Evaluating the wild Brauer group

Classifying elements of the Brauer group of a variety X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$X$\end{document} over a p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p$\end{document}-adic field by the p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p$\end{document}-adic accuracy needed to evaluate them gives a filtration on BrX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\operatorname{Br}X$\end{document}. We relate this filtration to that defined by Kato’s Swan conductor. The refined Swan conductor controls how the evaluation maps vary on p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p$\end{document}-adic discs: this provides a geometric characterisation of the refined Swan conductor. We give applications to rational points on varieties over number fields, including failure of weak approximation for varieties admitting a non-zero global 2-form.


Introduction
Let k be a p-adic field with ring of integers O k , uniformiser π and residue field F, and let X/k be a smooth geometrically irreducible variety. The most naïve filtration on the Brauer group Br X, and the one we aim to understand, is that given by evaluation of elements of Br X at the k-points of X. If A ∈ Br X has order coprime to p, then [7, §5] shows that the evaluation map for A factors through reduction to the special fibre. In this article, we describe the variation of the evaluation map in the considerably more complicated case of elements of order a power of p in Br X.
To define the evaluation filtration, fix a smooth model X /O k having geometrically integral special fibre Y /F with function field F . Given A ∈ Br X, one can ask whether the evaluation map |A| : X (O k ) → Br k factors through the reduction map X (O k ) → X (O k /π i ) for any i ≥ 1. We first define some notation.
Let k ′ be a finite extension of k of ramification index e(k ′ /k), with ring of integers O k ′ and uniformiser π ′ . For r ≥ 1 and P ∈ X (O k ′ ), let B(P, r) be the set of points Q ∈ X (O k ′ ) such that Q has the same image as P in X (O k ′ /(π ′ ) r ). Define Ev n Br X = {A ∈ Br X | ∀ k ′ /k finite, ∀ P ∈ X (O k ′ ), |A| is constant on B(P, e(k ′ /k)n + 1)} (n ≥ 0), Let {fil n Br X} n≥0 denote the filtration given by Kato's Swan conductor (see Definition 2.1) and, for n ≥ 1, write rsw n (A) for the refined Swan conductor of A ∈ fil n Br X (see Definition 2.5). We have rsw n (A) = [α, β] π,n for some (α, β) ∈ Ω 2 F ⊕ Ω 1 F . In Section 3 we will define a residue map ∂ : fil 0 Br X → H 1 (Y, Q/Z). Theorem A. Let k be a finite extension of Q p . Let X be a smooth, geometrically irreducible variety over k, and let X → Spec O k be a smooth model of X. Suppose that the special fibre Y of X is geometrically irreducible. Then (1) Ev −2 Br X = {A ∈ fil 0 Br X | ∂A = 0}; (2) Ev −1 Br X = {A ∈ fil 0 Br X | ∂A ∈ H 1 (F, Q/Z)}; (3) Ev 0 Br X = fil 0 Br X; (4) for n ≥ 1, Ev n Br X = {A ∈ fil n+1 Br X | rsw n+1 (A) ∈ [Ω 2 F , 0] π,n+1 }.
(1) By definition of the refined Swan conductor, fil n Br X ⊂ {A ∈ fil n+1 Br X | rsw n+1 (A) ∈ [Ω 2 F , 0] π,n+1 }, with equality if p ∤ n + 1. See Lemma 2.8 for more details. (2) In the case of H 1 (K) = H 1 (K, Q/Z), where K is the function field of X, Kato's filtration and the refined Swan conductor have been extensively studied in the literature, and are closely related to ramification theory (see Section 10). We believe that Theorem A is the first geometric characterisation of Kato's filtration on H 2 (K) = Br K, with Theorem B below giving a geometric description of the refined Swan conductor. The modified version of Kato's filtration featuring in Theorem A does not seem to have appeared elsewhere, though it is analogous to the "non-logarithmic" version of Kato's filtration on H 1 (K) defined by Matsuda [31] in equal characteristic. (3) The reason for considering points over finite extensions of k, instead of just over k itself, is that the filtration obtained is better behaved. (A function that is non-constant on points over some field extension can be constant on the rational points, simply because there are "too few" points of Y (F): see [7,Remark 5.20] for an example.) (4) A consequence of Theorem A is that the evaluation filtration does not change if Y is replaced by a non-empty open subset. (5) In fact, our proof of Theorem A shows that it remains true if we modify the definition of Ev n Br X by restricting to unramified finite extensions k ′ /k instead of all finite extensions. (6) The inclusion fil n Br X[p] ⊂ Ev n Br X[p] is implicit in work of Uematsu [39], at least in the case when k contains a primitive pth root of unity. (7) Yamazaki [41] has proved a result very similar to Theorem A in the case that X is a smooth proper curve. In that case, one can extend the Brauer-Manin pairing to the Picard group Pic X. Yamazaki defines a filtration on Pic X by considering the kernels of reduction modulo powers of π, and shows that the induced filtration on Br X coincides with Kato's filtration.
(When X is a curve, the group Ω 2 F is trivial, so our filtration in Theorem A also coincides with Kato's, by definition of the refined Swan conductor.) (8) Sato and Saito [33] have shown that Ev −2 Br X coincides with the image of Br X in Br X when X is regular and proper over O k , but without the assumption of smoothness. (They also assume that X satisfies purity for the Brauer group, but this is now known to hold for all regular schemes [40].) In Corollary 3.7, we will show how our results give a new proof of this when X is smooth over O k .
In order to prove Theorem A, we examine the behaviour of the evaluation maps on the graded pieces of Kato's filtration on the Brauer group. The results of this study for n ≥ 1 are summarised in Theorem B below. In order to state it, we need to introduce some more notation. Let P ∈ X (O k ) and let P 0 denote the image of P in Y (F). Elements in the image of the reduction map B(P, n) → X (O k /π n+1 ) can be identified with tangent vectors in T P0 Y (see Lemma 7.1). Write [ − − → P Q] n for the tangent vector corresponding to the image in X (O k /π n+1 ) of a point Q ∈ B(P, n). In the statement of the following theorem, we denote by x/p the image of x ∈ F p under the identification of Z/p with the p-torsion in Q/Z. The integer e is the absolute ramification index of k and we set e ′ = ep/(p − 1).
Theorem B. Let n > 0, let A ∈ fil n Br X, and let rsw n (A) = [α, β] π,n for some (α, β) ∈ Ω 2 F ⊕ Ω 1 F . Let P ∈ X (O k ), and let P 0 ∈ Y (F) be the reduction of P . Then α and β are regular at P 0 and we have the following description of the evaluation map |A| : X (O k ) → Br k.
Elements in Br X of order coprime to p have been thoroughly treated in the literature, in particular by Colliot-Thélène-Saito [8], Colliot-Thélène-Skorobogatov [10] and Bright [7]. The computation of the evaluation map in the coprime to p case is greatly aided by the fact that the map factors through reduction to the special fibre. In a similar way, Theorem B enables the computation of the evaluation map for Brauer group elements of order a power of p. Thus it facilitates a systematic treatment of Brauer-Manin obstructions, which will have both theoretical and computational implications for the study of rational points on varieties. Some first consequences of Theorem B are outlined below (see Theorems C and D).
Applications to the Brauer-Manin obstruction. Manin [30] introduced the use of the Brauer group to study rational points on varieties over number fields. Let V be a smooth, proper, geometrically irreducible variety over a number field L. The evaluation maps |B| : V (L v ) → Br L v for each B ∈ Br V and place v of L combine to give a pairing Informally: does the Brauer-Manin obstruction involve only the places of bad reduction and the Archimedean places?
In [10, Theorem 3.1] Colliot-Thélène and Skorobogatov prove that if the transcendental Brauer group of V (meaning the image of the natural map Br V → BrV ) is finite and S contains all the primes dividing its order then the answer to Question 1.3 is yes. Skorobogatov and Zarhin show in [37] that the transcendental Brauer groups of Abelian varieties and K3 surfaces over number fields are finite; their question as to whether this is true for smooth projective varieties more generally remains open (see [37,Question 1]).
Before we address Question 1.3, let us introduce one further question. Our next result owes its existence to Wittenberg's suggestion that our methods could be used to address Question 1.4. It gives a positive answer to that question, and also shows that the answer to Question 1.3 is no, in general. In fact, it shows that if H 0 (V, Ω 2 V ) = 0 then every prime of good ordinary reduction is involved in a Brauer-Manin obstruction over an extension of the base field.
Theorem C. Let V be a smooth, proper variety over a number field L with H 0 (V, Ω 2 V ) = 0. Let p be a prime of L at which V has good ordinary reduction, with residue characteristic p. Then there exist a finite extension L ′ /L, a prime p ′ of L ′ lying over p, and an element A ∈ Br V L ′ {p} such that the evaluation map It has been conjectured that a smooth, projective variety over a number field has infinitely many primes of good ordinary reduction: Joshi [23, Conjecture 3.1.1] attributes this conjecture to Serre. In the cases of Abelian surfaces and K3 surfaces, this is known to be true after a finite base extension [5,23].
The assumption that H 0 (V, Ω 2 V ) = 0 implies, via Hodge theory, that the second Betti number b 2 (V ) and geometric Picard number ρ(V ) are not equal. Since BrV contains a copy of (Q/Z) b2−ρ (see [11,Proposition 5.2.9]), this implies that there exists a finite extension L ′ /L such that the transcendental Brauer group of V L ′ is non-trivial. On the other hand, if the transcendental Brauer group is trivial then [8,Theorem 3.1] shows that the answer to Question 1.3 is yes.
Having seen in Theorem C that the places involved in the Brauer-Manin obstruction need not be confined to the places of bad reduction and the Archimedean places, one may be interested in the following question: Question 1.5. Let V be a smooth, proper, geometrically irreducible variety over a number field L such that PicV is finitely generated and torsion-free. Does there exist a finite set S of places of L and a closed set If so, can one give an explicit description of such a set S?
The assumption in Questions 1.3 and 1.5 that PicV be finitely generated and torsion-free is necessary: for example, in the case of an elliptic curve E over Q with finite Tate-Shafarevich group and trivial Mordell-Weil group, [36,Proposition 6.2.4] shows that where E(R) 0 denotes the connected component of the identity in E(R). This contradicts the description of (1.2). More generally, note that non-trivial torsion in PicV implies that the abelianisation of πé t 1 (V ) is non-trivial. For a smooth, proper, geometrically integral variety V over a number field L satisfying V (L) = ∅ and πé t 1 (V ) = 0, Harari [18, §2] has shown that for any finite set Σ of places of L, the variety V does not satisfy weak approximation outside Σ. The proof uses a descent obstruction which, in the case of an abelian covering, is coarser than the Brauer-Manin obstruction. It follows that in this setting the Brauer-Manin set is not of the form described in (1.2).
If one assumes that the transcendental Brauer group of V is finite in Question 1.5 then it follows from the Hochschild-Serre spectral sequence that the quotient of Br V by the image of Br L is finite; therefore the existence of the finite set S is a consequence of the Albert-Brauer-Hasse-Noether Theorem and the continuity of the Brauer-Manin pairing. The finiteness of the quotient of Br V by the image of Br L also implies that the Brauer-Manin set is open as well as closed. On the other hand, without the finiteness assumption on the transcendental Brauer group, the existence of the finite set S in Question 1.5 is not a priori obvious.
Theorem D below gives a positive answer to Question 1.5, without any finiteness assumption on the transcendental Brauer group of V . One consequence of this theorem is that for a K3 surface over Q the only places that play a rôle in the Brauer-Manin obstruction are the Archimedean places, the primes of bad reduction, and the prime 2 (see Remark 11.5).
Theorem D. Let L be a number field. Let V be a smooth, proper, geometrically irreducible variety over L. Assume PicV is finitely generated and torsion-free. Then there is a finite set of places S of L such that, for all A ∈ Br V and all places p / ∈ S, the evaluation map |A| : V (L p ) → Br L p is constant. Furthermore, the set S can be taken to consist of the following places of L: (1) Archimedean places; (2) places of bad reduction for V ; (3) places p satisfying e p ≥ p − 1, where e p is the absolute ramification index of p and p is the residue characteristic of p; (4) places p for which, for any smooth proper model Remark 1.6. Condition (4) is related to torsion in the crystalline cohomology of V(p). For example, suppose that O p is absolutely unramified, so that O p is the ring of Witt vectors of its residue field; then, by Deligne-Illusie [13], the Hodge-de Rham spectral sequence for V(p) degenerates, showing  (4) is always empty -see Remark 11.5.
Outline of the paper. Section 2 contains some technical results and background relating to Kato's refined Swan conductor. In Section 3 we define a residue map ∂ : fil 0 Br X → H 1 (Y, Q/Z) and use it to describe the evaluation maps for elements of fil 0 Br X. The main body of the paper, Sections 4-8, is concerned with the proof of Theorem B. Its proof will involve a chain of blowups with an associated decreasing sequence of Swan conductors at the exceptional divisors. Eventually, we will obtain Swan conductor zero, whereupon evaluations are controlled by the residue map, as in Section 3. Section 4 contains some technical results that will be used in Section 8 to relate the refined Swan conductor of a Brauer group element to that of its residue along the exceptional divisor. In Section 5, we show how information about the refined Swan conductor is retained under blowups. Section 6 gathers some calculations pertaining to the exceptional divisors of these blowups. Section 7 relates lifts of points to tangent vectors and shows how to keep track of them when blowing up. In Section 8 we bring everything together to complete the proof of Theorem B. In Section 9 we prove Theorem A. In Section 10 we compare various other filtrations in the literature with our modified version of Kato's filtration which gives rise to the evaluation filtration on the Brauer group. Section 11 is concerned with applications to the Brauer-Manin obstruction and contains the proofs of Theorems C and D.
Notation. If A is an Abelian group and n a positive integer, then A[n] and A/n denote the kernel and cokernel, respectively, of multiplication by n on A. If ℓ is prime, then A{ℓ} denotes the ℓ-power torsion subgroup of A. We use extensively the notation introduced in [27, §1]. In particular, the notation Z/n(r) has a particular meaning in characteristic p. Write n = p s m with p ∤ m. For any scheme S smooth over a field of characteristic p, the sheaf Z/n(r) on Sé t is defined by . For the definition of Ω r S,log , see [21, I.5.7]. We further use Kato's notation H q n (R) := H q (Ré t , Z/n(q − 1)), H q (R) := lim − → n H q n (R) whenever either n is invertible in R, or R is smooth over a field of characteristic p.
Acknowledgements. We thank Evis Ieronymou for many valuable comments and suggestions on previous drafts of this article. We thank Daniel Loughran for several helpful comments and references that have improved the introduction. We thank Bhargav Bhatt, Bas Edixhoven, Christopher Lazda, Alexei Skorobogatov and Olivier Wittenberg for useful conversations. We are very grateful to the anonymous referee who suggested the geometric approach that we have now adopted, following Kato in [27]. We made substantial progress during the workshops Rational Points 2017 and Rational Points 2019 at Franken-Akademie Schloss Schney, and during the 2019 trimester programmeÀ la redécouverte des points rationnels/Reinventing rational points at the Institut Henri Poincaré, and we are very grateful to the organisers of these events. Rachel Newton is supported by EPSRC grant EP/S004696/1 and UKRI Future Leaders Fellowship MR/T041609/1.

Kato's refined Swan conductor
In this section, we gather some technical results from [27] relating to Kato's refined Swan conductor, extending them as necessary. For this section only, K denotes a Henselian discrete valuation field of characteristic zero with ring of integers O K and residue field F of characteristic p. Let π be a uniformiser in O K and denote by m K the maximal ideal of O K .
2.1. Vanishing cycles and the Swan conductor. Let A be a ring over O K , and let i, j be the inclusions of the special and generic fibres, respectively, into Spec A.
. The construction is functorial in the following sense. If f : A → A ′ is a homomorphism of rings over O K , then we obtain a commutative diagram of schemes. Applying (i ′ ) * to the natural base-change map f * Rj * Z/n(q − 1) → Rj ′ * Z/n(q − 1) and using (i ′ ) * f * = g * i * gives a map g * i * Rj * Z/n(q − 1) → (i ′ ) * Rj ′ * Z/n(q − 1) and so, by adjunction, a natural map V q n (A) → V q n (A ′ ) for all q, n. Also, the natural map of sheaves gives a natural map for all q, n, which Gabber [15] has proved to be an isomorphism if (A, m K A) is Henselian. In that case, we define a product (χ, a 1 , . . . , a r ) → {χ, a 1 , . . . , a r } using the Kummer map (A ⊗ OK K) × → H 1 (A ⊗ OK K, Z/n(1)) and the cup product For general A, let A (h) denote the Henselisation at the ideal m K A; then the natural map V q n (A) → V q n (A (h) ) is an isomorphism, because the stalks of i * Rj * Z/n(q−1) do not change when A is replaced by A (h) . This allows us to define the product (2.2) for A as well. The products for different n are compatible and so give rise to a product We can now define Kato's Swan conductor.
The increasing filtration {fil n H q (K)} n≥0 is defined by For χ ∈ H q (K), define the Swan conductor sw(χ) to be the smallest n ≥ 0 satisfying χ ∈ fil n H q (K).  λ π : H q n (F ) ⊕ H q−1 n (F ) → H q n (K) for all positive integers n. Note that Kato makes no assumption on the characteristics of K and F : for him K is simply a Henselian discrete valuation field with residue field F . Let A be a ring smooth over O K , with R = A/m K A. In [27, §1.9], Kato defines another injective homomorphism , and states that it coincides with the previous definition (2.3) in the case A = O K and n = p. This is sufficient for the definition of the refined Swan conductor. However, in Section 3 we will need such an extension not only to H q p (R) but to H q p r (R) for all r ≥ 1. Returning to our setting, wherein the characteristics of K and F are zero and p, respectively, we now define such a homomorphism λ π : H q p r (R) ⊕ H q−1 p r (R) → V q p r (A) for all q ≥ 2 and r ≥ 1, which coincides, up to sign, with Kato's definition in [27, §1.9] when r = 1. In Lemma 2.4 we prove that our definition of λ π coincides with Kato's definition (2.3) in the case when A = O K and n = p r . We closely follow [27, §1.9] throughout.
Since R has p-cohomological dimension ≤ 1 (see [2, X, Théorème 5.1]), the spectral sequence calculating V q p r (A) reduces to a short exact sequence [4], write M q−1 r = i * R q−1 j * Z/p r (q − 1). By [4,Theorem 1.4], there is a finite decreasing filtration on M q−1 r with graded pieces as follows: there is an isomorphism (For i < 0 we set Ω i R = 0.) Now H 1 (R, Ω i R ) = 0 for i ≥ 0 because Ω i R is a coherent sheaf on the affine scheme Spec R. Also, if we let K m = ker(ρ m ), then H 2 (R, K m ) = 0 because cd p (R) ≤ 1. This shows H 1 (R, gr m (M q−1 r )) = 0 for m ≥ 1. It follows that the natural map is an isomorphism. The right-hand group is, by definition, H q p r (R) ⊕ H q−1 p r (R). Composing the inverse of this isomorphism with the map occurring in (2.4) defines (−1) q−1 λ π .
Following Kato, we sometimes use λ π to denote the composition In the case A = O K , theétale cohomology groups become Galois cohomology: the sequence (2.4) becomes where K nr is the maximal unramified extension of K. The definition of (−1) q−1 λ π factors as → H q (K, Z/p r (q − 1)).
(Here F s is a separable closure of F , and we identify Gal(K nr /K) ∼ = Gal(F s /F ) without further comment.) 2.3. Change of ring. Let K ′ /K be a finite extension of Henselian discrete valuation fields. Let O K ′ be the ring of integers of K ′ and let F ′ be the residue field. Suppose that we have a commutative diagram and let i ′ , j ′ be the inclusions of the special and generic fibres, respectively, of ) There are natural maps φ * : V q n (A) → V q n (A ′ ), constructed exactly as in the case K = K ′ of Section 2.1. Letφ : R → R ′ be the induced map on residue rings, and φ * : H q n (R) → H q n (R ′ ) andφ * : W r Ω q R → W r Ω q R ′ the induced maps. Let π ′ be a uniformiser in O K ′ . Lemma 2.3. In the situation described above, let e be the ramification index of K ′ /K and defineā ∈ F ′ to be the reduction of φ(π)(π ′ ) −e . Then, for all q ≥ 2 and r ≥ 1, the following diagram commutes: Here dlogā ∈ W r Ω 1 R ′ is as defined in [21, I.3.23]. Proof. We go through the steps of the construction of λ π . Let g : Spec R ′ → Spec R be the morphism corresponding toφ. The natural map where a = φ(π)(π ′ ) −e . Therefore the isomorphisms (2.5) for A and A ′ satisfy the following commutative diagram: Finally, forming the wedge product with dlogā commutes with δ r . To see this, one checks that this wedge product commutes with C −1 − 1, giving a commutative diagram of sheaves on Ré t (and R ′ et ) where the rows are the sequence (2.6) as follows: Taking cohomology now gives the required property. The formula on the middle vertical arrow of (2.7) comes from the last line of [ Proof. We will prove this by induction on q, by showing that our map λ π satisfies the characterisation given in [27, §1.4].
Firstly, let q ≥ 2 and let a be an element of O × K . Letā ∈ F × be the reduction of a and let {ā} be its class in H 1 (F, Z/p r (1)). We claim that the following diagram commutes. (2.8) Here the horizontal maps are as follows. The first horizontal map is cup product with the class of a in H 1 (K, Z/p r (1)). The second horizontal map is induced by cup product with the class of a in H 1 (K nr , Z/p r (1)). The notation dlogā is as in [21, §I.3.23], and the third horizontal map is that induced on cohomology by the homomorphism ω → dlogā ∧ ω on each factor. The fourth horizontal map is given by cup products as written.
The bottom left vertical map is induced by the isomorphism ǫ : Z/p r → W r Ω 0 log (F s ) (see [21,Proposition 3.28]). The first horizontal map is cup product with the class of a in H 1 (K, Z/p r (1)), the second horizontal map is that induced on cohomology by sending 1 to the class of a in H 1 (K nr , Z/p r (1)), the third horizontal map is that induced on cohomology by 1 → (dlogā, 0), and the fourth horizontal map is the cup product map as written.
Finally, we have a third diagram (2.10) in which the bottom horizontal map is induced by the isomorphism ǫ : We can now prove the lemma. Write λ ′ π for the map defined at the end of [27, §1.4]. The left-hand column of diagrams (2.9) and (2.10) is, by definition, the map ι 1 p r of [27, §1.4]. Given this, the definition of λ ′ π shows that for q = 2 the map −λ ′ π coincides with the right-hand column of (2.9) and (2.10), which is our −λ π . This is the base case for our proof of Lemma 2.4, which will proceed by induction.
By the induction hypothesis, and the definition of λ ′ π and ι q p r in [27, §1.4], we obtain p r is determined by its action on elements of the form {ψ, a 1 , . . . , a q } for ψ ∈ H 1 p r (F ) and a 1 , . . . , a q ∈ O × K , this shows the desired result for λ ′ π on H q+1 p r (F ) ⊕ 0. The result for λ ′ π on 0 ⊕ H q p r (F ) follows in a similar way.
2.4. The refined Swan conductor. Equipped with the map λ π for the ring O K [T ], we can now define Kato's refined Swan conductor. We follow the exposition in [6, 4.5-4.6]. Let ω 1 F denote the F -module of absolute Kähler differentials with respect to the log structure on F given by the reduction map Kato [27,Theorem 5.1] proves the following: if χ is an element of fil n H q (K) for n ≥ 1, then there exists a unique (α, where α, β are as in (2.11). We will often write [α, β] π,n as shorthand for π −n (α + β ∧ dlog π).
For n ≥ 1, the refined Swan conductor defines an injective homomorphism Remark 2.6. The pair (α, β) in (2.11) depends on the choice of uniformiser π, in a way that is made precise by the following lemma (in the special case K = K ′ ). This shows that rsw n (χ) is independent of the choice of π.
We now prove several auxiliary results about the refined Swan conductor.
Lemma 2.7. Let K ′ /K be a finite extension of Henselian discrete valuation fields of ramification index e. Let π ′ be a uniformiser in K ′ , let F ′ be the residue field of K ′ and defineā ∈ F ′ to be the reduction of π(π ′ ) −e . Let χ be an element of fil n H q (K), and let res: be the restriction map. Then res(χ) lies in fil en H q (K ′ ), and setting rsw n (χ) = [α, β] π,n we have Proof. That res(χ) lies in fil en H q (K ′ ) follows from the characterisation of fil n given in [27, Proposition 6.3]. Lemma 2.3 gives where a = π(π ′ ) −e . Applying Lemma 2.3 a second time to the automorphism of O K [T ] defined by T → a n T proves the claimed formula.
The following lemma is implicit in [27,Proposition 5.4], which is stated without proof. For completeness, we provide a proof here.
We first prove a lemma whose first part will be used in the proof of Lemma 2.8 and whose second part will be used in the proof of Lemma 2.10 below. We use C to denote the Cartier operator. Lemma 2.9. Let R = F (T ) and let α be an element of Ω q−1 F for some q ≥ 2.
Proof of Lemma 2.8. By definition of rsw n , we have {χ, . We would like to take the cup product with −π n T , but as this is not a unit in K[T ] we first have to pass to a larger ring. Let A be the Henselisation The last term in (2.12) is zero, from the formula in [27, §1.3] and dlog(−1) = 0. Let ι q p : H q p (F (T )) → V q (A) be the canonical lifting map of [27, §1.4], which is the first component of λ π . By Lemma 2.9(1) and [27, §1.4], the first term of (2.12) is For the middle term of (2.12) we have The result now follows from the injectivity of λ π and [27, Lemma 3.8].
We conclude this subsection with a description of the refined Swan conductor of pχ in the case that sw(χ) is sufficiently large. Lemma 2.10. Let e be the absolute ramification index of K, and set e ′ = ep/(p−1). Let u = p/π e and letū ∈ F × be its reduction. Let χ ∈ H q (K) have sw(χ) = n > 0 and write rsw n (χ) = [α, β] π,n .
2.5. The residue map. Let n ≥ 1. By [27, Proposition 6.1(1)], the image of to be the inverse of λ π followed by projection onto the second factor, c.f. [27, §7.5]. Its inductive limit is a homomorphism fil 0 H 2 (K) → H 1 (F ), which we also denote by ∂. We will refer to ∂ as the residue map.
2.6. Comparison with the classical residue map. For a Henselian discrete valuation field with perfect residue field, there is a standard definition of a residue map, as in for example [35,§XII.3], [12, §1.1], or [11, §1.4.3], where it is called the Witt residue. In our setting, this definition carries over unchanged to define a residue map ∂ ′ : Br(K nr /K) → H 1 (F, Q/Z). We will now recall this definition and verify that it is compatible with ours. First note that Br(K nr /K) = fil 0 H 2 (K), as follows from [27, Proposition 6.1(1)] and [11, p. 35].
Let δ : H 1 (F, Q/Z) → H 2 (F, Z) be the connecting map coming from the short exact sequence 0 → Z → Q → Q/Z → 0 of Galois modules. It is an isomorphism. Let ∂ ′ be the composite map where v : K × nr → Z is the valuation. Let A be the ring of integers in K, and let ι ′ be the composite of the natural maps By the same argument as [35, §XII.3, Theorem 2] and the remark following it, the sequence To state the following proposition, we make use of the exact triangle of complexes of sheaves on theétale site of any field, for any n ≥ 1. Recall also the canonical lifting map ι 2 n : H 2 (F, Z/n(1)) → fil 0 H 2 (K, Z/n(1)), which is the first component of λ π (see [27, §1.4]).
Proposition 2.12. For any integer n ≥ 1, the following diagram commutes: Here the two left-hand vertical maps come from the triangle (2.16), and the righthand one from the natural inclusion Z/n → Q/Z.
We first prove a lemma on cup products.
Lemma 2.13. Let L be a field, and let n be a positive integer. Let u : L × → H 1 (L, Z/n(1)) and t : H 2 (L, Z/n(1)) → Br L be the maps coming from the triangle (2.16). Let δ : H 1 (L, Z/n) → H 2 (L, Z) be the connecting map coming from the short exact sequence 0 → Z → Z → Z/n → 0 of Galois modules. For χ ∈ H 1 (L, Z/n) and a ∈ L × , we have δχ ∪ a = t(χ ∪ u(a)).
Note that this definition of δ agrees with the previous one when H 1 (L, Z/n) is considered as a subgroup of H 1 (L, Q/Z).
Proof. It suffices to prove the lemma separately for n invertible in L, and for n = p r where p > 0 is the characteristic of L and r ≥ 1. For n invertible in L, we have Z/n(1) = µ n , the triangle (2.16) is the Kummer sequence, and the lemma is proved in [16, proof of Proposition 4.7.1] .
For n = p r , let L s be a separable closure of L. The triangle (2.16) is the short exact sequence . Note that the above sequence is isomorphic to that obtained by taking the short exact sequence 0 → Z → Z → Z/p r → 0 and forming the tensor product with (L s ) × . The result then follows from [16,Proposition 3.4.8].
Proof of Proposition 2.12. We first express ι ′ in terms of Galois cohomology. The strict Henselisation A sh is the ring of integers in K nr and has residue field F s , a separable closure of F . The Hochschild-Serre spectral sequence, together with Pic(A sh ) = Br(A sh ) = 0 [32, Corollary IV.1.7], gives an isomorphism . Both rows of the diagram (2.17) are split exact sequences: the map χ → {ι 1 n (χ), π} is (by definition) a section of ∂; and the map χ → δχ ∪ π is a section of ∂ ′ . (Here we identify the absolute Galois group of F with Gal(K nr /K).) It is therefore enough to show that the following diagram commutes: .
That the right-hand square commutes follows from Lemma 2.13 applied to K. (Note that ι 1 n is simply the identification of Galois groups just mentioned) For the left-hand square, ι 2 n is defined separately in [27, §1.4] for n invertible in F , and for n = p r . If n is invertible in F , then the commutativity follows immediately from the definition and the Kummer sequence on A. For n = p r , it suffices to prove it for elements {χ,ā} where χ ∈ H 1 (F, Z/n) andā ∈ F × . By definition, we have {χ,ā} = χ ∪ u(ā), so Lemma 2.13 shows that the image of this element in H 2 (F, (F s ) × ) is equal to δχ ∪ā; applying ι ′ gives δχ ∪ a, where a ∈ A × is a lift ofā and we have as before identified Gal(F s /F ) with Gal(K nr /K). On the other hand, first applying ι 2 n gives {ι 1 n (χ), a} = χ ∪ u(a) and Lemma 2.13 again shows that the image in H 2 (K nr /K, K × nr ) is δχ ∪ a, as desired.

The tame part
We return to the situation of the introduction. Let k be a finite extension of Q p with ring of integers O k , uniformiser π and residue field F. Let X/k be a smooth, geometrically irreducible variety over k, and let X be a smooth O k -model of X having geometrically irreducible special fibre Y . Denote by K the function field of X and by F the function field of Y . Let K h be the field of fractions of a Henselisation of the discrete valuation ring O X ,Y .
The natural map Br X → Br K h allows us to pull back Kato's definition of the Swan conductor, and the associated filtration, to Br X. In this section we look at the smallest piece fil 0 Br X of Kato's filtration on Br X. By [27, Proposition 6.1] and [17,Corollaire 1.3], this is the same as the subgroup of Br X consisting of those elements whose image in Br K h is split by an unramified extension of K h . Equivalently, such an element is split by a finite extension L/K, where L is the field of fractions of a discrete valuation ringétale over O K = O X ,Y . To see this equivalence, note that the maximal unramified extension K h nr of K h is the field of fractions of a strict Henselisation of O K , and therefore is the colimit of all such extensions L/K. We will denote the composition fil 0 Br X → fil 0 H 2 (K h ) ∂ − → H 1 (F ) also by ∂. Recall that ∂ : Br k → H 1 (F, Q/Z) is an isomorphism, by a standard calculation of local class field theory. The main result of this section is the following.
Then the following diagram commutes: The following corollary is immediate.
(ii) For A ∈ fil 0 Br X and P ∈ X(k) reducing to a smooth point P 0 ∈ Y (F), the evaluation A(P ) depends only on P 0 . Proposition 3.1 will be used in the proof of Theorem B, in combination with the following lemma. We use 1 p to denote the map H 1 (F, Z/p) → H 1 (F, Q/Z) induced by identifying Z/p with the p-torsion in Q/Z.
On the other hand, the 1-cocycle δ 1 (x) is defined as follows: let y ∈F be such that y p − y = x; then, for σ ∈ Gal(F/F), we define δ 1 (x)(σ) = σ(y) − y ∈ Z/p. Combining these definitions gives In Section 3.1 we prove Proposition 3.1 for A ∈ Br X[p r ]. The result for Brauer group elements of order prime to p follows from comparison with the classical residue map (see Proposition 2.12) and well-known properties of the latter: see for example [7,  3.1. Evaluation of tame elements of p-power order. We first prove a lemma.
Proof. We use induction on r. For the case r = 1, it suffices to prove the statement after adjoining a pth root of unity to the base field k, and then this is [4, Proposition 6.1(i)].
For any q, m, the sheaf τ * i * R q j * Z/p r (m) on Fé t is the sheaf corresponding to the Gal(F s /F )-module H q (K h nr , Z/p r (m)). Consider the long exact sequence in cohomology on K h nr coming from the short exact sequence of Galois modules. We have a commutative diagram ) in which the vertical maps are the Galois symbols, which are surjective by [4, §5]; this shows that the bottom map is surjective. It follows that the long exact sequence of cohomology of (3.1) gives ). Consider this as a sequence of sheaves on Fé t . Applying τ * gives the bottom row of the following commutative diagram of sheaves on Y .
By induction, the two outer vertical maps are injective, and therefore the middle one is as well.
To prove Proposition 3.1(1), we will prove a result for general q, in the case that X is affine.
be the map of Section 2.2. The sequences (2.4) give a commutative diagram as follows.
By assumption χ = λ π (α, β) lies in the image of b. To show that (α, β) lies in the image of a, it is enough to prove that c is injective; but this follows from Lemma 3.4.
Proof of Proposition 3.1. We first prove part (1)  We conclude this section with an alternative description of the kernel of ∂. The natural map Br K → Br K h allows us to extend the definition of the classical residue map ∂ ′ to Br(K h nr /K) = fil 0 Br K. The following lemma is a generalisation of a result of [12, §1.1] to the case of imperfect residue field.
Proof. Let i : Spec F → Spec O X ,Y and j : Spec K → Spec O X ,Y be the inclusions of the special and generic points, respectively. As in [17, §2], where the case of perfect residue field is treated, the short exact sequence The Leray spectral sequence shows that H 2 (O X ,Y , j * G m ) is the kernel of the natural map Br K → Br K h nr . Applying the same construction to the Henselisation .
If α ∈ fil 0 Br K satisfies ∂ ′ (α) = 0, then the exact sequence (2.15) shows that the image of α in Br K h lies in the image of ι ′ , which is the image of Br A. From the above diagram it then follows that α lies in the image of Br O X ,Y .

Comparisons of some (refined) Swan conductors
In Section 8 we will prove Theorem B by induction on the Swan conductor. The induction will involve a chain of blowups with an associated decreasing sequence of Swan conductors at the exceptional divisors. Eventually, we will obtain Swan conductor zero, whereupon evaluations are controlled by the residue map as in Proposition 3.1. In order to relate these evaluations to the refined Swan conductor we started out with, we need to keep track of the refined Swan conductors at each stage and finally compute the residue of a Brauer group element along the last exceptional divisor in the chain. This last objective is served by Lemma 4.2 below and its application in the proof of Lemma 8.4. We begin by recalling some more notions from [27].
4.1. Unramified elements. Let X be a normal irreducible scheme with function field K. For x ∈ X, let K x be the field of fractions of the Henselisation O h X,x of O X,x . Following [27, §1.5], we say that an element χ of H q (K) is unramified on X if for any x ∈ X, the image of χ in H q (K x ) belongs to the image of the canonical lifting map ι q (O h X,x ) : H q (κ(x)) → H q (K x ), which is the first component of λ π (see [27, §1.4]). In the case q = 1, χ is unramified on X in this sense if and only if χ belongs to H 1 (X, Q/Z) ⊂ H 1 (K).

4.2.
(Refined) Swan conductors in a geometric setting. As in Subsection 4.1, let X be a normal irreducible scheme with function field K and let χ ∈ H q (K). For p ∈ X 1 = {x ∈ X | dim O X,x = 1}, the field K p = Frac(O h X,p ) is a Henselian discrete valuation field. Let χ p denote the image of χ in H q (K p ). Following [27, Part II], we denote by sw p χ the Swan conductor of χ p . Now suppose that χ p lies in fil n H q (K p ) for some n ≥ 1. We denote by rsw p,n χ the refined Swan conductor rsw n χ p .

4.3.
Comparisons. For the rest of this section, let A be an excellent regular local ring with field of fractions K and residue field ℓ of characteristic p > 0 such that [ℓ : ℓ p ] = p c . Let (π i ) 1≤i≤r be part of a regular system of parameters of A, let Lemma 4.1. Let j ∈ {1, . . . , r} and assume n j = 0. Write χ = λ πj (χ 1 , χ 2 ) in H q (K pj ), with χ 1 ∈ H q (κ(p j )) and χ 2 ∈ H q−1 (κ(p j )). Denote byp i the image of p i in A/p j . Then, for i = j, we have swp i (χ 1 ) ≤ n i and swp i (χ 2 ) ≤ n i .
Proof. As in Kato's proof of [27,Theorem 7.1], we reduce to the following situation: dim(A) = 2; A is complete; H c+1 p (ℓ) = 0; the order of χ is a power of p; and q ≤ c+2.

Blowing up
In this section, we show how information about the refined Swan conductor [α, β] π,n is retained under blowups. Namely, in Lemma 5.1, we show that after a blowup at a point P 0 on the special fibre, one can read off α P0 and β P0 from the residues at logarithmic poles of some relevant differentials.
First, we introduce some notation. . Let m be the maximal ideal of R. We begin by identifying some natural isomorphisms.
If we identify H 0 (Z, O Z (1)) m with H 0 (Z, O Z (1)) ⊗2 , then the map to H 0 (Z, O Z (2)) is given by x ⊗ y → xy, and so its kernel is naturally (up to a choice of sign) identified with 2 H 0 (Z, O Z (1)) under the embedding x ∧ y → (x ⊗ y) − (y ⊗ x). Combining this with the isomorphism ψ gives an isomorphism ). The proof of [19,Theorem II.8.13] leads to the following explicit description of ϕ: if x 1 , . . . , x m form a basis for m/m 2 , then We are now ready to state the main result of this section.
(1) The element π Z is a local parameter in OỸ ,Z .
(5) If furthermore α P0 = 0, then (f * α)/π r Z does not lie in Ω 2 Proof. Since the statements concern OỸ ,Z , we may work on the affine piece U of Y obtained by inverting x (1) . Denote by f U the restriction of f to U . We use standard facts about blowups: see, for example, [ i /x (1) . The ideal mR ′ is principal, generated by the image of x, proving (1). In what follows we will often abuse notation and identify R with its image in R ′ .

OỸ ,Z
; but this is not the case, since by 2 its residue is non-zero.
To prove (4), write α = i>j a ij dx i ∧ dx j with a ij ∈ R. We treat the terms separately.
For terms with j = 1, we have which has residue 1 ) 2 . This proves (4), and (5) follows as in the first case.

Some calculations for P n
This section collects some calculations for projective space. In Section 6.1 we study some spaces of differentials on projective space with poles of bounded (logarithmic) order along a hyperplane, and show to what extent they are determined by their residues. This will be used in Section 8 in the proof of Theorem B in conjunction with a result of Kato ([27, Proposition 7.3]) to transfer information about the refined Swan conductor along a chain of blowups. The relevant projective spaces will be the exceptional divisors of these blowups. In Section 6.2 we describe the graded pieces of Kato's filtration by Swan conductor on H 1 (E \ Z, Q/Z), where E is projective space over a field of characteristic p and Z is a hyperplane. This will be used in Section 8 in the proof of Theorem B once our successive blowups have reduced the Swan conductor to zero, and Proposition 3.1 has reduced our task to computing the residue map ∂ via Lemma 3.3.
6.1. Differentials with logarithmic poles along a hyperplane in P n . Let L be a field and let P n = P n L have coordinates X 0 , . . . , X n . Let H ⊂ P n be the hyperplane defined by X 0 = 0.
H 0 (P n , Ω 1 (2H)) has a basis consisting of the n(n + 1)/2 elements The n of these with i = 0 make up the aforementioned basis of the subspace H 0 (P n , Ω 1 (H + log H)). The natural morphism of sheaves Ω 1 P n → ι * Ω 1 H (where ι is the inclusion of H in P n ) gives rise to a short exact sequence in which, for 0 < i < j ≤ n, the basis element (X 2 i /X 2 0 )d(X j /X i ) maps to X 2 i d(X j /X i ) ∈ H 0 (H, Ω 1 (2)).  2), is an isomorphism. H 0 (P n , Ω 2 (2H + log H)) has a basis consisting of the n(n − 1)/2 elements Proof. We will repeatedly use the short exact sequence [19,Theorem II.8.13] which, together with the standard calculation of the cohomology groups H i (P n , O(r)), lets us calculate the cohomology of Ω 1 (r) (or, equivalently, Ω 1 (rH)). For r > 0, taking the tensor product with O(r) and taking cohomology gives the exact sequence where we have used H 1 (P n , O(r − 1)) = 0. Identifying H 0 (P n , O(r − 1)) with the space of homogeneous polynomials of degree r − 1, the map α is given by To prove (1), take r = 1; then α is an isomorphism, showing that H 0 (P n , Ω 1 (1)) and H 1 (P n , Ω 1 (1)) are both zero. It then follows easily that ρ : H 0 (P n , showing ρ(d(X i /X 0 )) = −X i | H . The elements −X i | H form a basis of H 0 (H, O(1)), showing that the d(X i /X 0 ) form a basis for H 0 (P n , Ω 1 (H + log H)).
To prove (2), take r = 2; then a basis for the kernel of α is given by the n(n+1)/2 elements having X i in the jth position and −X j in the ith position, for i < j. Therefore H 0 (P n , Ω 1 (2H)) has dimension n(n + 1)/2.
We can now prove by induction on n that the claimed elements do indeed form a basis. For n = 1 the dimension is 1 and it is clear. Assuming the statement to be true for P n−1 , the elements X 2 i d(X j /X i ) with 0 < i < j ≤ n form a basis for H 0 (H, Ω 1 (2)). Now, of the claimed basis elements (X 2 i /X 2 0 )d(X j /X i ), those with i = 0 form a basis for H 0 (P n , Ω 1 (H + log H)) by (1) and restrict to zero on H. Those with i > 0 map bijectively onto our basis for H 0 (H, Ω 1 (2)), showing that all together they form a basis for H 0 (P n , Ω 1 (2H)).
(2) For p ∤ n, there is an isomorphism is any lift ofF and is the Artin-Schreier map. The refined Swan conductor is given by Proof. There is an exact sequence whereĒ,Z are base changes of E, Z to a separable closure of F. By [27, Proposition 6.1], the piece fil 0 H 1 (E \ Z, Q/Z) is generated by the prime-to-p torsion in H 1 (E \ Z, Q/Z) together with the elements unramified at Z, that is, the image of H 1 (E, Q/Z). We have E ∼ = P m F and E \ Z ∼ = A m F . Therefore H 1 (Ē \Z, Q/Z) has no prime-to-p torsion, showing that the prime-to-p torsion in H 1 (E \ Z, Q/Z) all comes from H 1 (F, Q/Z). Moreover, H 1 (Ē, Q/Z) is trivial, showing that the image of H 1 (E, Q/Z) also coincides with H 1 (F, Q/Z). This proves (1).
Since the filtration on H 1 (E \ Z, Q/Z) is obtained by pulling back that on H 1 (κ(E) Z ), the resulting map is injective. By [27, Lemma 3.6], for p ∤ n there is a surjection By [27,Lemma 3.7], the resulting element has refined Swan conductor In particular, this shows that h is an isomorphism for p ∤ n. We claim that the image of H 0 (Z, O(n)) under the injective mapF →F /X n 1 corresponds under the isomorphism h to gr n H 1 (E \ Z, Q/Z).
Indeed, we have h(F /X n 1 ) = 1 p δ 1 (F/X n 0 ), which is unramified outside Z. On the other hand, if χ is an element of gr n H 1 (E \ Z, Q/Z) then we write χ = 1 p δ 1 (xπ −n ) and consider its refined Swan conductor. By [27, Theorem 7.1] applied to the local rings of all points in Z, we see thatx is regular on Z apart from a pole of order at most n along X 1 = 0. (For points where π is a local equation for Z this follows immediately; at other points of Z a simple change of variables is needed.) Thusx is of the formF /X n 1 for someF ∈ H 0 (Z, O(n)), as claimed.

Tangent vectors
We return to the setting and notation of Theorem B, the statement of which involves various tangent vectors. In Lemma 7.1 we collect some well-known facts relating lifts of points to tangent vectors. In Lemma 7.3 we show how to keep track of these tangent vectors when blowing up our scheme X at a point P 0 on the special fibre.
For r ≥ 1 we write q r for the reduction map X (O k ) → X (O k /π r O k ), where π denotes a uniformiser of k. For P ∈ X (O k ) we use B(P, r) to denote the set q −1 r (q r (P )) of points Q ∈ X (O k ) such that Q has the same image as P in X (O k /π r ). Lemma 7.1. There is a function B(P, r) → T P0 (Y ), which we denote as Q → [ − − → P Q] r , depending on the choice of uniformiser π and with the following properties.
(1) The function factors as q r+1 followed by a bijection from q r+1 (B(P, r)) to T P0 (Y ).
(2) For a point Q ∈ B(P, r) and a regular function f ∈ O X ,P0 , we have (3) Let k ′ /k be a finite extension, with F ′ /F the extension of residue fields, and let X ′ and Y ′ be the base changes of X to k ′ and Y to F ′ , respectively. Let P ′ ∈ X ′ (k ′ ) be the base change of P . Fix a uniformiser π ′ in k ′ and write π = c(π ′ ) e with c ∈ O × k ′ , so that e is the ramification index of k ′ /k. Letc denote the image of c in F × . Then the diagram q r+1 (B(P, r)) Proof. One explicit way to see this is as follows. Write d = dim X. Since X → O k is smooth at P 0 , there is a neighbourhood of P 0 that embeds into A n O k as the zero set of n − d polynomials f 1 , . . . , f n−d . Such an embedding induces an embedding of the tangent space is a vector. Using the Taylor expansion, the condition that q r+1 (Q) lie in X can be written as (7.2) (f 1 (Q), . . . , f n−d (Q)) = (f 1 (P ), . . . , f n−d (P )) + π r J(P )v ≡ 0 (mod π r+1 ), where J is the (n − d) × n Jacobian matrix of partial derivatives of the f i . Let v ∈ F n be the reduction of v modulo π; the reduction of J(P ) modulo π is J(P 0 ). The condition (7.2) is equivalent to J(P 0 )v = 0, which simply says thatv lies in the tangent space T P0 (Y ); because Y is smooth at P 0 , this is an F-vector space of dimension d. So every point Q ∈ B(P, r) gives rise to a vectorv ∈ T P0 (Y ), and we define [ − − → P Q] r =v. Conversely, everyv ∈ T P0 (Y ) gives a solution to (7.2), which by Hensel's Lemma lifts to a point of B(P, r). This defines the bijection of (1).
For (2), take Q ∈ B(P, r) and write as before Q = P + π r v, where v ∈ O n k has reductionv lying in T P0 (Y ). The function f extends to a regular function on a neighbourhood of P 0 in A n O k , and we denote the extension also by f . Taylor expansion gives This depends only onv, and the restriction of ∇f (P ) to T P0 (Y ) is df P0 , proving (2). Also, property (2) characterises the bijection and does not depend on the embedding used to define it, showing that the bijection itself does not depend on the embedding.
The statement (3) follows easily from the definitions using Remark 7.2. The canonical bijection is between q r+1 (B(P, r)) and the vector space , as can be seen by applying (3) with k = k ′ . See, for example, [1,§III.5]. That gives a bijection independent of the choice of π. However, since we will use the formula (2), we opt for the explicit rather than the canonical choice.

Tangent vectors and blowups.
Let m denote the maximal ideal of O Y,P0 . Let f :X → X be the blowup of X at P 0 and let E be the exceptional divisor, isomorphic to P m F . LetỸ denote the strict transform of Y . The linear form π (1) ∈ H 0 (E, O(1)) cuts out a hyperplane in E which is E ∩Ỹ . Its complement U is naturally isomorphic to Spec Sym(m/m 2 ), the affine space corresponding to the vector space m/m 2 . To make this explicit, choose a system of local parameters π, x 1 , . . . , x m in O X ,P0 ; then π (1) , x (1) If r > 1, thenQ ∈ B(P , r − 1) and the dual map θ ∨ : Proof. Work on the affine piece ofX corresponding to π (1) ; then f is defined by x i = πu i for 1 ≤ i ≤ m and (7.1) yields Hence, u i (Q) ≡ u i (P ) (mod π r−1 ) for 1 ≤ i ≤ m, wherebyQ ∈ B(P , r − 1). For r > 1, (7.1) yields and comparing (7.3) and (7.4) proves (1). To prove (2), observe that for r = 1, (7.3) gives since u i (P 0 ) = 0.

Proof of Theorem B
We continue with the setting and notation of Theorem B. Let f :X → X be the blowup of X at P 0 and let E be the exceptional divisor, isomorphic to P m F . LetỸ ,P ,Q,R denote the strict transforms of Y, P, Q, R and let Z = E ∩Ỹ . Let P 0 ,Q 0 ,R 0 ∈ E(F) be the reductions ofP ,Q,R, respectively. Let K denote the function field of X.
We begin by noting that the claim in Theorem B that α and β are regular at P 0 follows from [27,Theorem 7.1] m ∈ H 0 (X , O(1)) give a system of homogeneous coordinates on E ∼ = P m F , in which Z ⊂ E is cut out by π (1) = 0. Oñ X we have πx are local parameters at the divisorsỸ and E respectively The first two results in this section relate the Swan conductor and refined Swan conductor at E of f * A to those of A.
Since F is perfect, A is not strongly clean with respect to O X ,P0 , so [27,Theorem 8.1] shows sw E (f * A) < n. Write sw E (f * A) = n − r for some r ≥ 1.
If, on the other hand, we have β P0 = 0, then either r = 1 and the above calculation gives β E = 0; or r > 1 and α E , β E both vanish; in either case, the claimed equation for β E holds.
Taking s = 0 in Lemma 8.2 below shows that when β = 0 the Swan conductor drops further and the refined Swan conductor at E of f * A is related to α P0 . Lemma 8.2. Suppose n > 0 and p | n. Let s ≥ 0 and suppose sw(A) = n − 2s and rsw n−2s (A) = [α s , β s ] π,n−2s with α s = i>j a ij dx i ∧ dx j and β s = s i>j a ij x 2 In particular, if α s,P0 = 0 then sw E (f * A) = n − 2(s + 1).
It remains to prove the claims concerning the refined Swan conductor at level n − 2(s + 1). Write sw E f * A = n − 2(s + 1) − r for some r ≥ 0. Then (8.1) gives (8.5) whereupon Lemma 5.1 shows that if r > 0 then α s,P0 = 0. This implies that all the a ij (P 0 ) are zero, which proves the claimed equality in the case r > 0, since then α E and β E are zero by definition of the refined Swan conductor at level n − 2(s + 1).
Let us assume henceforth that α s,P0 = 0 and hence sw E (f * A) = n − 2(s + 1). Now (8.2) becomes (8.6) Applying [27,Theorem 7.1] to all local rings of E gives β E ∈ Ω 1 E (2Z + log Z) and α E ∈ Ω 2 E (2Z +log Z). Now we apply [27, Proposition 7.3] several times. Comparing the dlog π E ∧ dlog πỸ terms in (8.5) and (8.6) shows that the residue of π 2 Y β E along Z ⊂ E equals the residue of −π −2 E f * β s along Z ⊂Ỹ . We now show that this residue is zero. Recalling that f * x 1 = π E , we have which has zero residue along Z ⊂Ỹ . Therefore, π −2 E f * β s ∈ Ω 1 OỸ ,Z and β E ∈ Ω 1 E (2Z). Comparing the dlog π E terms in (8.5) and (8.6) shows that (π 2 Y β E )| Z equals the sum of the residue of π −2 E f * α s along Z ⊂Ỹ and π −2 E f * β s | Z . Lemma 5.1 shows that the residue of By Lemma 6.1 (2), this proves the claim regarding β E . Now compare the dlog πỸ terms in (8.5) and (8.6) to see that π −2 E f * β s | Z is equal to the sum of the residue of π 2 Y α E along Z ⊂ E and (π 2 Y β E )| Z . By our calculations above, this implies that the residue of π 2 (3) shows that this residue is equal to ρ(α E )/(x (1) ) 2 and hence α E = i>j a ij (P 0 )du i ∧ du j . The next two results deal with the end game, where the Swan conductor at E is zero and our task is to compute the residue ∂ E (f * A). Proof. We begin as in the proof of Lemma 8.1 to obtain sw E (f * A) = 0 and  Proof. We begin as in the proof of Lemma 8.2 to obtain sw E (f * A) = 0 and Write in whichω is equal to the residue of π −2 E f * α at Z, that is (2)) be any quadratic form such that dF restricts to χ(α P0 ) on Z. To see that such an F exists, write α = i>j a ij dx i ∧ dx j with a ij ∈ O Y,P0 . Then since we are in characteristic p = 2. Now let δ 1 : κ(E) Z → H 1 p (κ(E) Z ) denote the Artin-Schreier map. Note that δ 1 (F/(π (1) ) 2 ) is unramified outside Z and hence 1 p δ 1 (F/(π (1) ) 2 ) ∈ H 1 (E \ Z, Q/Z). By [27, Lemmas 3.6 and 3.7], 1 p δ 1 (F/(π (1) ) 2 ) ∈ H 1 (κ(E) Z ) has refined Swan conductor The result now follows from (8.8), (8.9), (8.10) and the injectivity of the refined Swan conductor on the graded pieces of Kato's filtration by Swan conductor.
We now prove Theorem B(1) in the case n = 1. This will form the basis for a proof of Theorem B(1) by induction.
We now prove Theorem B(1) by induction.
Proof of Theorem B(1). Let N ≥ 1 and suppose that we have proved Theorem B(1) for all n ≤ N . Our task is to prove it for n = N + 1. Suppose that A ∈ fil N +1 Br X, Now we turn our attention to the proof of Theorem B (2) and (3). We begin with the first statement of Theorem B(2), which is the content of the next lemma.
where A = f * A(Q) − f * A(P ) ∈ Br k. By Proposition 3.1, Lemma 8.5 and Lemma 6.2, A ∈ Br k [2] and (2)) is any quadratic form such that dF restricts to χ(α P0 ) on Z, G ∈ H 0 (E, O(1)) is a linear form, and δ 1 : is the Artin-Schreier map. Recalling that the characteristic is 2, we can take F = a ij x For the second claim, let Q(v) = a ij dx i (v)dx j (v) and let the associated bilinear form be Note that B(v, w) = (Q(v+w)+γ(v+w))−(Q(v)+γ(v))−(Q(w)+γ(w)). It suffices to show the existence of v, w ∈ T P0 (Y ) such that Tr F/F2 (B(v, w)) = 0, since then at least one of Now the non-degeneracy of the trace form shows the existence of λ ∈ F such that Tr F/F2 (λB(v ′ , w ′ )) = 0, whence the result.

Now we complete the proof of Theorem B(2).
Proof of Theorem B(2). We first introduce notation for successive blowups. Let − → X be the blowup of X at P 0 with exceptional divisor E 1 and let Z 1 = E 1 ∩Ỹ .
Let P 1 ∈ X 1 (O k ) be the section lifting P and let X 2 f2 − → X 1 be the blowup at the closed point P 1,0 of P 1 . Iterating this construction gives a sequence of blowups Write E i for the exceptional divisor of the ith blowup and let P i , Q i , R i ∈ X i (O k ) be the sections lifting P, Q, R, respectively. Let m be the maximal ideal of O X ,P0 and let π, x 1 , . . . , x m be a basis for m/m 2 . Let u 1,j = x (1) j /π (1) so that u 1,1 , . . . , u 1,m restrict to a system of affine coordinates on 1,j /π (1) so that u 2,1 , . . . , u 2,m restrict to a system of affine coordinates on E 2 \ Z 2 , and so on and so forth.
Write α = i>j a ij dx i ∧dx j for some a ij ∈ O Y,P0 . By Lemma 8.2, sw E1 (f * 1 A) ≤ n − 2, with equality if α P0 = 0 and, furthermore, rsw E1,n−2 (f * ). Let C 1 be the explicit Brauer group element constructed in Lemma 8.9 with sw E1 (C 1 ) ≤ n − 2 such that rsw E1,n−2 ( Now we blow up at P 1,0 = Q 1,0 . Applying Lemma 8.2 to B 1 and repeating the argument above, we can write f * Continuing in this way, after s blowups we obtain, on the generic fibre of some open neighbourhood V s of P s,0 in X s , where sw Es (B s ) ≤ n − 2s and rsw Es,n−2s (B s ) = [α Es , β Es ] π,n−2s with α Es = i>j a ij (P 0 )du s,i ∧ du s,j and β Es = s i>j a ij (P 0 )u 2 s,j d(u s,i /u s,j ). We have gathered all the terms coming from the C i 's into D s .
Note that R s ∈ B(Q s , n − 2s) so Theorem B(1) gives A calculation shows that β Es,Qs,0 ([ Define g A,Q (R) to be inv D s (R s ) − inv D s (Q s ). We claim that this does not depend on the choice of a suitable s. In other words, suppose that Q ∈ B(P, s) for some s with 1 < s < n/2. Then we can also consider Q as lying in B(P, s− 1). Our claim is that for R ∈ B(Q, n − s For R ∈ B(Q, n − s + 1), we have R s ∈ B(Q s , n − 2s + 1), whereby [ − −− → Q s R s ] n−2s = 0. Therefore, Theorem B(1) shows that inv C s (R s ) = inv C s (Q s ), whence the claim.
There exists an open neighbourhood V of P 0 in X with generic fibre V and an element C ∈ fil n Br V [p t+1 ] such that rsw E,n (C) = [0, β] π,n . Explicitly, write K for the function field of X and let if p | n − te.
Proof. By Lemma 6.1(1), and let π ′ be a uniformiser of k ′ . Let K denote the field of fractions of the Henselisation of the function field of X with respect to the valuation given by π E and let K ′ = Kk ′ . Let F denote the common residue field of K and K ′ and let K h and (K ′ ) h be their respective Henselisations. Let w = (π ′ ) −ε N k ′ /k π ′ and letw ∈ F × be its reduction. The Brauer group element C will be constructed via corestriction. We begin by proving the following claims.
Let ζ p t+1 ∈ k ′ be a fixed choice of primitive p t+1 th root of unity in k ′ . The choice of ζ p t+1 yields choices of primitive p s th roots of unity for 1 ≤ s ≤ t + 1 by setting . For 1 ≤ s ≤ t + 1 and x, y ∈ (K ′ ) × , let (x, y) p s ∈ Br K ′ [p s ] denote the class of the corresponding cyclic algebra, which depends on our chosen primitive p s th root of unity. Alternatively, as in [35,§XIV.2], (x, y) p s can be constructed as a cup product as follows: let δ : (K ′ ) × /(K ′ ) ×p s → H 1 (K ′ , µ p s ) be the Kummer isomorphism, and take the image of (δ(x), δ(y)) under the composition where we have used the choice of ζ p s to give an isomorphism µ ⊗2 First suppose 0 < n < e ′ and p ∤ n. Let By [27,Proposition 4.1 and Lemma 4.3], C ′ ∈ fil εn Br K ′ and rsw εn (C ′ ) = [0,ā −n β] π ′ ,εn , as desired. Now suppose 0 < n < e ′ and p | n.
has the desired Swan conductor and refined Swan conductor.
for some b i ∈ F and we set γ = − b i dx i . The proof of Lemma 8.7 goes on to show the existence of v, w ∈ T P0 Y such that Then |A| takes p distinct values on B(Q A , n − 1). We will show that |A| : B(Q A , n − te − 1) → p −(t+1) Z/Z is surjective. The case t = 0 follows immediately from (8.15) and (8.16). Now suppose we have proved the result for t 0 and we want to prove it for t 0 + 1. By Lemma 2.10, sw(pA) = n − e and rsw n−e (pA) = [ūα,ūβ] π,n−e . Applying Lemma 2.10 to f * A shows that sw(f * (pA)) = n − 2 − e and rsw E,n−2−e (f * (pA)) = [ūα E ,ūβ E ] π,n−2−e . The construction of γ detailed above shows that for Q ∈ B(P, 1) and R ∈ B(Q, n − e − 1) .

Proof of Theorem A
We now prove Theorem A. For ease of notation we define a modified version of Kato's filtration as follows.
For the purposes of the definition, K h could be replaced by any Henselian discrete valuation field of characteristic zero. Pulling back from Br K h to Br X gives a filtration on Br X whose pieces we denote by fil n Br X.
Lemma 9.1. For n ≥ −2, we have fil n Br X ⊂ Ev n Br X.
The reverse inclusions will be given by the following lemmas.
Lemma 9.2. Let n ≥ 1, and let A be an element of fil n Br X \ fil n−1 Br X. Then A does not lie in Ev n−1 Br X.
Proof. Since A ∈ fil n Br X, we have A ∈ fil n+1 Br X and rsw n+1 (A) = [α, 0] π,n+1 for some α ∈ Ω 2 F . By Theorem B , α lies in Ω 2 (Y ). Suppose first that α = 0. Let Z ⊂ Y be the zero locus of α, which by assumption is a strict closed subset of Y , and set U = Y \ Z. By the Lang-Weil estimates [29], there is a finite extension F ′ /F such that U (F ′ ) is non-empty. Let k ′ /k be the unramified extension of k having residue field F ′ . Choose any P 0 ∈ U (F ′ ) and lift it (by Hensel's Lemma) to a point P ∈ X (O k ′ ). By Lemma 2.7 we have res k ′ /k A ∈ fil n Br X k ′ and rsw n+1 (res k ′ /k A) = rsw n+1 (A). Since α P0 = 0, Theorem B(3) shows that there exists Q ∈ B(P, 1) such that |A| takes p distinct values on B(Q, n). It follows that A / ∈ Ev n−1 Br X. Now suppose that α = 0. Then A ∈ fil n Br X. Let rsw n (A) = [α ′ , β ′ ] π,n for some (α ′ , β ′ ) ∈ Ω 2 (Y ) ⊕ Ω 1 (Y ). Note that β ′ = 0 since A / ∈ fil n−1 Br X. Then by the same argument as above, there exists a finite extension F ′ /F and a point P 0 ∈ Y (F ′ ) satisfying β ′ P0 = 0. Let k ′ /k be the unramified extension with residue field F ′ and lift P 0 to a point P ∈ X (O k ′ ). Now Theorem B(1) shows that |A| takes p distinct values on B(P, n), whereby A / ∈ Ev n−1 Br X.
Lemma 9.3. For n ≥ 0, we have Ev n Br X ⊂ fil n Br X.
Proof. Take A ∈ Ev n Br X. Let r be the smallest non-negative integer such that A ∈ fil r Br X, and suppose that r > n. By Lemma 9.2, A does not lie in Ev r−1 Br X, which contains Ev n Br X, giving a contradiction. WriteȲ for the base change of Y to an algebraic closure of F. Since Y is geometrically connected, the Hochschild-Serre spectral sequence gives a short exact sequence 0 → H 1 (F, Z/p r ) → H 1 (Y, Z/p r ) → H 1 (Ȳ , Z/p r ). If ∂(A) lies in H 1 (F, Z/p r ), then Proposition 3.1 shows that the corresponding evaluation map X (O k ) → Br k is constant and non-zero. If X (O k ) is non-empty, then this proves A / ∈ Ev −2 Br X[p r ]. Otherwise, we can use Lang-Weil to pass to an extension k ′ /k of degree prime to p where X (O k ′ ) is non-empty, and we obtain the same result.
On the other hand, suppose that ∂(A) does not lie in H 1 (F, Z/p r ). To prove that A does not lie in Ev −1 Br X[p r ], we may change A by a constant algebra. Writē ∂(A) for the image of ∂(A) in H 1 (Ȳ , Z/p r ). Then∂(A) has order p s , with s ≥ 1, and p s ∂(A) is an element of order dividing p r−s in H 1 (F, Z/p r ), which is cyclic of order p r . Therefore there exists α ∈ H 1 (F, Z/p r ) satisfying p s α = p s ∂(A). Let A ′ ∈ Br k[p r ] satisfy ∂(A ′ ) = α. Replacing A by A − A ′ , we reduce to the case where ∂(A) and∂(A) have the same order p s .
The class ∂(A) lies in the subgroup H 1 (Y, Z/p s ) ⊂ H 1 (Y, Z/p r ). Let T → Y be a Z/p s -torsor representing this class; since its image in H 1 (Ȳ , Z/p s ) also has order p s , [7,Lemma 5.15] shows that the variety T is geometrically connected. As it is smooth, it is also geometrically irreducible. The image of T (F) → Y (F) consists of those points P 0 ∈ Y (F) such that ∂(A) maps to 0 under the induced map Similarly, for any a ∈ H 1 (F, Z/p s ), let T a → Y be a torsor representing the class ∂(A) − a; then the image of T a (F) → Y (F) consists of those P 0 satisfying P * 0 (∂(A)) = a. For any fixed a, it follows from Lang-Weil that T a has points over any sufficiently large extension of F. Therefore, for some extension F ′ /F, there exist P 0 , Q 0 ∈ Y (F ′ ) satisfying P * 0 (∂(A)) = Q * 0 (∂(A)) in H 1 (F ′ , Z/p s ). Let k ′ /k be the unramified extension with residue field F ′ , and let P, Q be lifts of P 0 , Q 0 to X (O k ′ ). By Proposition 3.1, we have This completes the proof of Theorem A.

Comparison with other filtrations
Throughout this section, let K denote a Henselian discrete valuation field of characteristic zero.
There are several other constructions in the literature which give rise to filtrations on Br K, and the question naturally arises as to whether our filtration { fil n Br K}, as defined at the beginning of Section 9, coincides with any of these. In this section we look at the relationships between several existing filtrations and ours. We consider two sources of filtrations: existing filtrations on H 1 (K), which give rise to filtrations on Br K via the cup product; and ramification filtrations on the absolute Galois group of K, which give rise to filtrations on Br K by considering those elements in the kernel of restriction to the subgroups in the filtration.
In Proof. We use Bloch-Kato's explicit description of the graded pieces of the filtration, as described in [27,Theorem 4.1(6)]. Fixing a primitive pth root of unity in K gives an isomorphism H 1 p (K) ∼ = K × /(K × ) p , under which Kato's filtration on H 1 p (K) corresponds to the reverse of the natural filtration on K × . There are now several cases to consider.
• If n = 0, then χ ∈ fil 0 H 1 p (K) and it follows that sw({χ, y}) = 0 for all y ∈ K × . • If 0 < n < e ′ , then χ corresponds to an element (1 + xπ e ′ −n ) ∈ K × /(K × ) p with x ∈ O × K . Letx ∈ F be the reduction of x. First suppose that p ∤ n. Let y ∈ O × K be an element satisfying dȳ = 0; such an element exists since F is not perfect. Thenx dȳ y ∈ Ω 1 F is non-zero, and by the first isomorphism of [27, (4 G, we can obtain an ascending filtration on H q n (K) by taking the kernels of the restriction maps H q n (K) = H q (G, Z/n(q − 1)) → H q (G i , Z/n(q − 1)). In the case of perfect residue field, the ramification groups with the upper numbering give a well-studied filtration on G: see [35,Ch. IV]. In the general setting, Abbes and Saito [3] made two definitions of ramification groups, (G a ) a∈Q ≥0 and (G a log ) a∈Q ≥0 , called "non-logarithmic" and "logarithmic". In the case of perfect residue field, these coincide (up to a shift in numbering) but in general they are different.
Each of these ramification filtrations gives a filtration on H 1 (K) = Hom(G, Q/Z), and one might naturally ask whether those filtrations are related to those described in Section 10.1. This is indeed the case: Kato and Saito [28] have proved that Kato's filtration on H 1 (K) coincides with that induced by the logarithmic ramification filtration; and Saito [34] has proved in the case of positive characteristic that Matsuda's non-logarithmic variant of Kato's filtration on H 1 (K) coincides with that induced by the non-logarithmic ramification filtration. We will show that our modified Kato filtration on Br K[p] = H 2 p (K) is not induced by either of the Abbes-Saito filtrations (where the numbering of the non-logarithmic filtration is shifted by 1).
Proof. Since K contains a primitive pth root of unity, Kato's filtration on H 1 p (K) coincides with that of Bloch-Kato (see [27,Proposition 4.1(6)]). This gives explicit generators for the graded pieces of the right-hand filtration, so it is just a case of calculating the conductors of the corresponding cyclic extensions. This is accomplished in the following series of lemmas by finding the minimal polynomial of a generator for the ring of integers in each extension and applying [3, Lemma 6.6].
The calculations in the following lemmas are standard and probably well known. Lemma 10.6. Suppose that K contains a primitive pth root of unity. Let χ ∈ H 1 (K, Z/p) correspond to the extension K( p √ 1 + xπ m )/K, where x ∈ O × K , 0 < m < e ′ and p ∤ m. Then f K (χ) = e ′ + 1 − m.
Proof. Let L = K( p √ 1 + xπ m ) and let ̟ = p √ 1 + xπ m − 1. Write 1 = rm + sp for r, s ∈ Z. Considering the terms of smallest valuation in the minimal polynomial of ̟ shows that ̟ r π s is a uniformiser for L and hence O L = O K [̟ r π s ]. Now apply [3, Lemma 6.6].
Lemma 10.7. Suppose that K contains a primitive pth root of unity. Let x ∈ O × K be such thatx ∈ F is not a pth power. Let χ ∈ H 1 (K, Z/p) correspond to the extension K( p √ 1 + xπ np )/K, where 0 < np < e ′ . Then f K (χ) = e ′ − np.
We now move to q = 2 and show that the filtration { fil n Br K[p]} is not in general induced by either of the Abbes-Saito ramification filtrations, beginning with the non-logarithmic filtration.
Proposition 10.8. Suppose that K contains a primitive pth root of unity and that the residue field F of K is not perfect. Then it is not true that, for all n ≥ 0, Proof. We will show that the equality does not hold for n = e ′ . Let x be an element of F \ F p , letx ∈ O K be a lift of x and let ψ ∈ H 1 (K, Z/p) correspond to the extension K( p Now we treat the logarithmic filtration, by showing that its behaviour under field extension differs from that of our filtration. For each finite extension L of K contained inK, let {G a L,log } be the logarithmic filtration on G L = Gal(K/L). Proposition 10.9. Suppose Ω 2 F = 0. It is not true that, for all finite extensions L/K, we have fil n H 2 p (L) = {χ ∈ H 2 p (L) | f log L (χ) ≤ n}. Proof. Suppose for contradiction that the statement is true. We may assume that K contains a primitive pth root of unity. Let x, y ∈ F be such that ω = dx x ∧ dy y = 0, and letx,ỹ ∈ O K be lifts of x, y respectively. Define A = (x,ỹ) p ∈ Br K. By [27, Proposition 4.1 and Lemma 4.3], we have A ∈ fil e ′ Br K[p], and rsw e ′ (A) = [cω, 0] π,e ′ wherec ∈ F is non-zero. Therefore A lies in fil e ′ −1 Br K[p], and by assumption f log K (A) ≤ e ′ − 1. Let L/K be any wildly ramified extension of degree p. The inclusion G pa L,log ⊂ G a log for all a ≥ 0 (see [3]) implies f log L (A) ≤ p(e ′ − 1), and so the image of A in Br L lies in fil p(e ′ −1) Br L[p]. However, the same calculation as before shows that rsw e ′ L (A) = [cω, 0] π,e ′ L , with e ′ L = pe ′ , and so the image of A lies in fil pe ′ −1 Br L[p] but not fil pe ′ −2 Br L[p], giving a contradiction.

Applications to the Brauer-Manin obstruction
Let V be a smooth, proper, geometrically irreducible variety over a number field L such that V (A L ) = ∅. The surjectivity results described in Theorem B have implications for the existence of Brauer-Manin obstructions to the Hasse principle and weak approximation on V , as follows. Suppose that B has order n in Br V , and that p is a finite place of L such that the evaluation map |B| : V (L p ) → Br L p [n] is surjective. Write V (A L ) B for the subset of adèlic points of V that are orthogonal to B under the Brauer-Manin pairing; this contains V (A L ) Br . Our freedom to adjust the value taken by B at the place p shows that In other words, B does not obstruct the Hasse principle on V , but it does obstruct weak approximation on V . Note that in order to show that B obstructs weak approximation on V , it suffices that |B| : V (L p ) → Br L p be non-constant. The existence of Brauer group elements with non-constant evaluations at primes of good ordinary reduction is the subject of Theorem C, which we now prove.
Proof of Theorem C. Let V p be the base change of V to L p , and choose a smooth model V of V p over the ring of integers of L p such that the special fibre Y is ordinary. LetV p denote the base change of V p to an algebraic closure of L p . The spectral sequences [4, 0.2] E s,t 2 = H s (Ȳ ,ī * R tj * Z/p r ) =⇒ H s+t (V p , Z/p r ) define decreasing filtrations on H q (V p , Z/p r ) for all r, and also on H q (V p , Z p ) and H q (V p , Q p ). For any of these filtrations, let gr i denote the graded pieces. By [4, Theorem 0.7(iii)], we have gr 0 H 2 (V p , Q p ) = 0. Therefore gr 0 H 2 (V p , Z p ) is also non-zero, and so gr 0 H 2 (V p , Z/p r ) is non-zero for some r ≥ 1.
LetL be the algebraic closure of L inside our chosen algebraic closure of L p , and letV be the base change of V toL. By proper base change [32, Corollary VI.2.6], the natural map H 2 (V , Z/p r ) → H 2 (V p , Z/p r ) is an isomorphism. Let α ∈ H 2 (V , Z/p r ) have non-zero image in gr 0 H 2 (V p , Z/p r ). Replacing L by a finite extension, we may assume that α is defined over L and that L contains the p r th roots of unity. We fix an isomorphism Z/p r ∼ = Z/p r (1) on V , and view α as an element of H 2 (V, Z/p r (1)).
We will show that the image of α in Br V p does not lie in fil 0 Br V p . Let K h be the Henselisation of the function field K = L p (V ) at the discrete valuation corresponding to Y , and let K h nr be its maximal unramified extension. Comparing the spectral sequences of vanishing cycles for V p and K h gives a commutative diagram ) in which gr 0 H 2 (V p , Z/p r (1)) is the image of f , and fil 0 H 2 p r (K h ) is the kernel of res. By construction, f (α) is non-zero. By Lemma 3.4, g is injective, showing that g(f (α)) is non-zero. So the image of α in H 2 (K h ) does not lie in fil 0 H 2 (K h ).
Let A be the image of α in Br V . By Theorem A, after possibly replacing L by a further finite extension, the evaluation map |A| : V (L p ) → Br L p is non-constant, showing that A obstructs weak approximation on V .
Our final task is to prove Theorem D. We begin by gathering some criteria which can be used to show that various graded pieces of the filtration on Br X vanish. Lemma 11.1 is not actually used in the proof of Theorem D but is included as a first example of how one can deduce information about Br X from properties of the special fibre. Proof. If A is an element of fil n Br X for n ≥ 1, then [27,Theorem 7.1] shows that rsw n,π (A) actually lies in H 0 (Y, Ω 2 Y ) ⊕ H 0 (Y, Ω 1 Y ) = 0. This shows fil n Br X = fil n−1 Br X for all n ≥ 1, and so fil 0 Br X = Br X.
Thus we have fil n Br X = fil n−1 Br X for all n ≥ 1, and so fil 0 Br X = Br X. Proof. Firstly, the group H 1 (Ȳ , Z/p r ) is trivial for all r: it is an Abelian p-group and its p-torsion subgroup H 1 (Ȳ , Z/p) is trivial. Now, for every r, the Hochschild-Serre spectral sequence gives a short exact sequence showing that the natural map H 1 (F, Z/p r ) → H 1 (Y, Z/p r ) is an isomorphism. The result then follows from Proposition 3.1.
Lemma 11.4. Let X → O k be a smooth proper morphism such that the generic fibre X is geometrically integral. Let n be a positive integer and suppose H 1 (X, Z/n) = 0. Then the special fibre Y satisfies H 1 (Ȳ , Z/n) = 0.
Proof. Let k ′ be a finite extension of k, with ring of integers O k ′ and residue field F ′ . Let X ′ = X × O k O k ′ be the base change and denote its special and generic fibres by X ′ and Y ′ respectively. X ′ is proper over O k ′ , so the proper base change theorem gives an isomorphism H 1 (Y ′ , Z/n) ∼ = H 1 (X ′ , Z/n). On the other hand, X ′ is an open subset of the normal integral scheme X ′ , so the natural map H 1 (X ′ , Z/n) → H 1 (X ′ , Z/n) is injective. We deduce that H 1 (Y ′ , Z/n) injects into H 1 (X ′ , Z/n). Taking the limit over all finite extensions k ′ /k shows that H 1 (Ȳ , Z/n) injects into H 1 (X, Z/n) = 0.
Proof of Theorem D. Since V is smooth and proper over L, there exists a smooth proper model V → Spec O S for some finite set S of places of L containing all the infinite places. The assumption that PicV be torsion-free implies H 1 (V, O V ) = 0 and hence, by Hodge theory, H 0 (V, Ω 1 V ) = 0. For a finite place p / ∈ S, denote by V(p) the fibre V × OS k(p). Semi-continuity shows that, after possibly enlarging S, we have H 0 (V(p), Ω 1 V(p) ) = 0 for all p / ∈ S. Let n be any positive integer. Since PicV is torsion-free, the Kummer sequence gives H 1 (V , Z/n) ∼ = H 1 (V , µ n ) = 0. Suppose that p is a place of L not contained in S. By [32, VI.2.6], we have H 1 (V ×LL p , Z/n) = 0, and Lemma 11.4 applied to V × OS O Lp shows H 1 (V(p), Z/n) = 0.
We enlarge S to include all finite places p whose absolute ramification index e p satisfies e p ≥ p−1, where p is the residue characteristic of p. (It is enough to include all primes ramified in L and all primes above 2.) Let p be a place not in S, of residue characteristic p. Lemma 11.2 and Lemma 11.3 show that, for any A ∈ Br V {p}, the evaluation map |A| : V (L p ) → Br L p is constant. [10, Proposition 2.4] proves the same for A ∈ Br V of order prime to p, completing the proof.