X ( k ) be a k -point of X thatextends to an O -point of X . Replacing A by A − A ( P ), which affects neitherthe hypotheses nor the conclusion of the corollary, we reduce to the case in which A ( P ) = 0, that is, A is normalised at P . Let k d be the unramified extension of k of degree d . Corollary 5.18 shows that, assuming d to be sufficiently large, theevaluation map A : X ( O k d ) → Br k d [ n ] is surjective. It is straightforward to check that evaluating A at a point of X ( k d ) gives the same element of Br k d as evaluating A at the corresponding closed point of X . That the corestriction map from Br k d to Br k is an isomorphism follows easily from [33, Chapter XIII, Proposition 7]. Wededuce that the evaluation map associated to A takes the subset of zero-cycles ofdegree d in Z ( X, Z ) surjectively onto Br k [ n ]. Subtracting the zero-cycle dP givesthe same result for zero-cycles of degree zero, as claimed. (cid:3) Remark . As we have seen, an element of Br X can be evaluated both at pointsof X and at zero-cycles. The combined message of Theorem 5.16 and Corollary 5.19is that the images of these two evaluation maps can only differ when the residuefield is small. Applying this to varieties over global fields, we see that any differencebetween the Brauer–Manin obstruction to rational points and that to zero-cyclesof degree one happens at small primes. As an illustration, consider the diagonalquartic surface X ⊂ P Q , defined by the equation X + 47 X = 103 X + (17 × × X , described in Proposition 3.3 of [1]. There, it is shown that X has points everywherelocally, that Br X/ Br Q has order 2, and that the non-trivial class gives an obstruc-tion to the existence of rational points on X . More explicitly, the non-constant classin Br X is represented by the algebra A = (17 , f /X ), where f is the polynomial(20 X + (47 × X + (103 × X ). It is easy to check that the invariant mapof A is zero at all places apart from 17. At 17, the special fibre X of the givenmodel is the cone over a plane quartic curve; we find that ∂ A ∈ H ( X ◦ , Z / Z ) isnon-constant. Let ˜ f denote the reduction of f modulo 17; the variety defined over( X \ { ˜ f = 0 } ) by T = ˜ f extends to a torsor Y → X representing the class ∂ A .One can verify that Y ( F ) is empty, showing that the invariant of A is never zeroon X ( Q ); so the invariant map is constant with value . Indeed, all twelve pointsof X ( F ) lift to points on the quadratic twist of Y . However, the Hasse–Weilbounds show that Y does admit points over every non-trivial extension of F . Itfollows that X is a variety with no Q -rational points, but no Brauer–Manin ob-struction to the existence of a rational zero-cycle of degree one. Unfortunately, Ihave been unable to find a rational zero-cycle of degree one on X .As a final corollary to Theorem 5.16, we give an alternative proof of the followingresult of Colliot-Th´el`ene and Saito [5]. They used the Chebotarev density theoremin their proof; in ours, this is replaced by the appeal to the Weil conjectures in theproof of Theorem 5.16. Corollary 5.21.
Let A be an element of Br X ( p ′ ) . The following statements areequivalent:(i) A ∈ (Br X + Br k ) ⊂ Br X ;(ii) There exists α ∈ Br k such that ( A − α )( z ) = 0 for all z ∈ Z ( X, X ) ;(iii) A ( z ) = 0 for all z ∈ Z ( X, X ) ;(iv) A ( z ) is constant for z ∈ Z ( X, X ) of degree .In particular, the left kernel of the evaluation pairing (Br X/ Br k ) × Z ( X, X ) → Br k consists of the image of Br X in Br X/ Br k .Proof. The implication (i) ⇒ (ii) follows immediately from the fact that, if A liesin Br X , then we have A ( z ) = 0 for all z ∈ Z ( X, X ). The implications (ii) ⇒ (iii) AD REDUCTION OF THE BRAUER–MANIN OBSTRUCTION 17 is trivial. Given that, by assumption, X ( O ) is non-empty, the equivalence of (iii)and (iv) is also straightforward.It remains to prove that (iii) implies (i). Suppose that A ∈ Br X does notsatisfy (i); by Proposition 5.9, we have ¯ ∂ ( A ) = 0. Then Corollary 5.19 shows the A ( z ) takes several different values for z ∈ Z ( X, X ), contradicting statement (iii). (cid:3) Generalisation to several algebras.
A natural generalisation of Theo-rem 5.16 to several elements of Br X would say the following: given a collection of“independent” elements of Br X , they should “independently” take all possible val-ues when evaluated at points of X ( k ), assuming that the residue field is sufficientlylarge. In this section we make this statement precise and prove it. Definition 5.22.
Let A be a torsion Abelian group, and let a , . . . , a r be ele-ments of A with orders n , . . . , n r respectively. We will say that the a i are linearlyindependent if any of the following, clearly equivalent, conditions hold:(i) The size of the subgroup generated by the a i is Q i n i ;(ii) The homomorphism Q i ( Z /n i ) → A , defined on the i th factor by 1 a i , isinjective;(iii) Whenever P i λ i a i = 0 holds with λ i ∈ Z , then we have n i | λ i for all i . Lemma 5.23.
Let G be a group, and let χ , . . . , χ r ∈ Hom( G, Q / Z ) be charactersof G with orders n , . . . , n r respectively. Then the χ i are linearly independent ifand only if the product homomorphism Y i χ i : G → Y i (cid:0) n i Z (cid:1) / Z is surjective.Proof. Replacing G by its image under Q i χ i , we may assume that G is finite andAbelian. Denote by φ the homomorphism Q i ( Z /n i ) → Hom( G, Q / Z ) defined onthe i th factor by 1 χ i . As remarked in Definition 5.22, the χ i are linearlyindependent if and only if φ is injective. But the homomorphism Q i χ i is thePontryagin dual of φ , so is surjective if and only if φ is injective. (cid:3) Theorem 5.24.
Let c , . . . , c r be linearly independent classes in H ( ¯ X ◦ , Q / Z ) oforders n , . . . , n r respectively, all coprime to p . Then there is some constant B ,depending only on the c i , such that, whenever | F | > B and for all r -tuples ofnormalised elements A i , . . . , A r ∈ Br X with ¯ ∂ A i = c i , the product of the evaluationmaps Y A i : X ( O ) → Y i Br k [ n i ] is surjective.Proof. Since the A i are normalised, Lemma 5.11 shows that each ∂ A i is of order n i in H ( X ◦ , Q / Z ). For each i = 1 , . . . , r , let Y i → X ◦ be a torsor under Z /n i representing the class of ∂ ( A i ) in H ( X ◦ , Z /n i ). Let Y be the fibre product Y = Y × X ◦ · · · × X ◦ Y r . Let ¯ Y be the base change of Y to ¯ F . Then ¯ Y represents the class Q i c i inH ( ¯ X ◦ , Q i Z /n i ). Now, identifying that cohomology group with Hom( π ( ¯ X ◦ ) , Q i Z /n i ),Lemma 5.23 shows that the corresponding homomorphism is surjective; by Lemma 5.15, ¯ Y is connected. Then Fact 5.13 shows that whenever | F | > B ( ¯ Y ), any variety over F geometrically isomorphic to ¯ Y admits an F -point.Now let ( α , . . . , α r ) be an element of Q i Br k [ n i ]; applying the residue map gives( β , . . . , β r ) ∈ Q i H ( F , Z /n i ). Thinking of β = Q i β i as a class in H ( F , Q i Z /n i ),we have the Galois twist Y β = Y β × X ◦ · · · × X ◦ Y β r r . So Y β ( F ) = ∅ if and only if Y β i i ( F ) = ∅ for all i , which by Lemma 5.12 happens ifand only if there is a point P ∈ X ( O ) satisfying A i ( P ) = − α i for all i . Because Y β is geometrically isomorphic to ¯ Y , this will happen if | F | > B irrespective of thechoice of ( α , . . . , α r ). (cid:3) Application: reduction to a cone
In this section, we look at the situation where X is the projective cone over asmooth variety. The purpose is to demonstrate how the condition Br ¯ X = 0 allowsus to deduce surjectivity of the evaluation maps for any linearly independent set ofelements of Br X/ Br k ; under additional hypotheses, we can go further and provevanishing of the Brauer–Manin obstruction over a global field. Let us first gathersome geometric information about cones. Proposition 6.1.
Let K be an algebraically closed field, and let Y be the projectivecone in P dK over a smooth subvariety Z of P d − K . Suppose that Y is normal. Then:(i) the Picard group of Y is isomorphic to Z , and is generated by the class of ahyperplane section;(ii) the prime-to- p torsion in the Brauer group of Y is trivial, where p is thecharacteristic of K .Proof. Let U ⊂ Y be the affine cone over Z . Hoobler [22] studied graded rings S and gave a list of functors F which satisfy F ( S ) = F ( S ), where S is the degree-zero part of S . In particular, he showed that Pic S = Pic S when S is normal, andthat Br S ( p ′ ) = Br S ( p ′ ). We apply this with S being the affine coordinate ring of U and S = K to show that Pic U and Br U ( p ′ ) are both trivial. Because U is thecomplement of a hyperplane section in Y , we obtain (i).To complete the calculation of Br Y , consider the open set V ⊂ Y obtainedby removing the vertex. The Mayer–Vietoris sequence (see [28, III.2.24]) for thecovering Y = U ∪ V contains the exact sequencePic U ⊕ Pic V → Pic( U ∩ V ) → Br Y → Br U ⊕ Br V → Br( U ∩ V ) . Because V is smooth, Pic V surjects onto Pic( U ∩ V ) and Br V injects into Br( U ∩ V ).We deduce Br Y ( p ′ ) = 0, proving (ii). (cid:3) Remark . In the case of a cone over a smooth projective curve in characteristic p , Gordon [16] showed that Br Y is isomorphic to the set of K -points of a unipotentalgebraic group. In particular, Br Y is p -torsion.We obtain the following consequence. Lemma 6.3.
Let X be a regular scheme, proper and flat over O , such that thegeneric fibre X is a smooth variety over k and the special fibre X is the projectivecone over a smooth projective variety. Then the kernel of the map ¯ ∂ : Br X ( p ′ ) → H ( ¯ X ◦ , Q / Z ) is Br k ( p ′ ) . AD REDUCTION OF THE BRAUER–MANIN OBSTRUCTION 19
Proof.
We first show that Br X ( p ′ ) is trivial, as follows. The Hochschild–Serrespectral sequence for ¯ X → X gives an exact sequenceBr F → ker(Br X → Br ¯ X ) → H ( F , Pic ¯ X ) . Since F is finite, Br F is trivial; and, since Pic ¯ X is isomorphic to Z with trivialGalois action, H ( F , Pic ¯ X ) is trivial. Therefore Br X injects into Br ¯ X , and soBr X ( p ′ ) is trivial. The result now follows from Proposition 3.4 and a diagram-chase using Lemma 5.5. (cid:3) Corollary 6.4.
Under the conditions of Lemma 6.3, suppose that A , . . . , A r ∈ Br X ( p ′ ) are linearly independent when considered as elements of (Br X/ Br k ) .Then the conclusion of Theorem 5.24 applies to A , . . . , A r . In particular situations, we can make the bounds of Theorem 5.24 explicit. Onesuch case is when X is the projective cone over a smooth projective curve C ofgenus g . Theorem 6.5.
Let X be a regular scheme, proper and flat over O , such that thegeneric fibre X is a smooth surface over k and the special fibre X is the projectivecone over a smooth projective curve of genus g . Let A , . . . , A r be normalisedelements of Br X , of orders coprime to p , the images of which in Br X/ Br k arelinearly independent of orders n , . . . , n r , and let N = n · · · n r . Suppose that theresidue field F satisfies (8) | F | > (cid:0) g ′ + p ( g ′ ) − (cid:1) , where g ′ = N ( g −
1) + 1 . Then the product of the evaluation maps Y A i : X ( k ) → Y i Br k [ n i ] is surjective.Proof. As in the proof of Theorem 5.24, it will be enough to show that any ´etalecover of X ◦ of degree N admits an F -rational point. Suppose that X is the projec-tive cone over a smooth projective curve C of genus g . After removing the vertex, X ◦ is a fibration over C , with fibres isomorphic to the affine line, and every ´etalecover of X ◦ arises by pulling back an ´etale cover D → C . If D → C is of degree N , then the Riemann–Hurwitz formula shows that the genus of D is g ′ . Now theHasse–Weil bounds show that, under the condition (8), every smooth projectivecurve of genus g ′ over F admits an F -rational point, and hence every ´etale cover of X ◦ of degree N admits an F -rational point. (cid:3) Remark . If g = 1, then the condition (8) is vacuous.7. Vanishing of Brauer–Manin obstructions
An application of the surjectivity results of Sections 5.2 and 5.3 is to showthat certain varieties over global fields have no Brauer–Manin obstruction to theexistence of a rational point.Let us first make a useful definition. Let X be a variety over a number field K ,and let v be a place of K . Suppose that X ( K v ) is non-empty, and let P be a pointof X ( K v ). Using P , we can define a pairingBr X/ Br K × X ( K v ) → Q / Z by ( A , Q ) inv v A ( Q ) − inv v A ( P ) . Indeed, for any Q , this defines a homomorphism Br X → Q / Z which is trivial onBr K , so factors through Br X/ Br K . Definition 7.1.
Let B be a subgroup of Br X/ Br K . We say that B is prolific atthe place v if the map X ( K v ) → Hom( B, Q / Z )induced by the above pairing is surjective. Remark . (1) The definition does not depend on the choice of P , sincechanging P simply translates the image by an element of Hom( B, Q / Z ).(2) Suppose that B is generated by a single algebra A of order n , which we mayassume to be normalised. Then B is prolific if and only if the evaluationmap A : X ( K v ) → Br k [ n ] is surjective.(3) In order for B to be prolific, the localisation map B → (Br X K v / Br K v )must be injective.The motivation for making this definition is the following easy proposition. Proposition 7.3.
Let X be a variety over a number field K . Let B be a subgroupof Br X/ Br K , and suppose that B is prolific at a place v of K . Then there is noBrauer–Manin obstruction to the existence of rational points on X coming from B .Proof. Let Ω denote the set of places of K ; to prove the proposition, we must findan adelic point ( P w ∈ X ( K w )) w ∈ Ω satisfying P w ∈ Ω inv w A ( P w ) = 0 for all A ∈ B .Let ( P w ) w ∈ Ω be any adelic point of X . Define a homomorphism φ : B → Q / Z by φ ( A ) = X w ∈ Ω inv w A ( P w ) . Since B is prolific at v , there is a point Q ∈ X ( K v ) such that − φ ( A ) = inv v A ( Q ) − inv v A ( P v ) for all A ∈ B. Replacing P v by Q then gives an adelic point satisfying P w ∈ Ω inv w A ( P w ) = 0for all A ∈ B , and so there is no Brauer–Manin obstruction to the existence of arational point on X . (cid:3) Let us now use the calculations of Section 6 to show vanishing of the Brauer–Manin obstruction on some varieties reducing to a cone at a bad prime.
Theorem 7.4.
Let X be a smooth, projective, geometrically integral surface overa number field K , with points in every completion of K . Let ¯ X denote the basechange of X to an algebraic closure ¯ K of K . Suppose that H ( X, O X ) is trivial,that Pic ¯ X is torsion-free and that Br X/ Br K is finite of order N . Suppose thatthere is a prime p of K such that(i) p does not divide N ;(ii) X extends to a regular scheme, projective and flat over the ring of integersat p , the special fibre of which is the projective cone over a smooth projectivecurve of genus g ; and(iii) the residue field at p satisfies condition (8) of Theorem 6.5.Then Br X/ Br K is prolific at p , and so there is no Brauer–Manin obstruction tothe existence of rational points on X . AD REDUCTION OF THE BRAUER–MANIN OBSTRUCTION 21
Before proving Theorem 7.4, we first prove a series of general lemmas.
Lemma 7.5.
Let G be a finite group and H ⊆ G a subgroup. Let M be a G -module which is torsion-free as a Z -module, and suppose that M H is the same as M G . Then the restriction map H ( G, M ) → H ( H, M ) is injective.Proof. Let us first notice that ( M ⊗ Q ) H is the same as ( M ⊗ Q ) G ; this follows fromthe observation that ( M ⊗ Q ) G = M G ⊗ Q , and similarly for H , and the fact thattaking the tensor product with Q preserves surjectivity of maps. Now consider theshort exact sequence of G -modules0 → M → M ⊗ Q → M ⊗ ( Q / Z ) → . Taking cohomology of both G and H results in the following diagram, which iscommutative with exact rows.( M ⊗ Q ) G −−−−→ ( M ⊗ Q / Z ) G −−−−→ H ( G, M ) −−−−→ y y res y y ( M ⊗ Q ) H −−−−→ ( M ⊗ Q / Z ) H −−−−→ H ( H, M ) −−−−→ now follows from the Five Lemma. (cid:3) Remark . If H is a normal subgroup of G , then the lemma follows immediatelyfrom the inflation-restriction exact sequence. Lemma 7.7.
Let X be a variety over a number field K , and suppose that X haspoints in each real completion of K . Then, for any integer n , the natural map Br X [ n ] → (Br X/ Br K )[ n ] is surjective.Proof. The fundamental exact sequence of class field theory0 → Br K → M v Br K v P v inv v −−−−−→ Q / Z → , being split, remains exact upon taking the tensor product with Z /n ; we see that theinduced map (Br K ) /n → L v (Br K v ) /n is an isomorphism. Therefore an elementof Br K is divisible by n if and only if it is divisible by n in each Br K v . For eachfinite place v , we know that Br K v ∼ = Q / Z is divisible; the same is trivially true atcomplex places. So whether an element of Br K is divisible by n is determined atthe real places.Let A ∈ Br X be such that n A is equivalent to a constant class α ∈ Br K .We must show that we can change A by a constant class to obtain an element ofBr X [ n ]. We have local points P v ∈ X ( K v ) for each real place v . It follows that,for each real place v , the image of α in Br K v is divisible by n , for α ( P v ) = n A ( P v ).So α ∈ Br K , being divisible by n in each Br K v , is also divisible by n in Br K .Let β ∈ Br K satisfy nβ = α . Then we have n ( A − β ) = 0, that is, A − β lies inBr X [ n ]. (cid:3) Lemma 7.8.
Let L ⊂ M be an extension of algebraically closed fields of character-istic zero, and let X be a smooth, proper variety over L . Denote by X M the basechange of X to M . Then the natural maps NS( X ) → NS( X M ) and Br X → Br X M are isomorphisms. Proof.
We can interpret NS( X ) as the group of connected components of the Picardscheme of X . Since L is algebraically closed, the base change M/L induces abijection on connected components; so the map on N´eron–Severi groups NS( X ) → NS( X M ) is an isomorphism. Now, for each integer n , the Kummer sequence givesa commutative diagram as follows:0 −−−−→ NS( X ) /n −−−−→ H ( X, µ n ) −−−−→ Br X [ n ] → y y y −−−−→ NS( X M ) /n −−−−→ H ( X M , µ n ) −−−−→ Br X M [ n ] → . The middle vertical arrow is an isomorphism by [28, VI.2.6], and so the right-handarrow is also an isomorphism. Because Br X and Br X M are torsion groups, theresult follows. (cid:3) Lemma 7.9.
Let X be a smooth, geometrically irreducible, projective variety overa global field K . Let v be a place of K and write X v for the base change of X to K v . Suppose that X ( K v ) is non-empty, and that Pic X v is generated by the classof a hyperplane section. Then the natural map (Br X/ Br K ) → (Br X v / Br K v ) isinjective.Proof. Let ¯ K be an algebraic closure of K , and denote by ¯ X the base change of X to ¯ K . From the Hochschild–Serre spectral sequence for ¯ X → X we deduce thefollowing standard exact sequence:0 → ker(H ( K, Pic ¯ X ) → H ( K, ¯ K × )) → Br X/ Br K → Br ¯ X. Let ¯ K v be an algebraic closure of K v containing ¯ K . Writing ¯ X v for the basechange of X to ¯ K v , we obtain a similar exact sequence for X v . Since H ( K, ¯ K × )and H ( K p , ¯ K × v ) are both trivial, it will be sufficient to show that the two mapsH ( K, Pic ¯ X ) → H ( K v , Pic ¯ X v ) and Br ¯ X → Br ¯ X v are both injective; the result will then follow by the Five Lemma. The mapBr ¯ X → Br ¯ X v is actually an isomorphism, by Lemma 7.8; so it remains to showthat H ( K, Pic ¯ X ) → H ( K v , Pic ¯ X v ) is injective.Since Pic X v is isomorphic to Z , it follows that the Abelian variety A = P ic X istrivial: otherwise, A ( K v ) would be uncountable, and because X admits a K v -pointwe have A ( K v ) = Pic ( X v ), so that Pic X v would also be uncountable. ThereforePic X is the same as the N´eron–Severi group NS( X ), and similarly for ¯ X and ¯ X v .By Lemma 7.8, the natural map Pic ¯ X → Pic ¯ X v is an isomorphism. Take L ⊂ ¯ K to be a finite Galois extension over which a set of generators for Pic ¯ X isdefined, so that Pic X L → Pic ¯ X is also an isomorphism. Let L v be the completionof L in ¯ K v ; then Hilbert’s Theorem 90 shows that Pic X L v → Pic ¯ X v is injective,and therefore also an isomorphism. Let G denote the Galois group of L/K . Theinclusion L ⊂ L v identifies Gal( L v /K v ) with the decomposition group G v ⊂ G , andthe two inflation-restriction sequences form a commutative diagram0 −−−−→ H ( G, Pic X L ) −−−−→ H ( K, Pic ¯ X ) −−−−→ H ( L, Pic ¯ X ) y y y −−−−→ H ( G v , Pic X L v ) −−−−→ H ( K v , Pic ¯ X v ) −−−−→ H ( L v , Pic ¯ X v ) AD REDUCTION OF THE BRAUER–MANIN OBSTRUCTION 23
Because Pic ¯ X is a finitely generated free Z -module on which the absolute Galoisgroup of L acts trivially, the group H ( L, Pic ¯ X ) vanishes, as does H ( L v , Pic ¯ X v ).So the map H ( K, Pic ¯ X ) → H ( K v , Pic ¯ X v ) can be identified with the restrictionmap H ( G, Pic X L ) → H ( G v , Pic X L ). Because X admits a K v -point, we have(Pic X L v ) G v ∼ = Pic X v , and so (Pic X L v ) G v is generated by the class of a hyperplanesection. This class is defined over K , and so (Pic X L ) G and (Pic X L v ) G v coincide;by Lemma 7.5 we deduce that H ( K, Pic ¯ X ) → H ( K v , Pic ¯ X v ) is an injection,completing the proof. (cid:3) of Theorem 7.4. Let X be the given model over the ring of integers of K p , and let X be the special fibre. The restriction map Pic X →
Pic X p is surjective because X is regular, and its kernel is generated by the class of the special fibre, whichis principal; so we have Pic X ∼ = Pic X p . By Proposition 6.1, Pic ¯ X is generatedby the class of a hyperplane section. It follows that H ( X , O X ) = 0. By [17,Corollaire 3 to Th´eor`eme 7], the map Pic X →
Pic X is injective, and so Pic X p is also generated by the class of a hyperplane section. Now Lemma 7.9 shows thatthe natural map (Br X/ Br K ) → (Br X p / Br K p ) is injective.Let A , . . . , A r be a minimal set of generators for Br X/ Br K . Then A , . . . , A r are also linearly independent as elements of Br X p / Br K p . By Lemma 7.7, we canchoose the A i such that each has the same order n i in Br X as it does in Br X/ Br K .Further changing each A i by an element of Br K [ n i ], we may assume that they arenormalised. Applying Theorem 6.5, we find that the product of the evaluationmaps Y A i : X ( K p ) → Y i Br K p [ n i ]is surjective, and therefore Br X/ Br K is prolific. By Proposition 7.3, there is noBrauer–Manin obstruction to the existence of rational points on X . (cid:3) As special cases of Theorem 7.4, we recover the following results from the liter-ature.(1) (Colliot-Th´el`ene, Kanevsky, Sansuc [4]) Let X ⊂ P K be a smooth diagonalcubic surface, X : a X + a X + a X + a X = 0 , with a i coprime integers, having points over every completion of K . Sup-pose that p is a prime such that v p ( a a a a ) = 1. Then there is noBrauer–Manin obstruction to the existence of rational points on X .(2) (Bright [2]) Let X ⊂ P K be a smooth cubic surface defined by an equationof the form f ( X , X , X ) + pg ( X , X , X , X ) = 0with f, g cubic forms with integer coefficients and p ∤ g (0 , , , K . (The condition p ∤ g (0 , , ,
1) ensuresthat the given model is regular at p , and conversely any cubic surface,regular at p , and having reduction modulo p a cone may be put into thisform.) Then there is no Brauer–Manin obstruction to the existence ofrational points on X .(3) (Bright [1]) Let X ⊂ P K be a smooth diagonal quartic surface, X : a X + a X + a X + a X = 0 , with a i coprime integers, having points over every completion of K . Let H be the subgroup of Q × / ( Q × ) generated by −
1, 4 and the quotients a i /a j ; suppose that H has order 256 (the most general case) and that H ∩{ , , } = ∅ . (These conditions ensure that Br X has order 2.) Supposethat there exists a prime p such that p >
100 and v p ( a a a a ) = 1. Thenthere is no Brauer–Manin obstruction to the existence of rational points on X .Similar applications should exist whenever a bound on the order of the Brauergroup is known. That is in general a difficult problem, but the case of diagonalquartic surfaces is well understood thanks to work of Ieronymou, Skorobogatov andZarhin. To conclude, we deduce from Theorem 7.4 a general result for diagonalquartics. Corollary 7.10.
Let X ⊂ P Q be the smooth diagonal quartic surface defined by theequation a X + a X + a X + a X = 0 with a i coprime integers, and suppose that X has points over every completion of Q . Suppose that there exists a prime p > (2 +2 +1) satisfying v p ( a a a a ) = 1 .Then there is no Brauer–Manin obstruction to the existence of rational points on X .Proof. Because X is a K3 surface, the condition H ( X, O X ) = 0 is satisfied andPic ¯ X is torsion-free (in fact, free of rank 20). The condition v p ( a a a a ) = 1ensures that the given equation defines a model for X over Z p that is regular; thespecial fibre is the projective cone over a smooth diagonal quartic curve, of genus3. Ieronymou and Skorobogatov have recently shown [23, Theorem 1.1] that thiscondition also implies the absence of any odd torsion in Br X/ Br Q ; and the sameauthors with Zarhin showed in [24] that the 2-torsion in Br X/ Br Q has order atmost 2 . So, for any p larger than the claimed bound, Theorem 7.4 applies andshows that there is no Brauer–Manin obstruction. (cid:3) It should be noted that Ieronymou and Skorobogatov [23] have shown that odd-order elements in the Brauer groups of diagonal quartic surfaces over Q , when theydo exist, can never obstruct the Hasse principle. They do this by calculating that,if A has order p = 3 or 5 in Br X/ Br Q , then the corresponding evaluation map A : X ( Q p ) → Br Q p [ p ] is surjective. Because this involves studying elements oforder p in the Brauer group at the prime p , the methods of this paper are uselessin understanding that result.acknowledgements. Substantial parts of this work were carried out during the se-mester on ‘Rational points and algebraic cycles’ at the Centre Interfacultaire Bernoulliof the ´Ecole Polytechnique F´ed´erale de Lausanne in 2012; I thank the organisersand staff for their support and hospitality. I also thank the Center for AdvancedMathematical Sciences of the American University of Beirut for their support. References [1] M. J. Bright. The Brauer–Manin obstruction on a general diagonal quartic surface.
ActaArith. , 147(3):291–302, 2011.[2] M. J. Bright. Evaluating Azumaya algebras on cubic surfaces.
Manuscripta Math. , 134(3-4):405–421, 2011.
AD REDUCTION OF THE BRAUER–MANIN OBSTRUCTION 25 [3] M. J. Bright. Brauer groups of singular del Pezzo surfaces.
Michigan Math. J. , 62(3):657–664,2013.[4] J.-L. Colliot-Th´el`ene, D. Kanevsky, and J.-J. Sansuc. Arithm´etique des surfaces cubiquesdiagonales. In
Diophantine approximation and transcendence theory (Bonn, 1985) , volume1290 of
Lecture Notes in Math. , pages 1–108. Springer, Berlin, 1987.[5] J.-L. Colliot-Th´el`ene and S. Saito. Z´ero-cycles sur les vari´et´es p -adiques et groupe de Brauer. Internat. Math. Res. Notices , (4):151–160, 1996.[6] J.-L. Colliot-Th´el`ene and A. N. Skorobogatov. Descente galoisienne sur le groupe de Brauer.
J. Reine Angew. Math. , 682:141–165, 2013.[7] J.-L. Colliot-Th´el`ene and A. N. Skorobogatov. Good reduction of the Brauer-Manin obstruc-tion.
Trans. Amer. Math. Soc. , 365(2):579–590, 2013.[8] D. F. Coray and M. A. Tsfasman. Arithmetic on singular Del Pezzo surfaces.
Proc. LondonMath. Soc. (3) , 57(1):25–87, 1988.[9] P. Corn. The Brauer-Manin obstruction on del Pezzo surfaces of degree 2.
Proc. Lond. Math.Soc. (3) , 95(3):735–777, 2007.[10] P. Deligne.
Cohomologie ´etale . Number 569 in Lecture Notes in Mathematics. Springer-Verlag,Berlin, 1977. S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie SGA 4 . Avec la collabora-tion de J. F. Boutot, A. Grothendieck, L. Illusie et J. L. Verdier.[11] P. Deligne. La conjecture de Weil. II. Inst. Hautes ´Etudes Sci. Publ. Math. , (52):137–252,1980.[12] F. R. DeMeyer and T. J. Ford. On the Brauer group of surfaces.
J. Algebra , 86(1):259–271,1984.[13] A.-S. Elsenhans and J. Jahnel. Cubic surfaces with a Galois invariant double-six.
Cent. Eur.J. Math. , 8(4):646–661, 2010.[14] A.-S. Elsenhans and J. Jahnel. Cubic surfaces with a Galois invariant pair of Steiner trihedra.
Int. J. Number Theory , 7(4):947–970, 2011.[15] K. Fujiwara. A proof of the absolute purity conjecture (after Gabber). In
Algebraic geometry2000, Azumino (Hotaka) , volume 36 of
Adv. Stud. Pure Math. , pages 153–183. Math. Soc.Japan, Tokyo, 2002.[16] W. J. Gordon. Brauer groups of local rings with conelike singularities.
J. Algebra , 76(2):353–366, 1982.[17] A. Grothendieck. G´eom´etrie formelle et g´eom´etrie alg´ebrique. In
Fondements de la g´eom´etriealg´ebrique. [Extraits du S´eminaire Bourbaki, 1957–1962.] . Secr´etariat math´ematique, Paris,1962.[18] A. Grothendieck. Le groupe de Brauer I. In J. Giraud et al., editors,
Dix Expos´es sur laCohomologie des Sch´emas , volume 3 of
Advanced studies in mathematics , pages 46–65. North-Holland, Amsterdam, 1968.[19] A. Grothendieck. Le groupe de Brauer II. In J. Giraud et al., editors,
Dix Expos´es surla Cohomologie des Sch´emas , volume 3 of
Advanced studies in mathematics , pages 66–87.North-Holland, Amsterdam, 1968.[20] A. Grothendieck. Le groupe de Brauer III. In J. Giraud et al., editors,
Dix Expos´es surla Cohomologie des Sch´emas , volume 3 of
Advanced studies in mathematics , pages 88–188.North-Holland, Amsterdam, 1968.[21] A. Grothendieck.
S´eminaire de G´eometrie Alg´ebrique du Bois-Marie SGA 1: Revˆetements´Etales et Groupe Fondamental . Number 224 in Lecture Notes in Mathematics. Springer-Verlag, 1971.[22] R. T. Hoobler. Functors of graded rings. In
Methods in ring theory (Antwerp, 1983) , volume129 of
NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. , pages 161–170. Reidel, Dordrecht,1984.[23] E. Ieronymou and A. N. Skorobogatov. Odd order Brauer–Manin obstruction on diagonalquartic surfaces.
Adv. Math. , 270:181–205, 2015.[24] E. Ieronymou, A. N. Skorobogatov, and Y. G. Zarhin. On the Brauer group of diagonalquartic surfaces.
J. London Math. Soc. , 83:659–672, 2011.[25] S. Lang and A. Weil. Number of points of varieties in finite fields.
Amer. J. Math. , 76:819–827,1954.[26] J. Lipman. Rational singularities, with applications to algebraic surfaces and unique factor-ization.
Inst. Hautes ´Etudes Sci. Publ. Math. , (36):195–279, 1969. [27] Yu. I. Manin. Le groupe de Brauer–Grothendieck en g´eom´etrie diophantienne. In
Actes duCongr`es International des Math´ematiciens (Nice, 1970), Tome 1 , pages 401–411. Gauthier-Villars, Paris, 1971.[28] J. S. Milne.
Etale Cohomology . Number 33 in Princeton mathematical series. Princeton Uni-versity Press, 1980.[29] J. S. Milne. Lectures on etale cohomology (v2.20). Available at ,2012.[30] J. Riou. Classes de Chern, morphismes de Gysin, puret´e absolue. In L. Illusie, Y. Laszlo, andF. Orgogozo, editors,
Travaux de Gabber sur l’uniformisation locale et la cohomologie ´etaledes sch´emas quasi-excellents. S´eminaire `a l’ ´Ecole polytechnique 2006–2008 , volume 363–364of
Ast´erisque . Soc. Math. France, 2014.[31] C. Salgado and R. van Luijk. Density of rational points on del Pezzo surfaces of degree one.
Adv. Math. , 261:154–199, 2014.[32] J.-P. Serre.
Corps Locaux , volume VIII of
Publications de l’Institut de Math´ematique del’Universit´e de Nancago . Hermann, Paris, 1968.[33] J.-P. Serre.
Local fields , volume 67 of
Graduate Texts in Mathematics . Springer-Verlag, NewYork, 1979. Translated from the French by Marvin Jay Greenberg.[34] A. Skorobogatov.
Torsors and rational points , volume 144 of
Cambridge Tracts in Mathe-matics . Cambridge University Press, Cambridge, 2001.[35] Sir Peter Swinnerton-Dyer. The Brauer group of cubic surfaces.
Math. Proc. Camb. Phil.Soc. , 113:449–460, May 1993.[36] E. Witt. Schiefk¨orper ¨uber diskret bewerteten K¨orpern.
J. Reine Angew. Math. , 176:153–156,1937.
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