Under consideration for publication in Math. Proc. Camb. Phil. Soc.  1

Efficient evaluation of the Brauer–Manin obstruction

By MARTIN BRIGHT

Department of Mathematics, University of Liverpool, Peach Street, Liverpool L69 7ZL

                         mjbright@liverpool.ac.uk

(Received )

Abstract

The Brauer–Manin obstruction is a concept which has been very effective in finding counter-examples to the Hasse principle, that is, sets of polynomial equations which have solutions in every completion of the rational numbers but have no rational solutions. The standard way of calculating the Brauer–Manin obstruction involves listing all the p-adic solutions to some accuracy, at finitely many primes p; this is a process which may be time-consuming. The result described in this article shows that, at some primes, we do not need to list all p-adic solutions, but only those lying over a closed subset; and, at other primes, we need only to list solutions modulo p.

1. Introduction

The Brauer–Manin obstruction was introduced by Manin [12, 13] to explain why some systems of algebraic equations over the rational numbers ℚ, while having solutions everywhere locally, fail to have global solutions. The obstruction is based on the Brauer group of the associated variety [see 8]. If we denote by V (𝔸_{ℚ}) the set of adelic points of the variety V, then we may define

V (𝔸_{ℚ})^{B} := {(x_{v}) ∈ V (𝔸_{ℚ}) | ∑_{v} inv_{v} A(x_{v}) = 0 for all A in B}

to be the set of adelic points orthogonal to a subset B of the Brauer group Br V, the sum here being taken over all places v of ℚ. Manin’s observation was that, by a standard result in class field theory, any rational points must be contained in V (𝔸_{ℚ})^{Br V}, through the usual diagonal inclusion of ℚ in 𝔸_{ℚ}. The point is that V (𝔸_{ℚ})^{Br V} is sometimes amenable to computation, giving a potential way to prove the non-existence of rational points on V.

One class of varieties on which one can often compute the Brauer–Manin obstruction is that of surfaces with finitely generated Picard group. This article is primarily aimed at computing the obstruction on such surfaces, though the results are true in a wider context. This class includes rational varieties such as cubic surfaces [16], pencils of Severi–Brauer varieties [3], intersections of two quadrics [4, 5] and some pencils of curves of genus 1 [6]. For some of these surfaces, it is known that the Brauer–Manin obstruction is the only obstruction to the existence of rational points: in those cases, computing the obstruction gives a definite answer to the question of whether there exist any rational points on the

2 Martin Bright

surface. The Brauer–Manin obstruction has also recently been used to obtain interesting results on curves [7].

In this article we address the problem of evaluating the obstruction associated to a given Azumaya algebra A. That is, we wish to compute the set V (𝔸_{ℚ})^{A}. The usual method for this is as follows. We consider the local invariant function

V (ℚ_{v}) → ℚ/ℤ : x ↦ inv_{v} A(x)

separately for each v. For all but finitely many v, depending on V and A, this function is zero. At each remaining place, the function is continuous, hence locally constant. By listing all the points of V (ℚ_{v}) to sufficient accuracy, we can evaluate the local invariant everywhere. We then combine this information to obtain V (𝔸_{ℚ})^{A}.

However, listing all points on V (ℚ_{v}) to sufficient accuracy to compute the local invariant at each is a time-consuming process. The result proved in this article drastically reduces the amount of work needed to compute the Brauer–Manin obstruction. We assume that the Azumaya algebra is split by a finite extension of the base field – that is, it belongs to the so-called algebraic Brauer group Br_{1}(V). The reduction of V modulo p falls into one or more irreducible components, which we assume to be reduced.

Suppose that A is split by an unramified extension of ℚ_{v}; then we prove that, at most points P of V (ℚ_{p}) lying above only one component, the invariant depends only on which component that is. This means that the listing of points is now only needed on the intersections of distinct components, and possibly on the non-regular subschemes of components. In this case, the arguments leading to this result are very similar to those described by Harari [10, Section 2.4].

At primes where the splitting extension is tamely ramified, we show that this approach also gives a very useful result: at most of the points, it is enough to know the point modulo p in order to calculate the invariant there.

Theorem 1. Let X be a smooth complete variety over ℚ_{p} and A an Azumaya algebra on X split by a finite Galois extension K/ℚ_{p}. Fix a proper model X /ℤ_{p} of X and let

q : X(ℚ_{p}) → X (𝔽_{p})

be the reduction map. Let Y be an irreducible component of multiplicity 1 of the special fibre X̃. Then there exists a dense open subset U of Y such that

• if K/ℚ_{p} is unramified, then inv_{p} A is constant on q^{−1}U(𝔽_{p});

• if K/ℚ_{p} is tamely ramified, then inv_{p} A is constant on q^{−1}{x}, for each x ∈ U(𝔽_{p}).

We will define the subscheme U as follows:

• if K/ℚ_{p} is unramified, U consists of those points on Y which are regular as points of X;

• if K/ℚ_{p} is ramified, U consists of the smooth points of Y.

It may be possible, by looking more closely at the geometry of X, to take a larger U than that defined here.

2. Proof of Theorem 1

If X is a variety over ℚ_{p} and K/ℚ_{p} a finite field extension, then X_{K} denotes the base extension of X to K. Similarly, if X is a scheme over ℤ_{p}, then X_{K} denotes the base extension of X to O_{K}.

Efficient evaluation of the Brauer–Manin obstruction 3

If X is any scheme, then Div X denotes the group of Cartier divisors on X. Let Y be a reduced closed subscheme of X; then Div_{Y} X denotes the subgroup of Cartier divisors whose support is contained in Y. There is an exact sequence

0 → Div_{Y} X → Div X → Div(X \ Y)

in which the rightmost map is in general not surjective. In particular, we will apply this when Y is the special fibre of X; in that case, the notation Div′ X will be used for the subgroup of Cartier divisors supported on the special fibre. The exact sequence becomes

0 → Div′ X → Div X → Div X.

We now state a hypothesis which will be satisfied by all the schemes we will consider.

Condition (*). X is an integral Noetherian separated scheme over ℤ_{p} such that

(i) the generic fibre X is a smooth variety; (ii) the special fibre X̃ is integral; (iii) the sequence

H^{2}(K/ℚ_{p}, Div′ X_{K}) → H^{2}(K/ℚ_{p}, Div X_{K}) → H^{2}(K/ℚ_{p}, Div X_{K})  (2·1)

is exact.

In particular, the sequence (2·1) is exact if X_{K} is regular, for then Div X_{K} is equal to the Weil divisor group and the sequence splits as a direct sum.

The scheme X which we will be studying to prove Theorem 1 is constructed as follows: take the given proper model for X over ℤ_{p}, and then remove from the special fibre everything outside the open set U. This gives a scheme over ℤ_{p} which is regular, and so satisfies Condition (*). For, if K/ℚ_{p} is unramified, then the morphism X_{K} → X is étale and therefore the inverse image on X_{K} of a regular point on X is again regular [see 9, I, 9.2]. On the other hand, suppose that K/ℚ_{p} is ramified but instead that U = X̃ is smooth over 𝔽_{p}; then X is smooth over ℤ_{p} and so X_{K} is smooth over O_{K} [9, II, 1.3]. As O_{K} is regular, it follows that X_{K} is regular [9, II, 3.1].

If X is a scheme which is Noetherian, separated, integral and regular in codimension 1, then Div X is a subgroup of the group Z^{1}X of Weil divisors [see 11, II, 6.11.2]. This will be true in particular for schemes satisfying Condition (*). For all codimensionone subschemes of the generic fibre certainly have regular local rings; the only other codimension-one subscheme is the special fibre, which has a local ring generated by p, so is also regular.

The proof of Theorem 1 consists of the following steps.

(i) We will look at the map

H^{2}(K/ℚ_{p}, k(X_{K})^{×}) → H^{2}(K/ℚ_{p}, Div X_{K})  (2·2)

and show that an algebra whose image is 0 under this map, when evaluated at a point, gives rise to a 2-cocycle with values in O_{K}^{×}.

(ii) We will show that every Azumaya algebra on X is equivalent to a constant algebra

plus an algebra giving 0 under the map (2·2). It is only a short argument to proceed from these results to the theorem.

2·1. The divisor map on Azumaya algebras

Let X be a smooth, irreducible variety over ℚ_{p}. Let R_{X} and D_{X} denote the sheaves of rational functions and Cartier divisors respectively. Any Azumaya algebra on X can

4 Martin Bright

be represented by an element of H^{2}(X, R_{X}) which maps to 0 in H^{2}(X, D_{X}). If A is an Azumaya algebra split by a finite field extension K/ℚ_{p}, then a stronger statement is true: A is represented by an element of H^{2}(K/ℚ_{p}, k(X_{K})^{×}) which maps to 0 in H^{2}(K/ℚ_{p}, Div X_{K}). This follows from the Hochschild–Serre spectral sequence and the fact that H^{1}(X, D_{X}) = 0.

Now let X be our model of X over ℤ_{p} satisfying Condition (*). Let Br(X_{K}/X) denote the subgroup of Br X consisting of algebras which split over the extension K/ℚ_{p}. Since the sequence (2·1) is exact, the map (2·2) gives rise to a map

ϕ_{K} : Br(X_{K}/X) → H^{2}(K/ℚ_{p}, Div′ X_{K}).

When X is proper and regular and K/ℚ_{p} is unramified, the purity theorem for the Brauer group (discussed by Grothendieck [8, III,6] and, we understand, proved by Gabber) states that an Azumaya algebra A on X, split by K, extends to an Azumaya algebra on X if and only if ϕ_{K}(A) = 0. In this case, evaluating A at a ℚ_{p}-point of X always gives a trivial algebra: for the evaluation map factors through Br ℤ_{p} = 0.

We will relax these hypotheses and see what happens even when K/ℚ_{p} is ramified.

Lemma 2. Let V → U be a finite surjective morphism of integral schemes, acted on by a group G such that U = V/G and G = Gal(k(V)/k(U)). Let α be an element of H^{2}(G, k(V)^{×}) which maps to 0 in H^{2}(G, Div V). Suppose that Pic V = 0. Then there exists a cocycle cohomologous to α taking values in k[V]^{×}.

Proof. As Pic V = 0, the sequence of G-modules

0 → k[V]^{×} → k(V)^{×} → Div V → 0

is exact. Part of the long exact sequence in cohomology is

H^{2}(G, k[V]^{×}) → H^{2}(G, k(V)^{×}) → H^{2}(G, Div V)

which gives the result.

Proposition 3. Let X satisfy Condition (*). Let P be a point of X(ℚ_{p}) extending to a ℤ_{p}-valued point of X. Let A be an Azumaya algebra on X split by a finite Galois extension K/ℚ_{p}, and suppose that ϕ_{K}(A) = 0. Then A(P) is represented by a cocycle in Z^{2}(K/ℚ_{p}, O_{K}^{×}).

Proof. Let Q be the unique point of X_{K} lying above P; let P̃ and Q̃ be the closed points associated to P and Q respectively. Consider the commutative diagram

O_{X_{K},Q̃}^{×} → O_{K}^{×}
∩  ∩
O_{X_{K},Q}^{×} ⟶^{Q} K^{×}
∩
k(X_{K})^{×}

where the lower horizontal map is evaluation at Q, and the upper horizontal map is evaluation at the corresponding O_{K}-point. We will apply Lemma 2 to the covering Spec O_{X_{K},Q̃} → Spec O_{X ,P̃}. The hypotheses are satisfied because

• A is Azumaya on X, so maps to 0 in H^{2}(K/ℚ_{p}, Div X_{K});

Efficient evaluation of the Brauer–Manin obstruction 5

• ϕ_{K}(A) = 0; combined with the previous fact, this shows that A maps to 0 in H^{2}(K/ℚ_{p}, Div X_{K}), using the exactness of the sequence (2·1);

• Spec O_{X_{K},Q̃} is local and so has trivial Picard group.

The conclusion is that we can represent A by a cocycle taking values in O_{X_{K},Q̃}^{×}; the diagram above shows that evaluating this cocycle at Q gives a cocycle representing A(P) which takes values in O_{K}^{×}.

Corollary 4. If K/ℚ_{p} is unramified, then inv_{p} A(P) = 0.

Proof. In this case, H^{2}(K/ℚ_{p}, O_{K}^{×}) = 0. See Serre [15, V,2].

Corollary 5. If K/ℚ_{p} is tamely ramified, then inv_{p} A(P) depends only the reduction of P modulo p.

Proof. In this case H^{2}(K/ℚ_{p}, U_{K}^{1}) = 0, where U_{K}^{1} is the group of units in O_{K} congruent to 1 modulo the prime ideal 𝔭_{K}. This fact follows from the observation that U_{K}^{1} is a prop-group and that [K : ℚ_{p}] is coprime to p. So the reduction map gives an isomorphism

H^{2}(K/ℚ_{p}, O_{K}^{×}) ≅ H^{2}(K/ℚ_{p}, (O_{K}/𝔭_{K})^{×})

through which evaluation at P factors.

2·2. The divisor map on constant algebras

In this section we will show that, under hypotheses slightly stronger than Condition (*), any Azumaya algebra on X split by K can be made to satisfy the hypotheses of Proposition 3 by adding a constant algebra, that is, an element of Br(K/ℚ_{p}).

Proposition 6. Let X satisfy Condition (*). Let A be an Azumaya algebra on the generic fibre X which splits in a finite Galois extension K of ℚ_{p}. Suppose that X has a ℤ_{p}-point, and that either

• the special fibre of X_{K} is irreducible; or

• X_{K} is regular; or

• K/ℚ_{p} is cyclic.

Then there exists a constant algebra B in Br(K/ℚ_{p}) such that ϕ_{K}(A − B) = 0.

Proof. Let P be the closed point associated to the ℤ_{p}-point on X. There is a unique closed point Q of X_{K} lying over P. Write S for Spec O_{X_{K},Q}.

The short exact sequence

0 → O_{X_{K},Q}^{×} → k(X_{K})^{×} → Div S → 0

gives rise to an exact sequence

H^{2}(K/ℚ_{p}, k(X_{K})^{×}) → H^{2}(K/ℚ_{p}, Div S) → H^{3}(K/ℚ_{p}, O_{X_{K},Q}^{×})

of cohomology groups. Br(X_{K}/X) is contained in this left-hand group, and its image in the middle group factors through H^{2}(K/ℚ_{p}, Div′ X_{K}). It follows that the bottom row of the following commutative diagram is a complex.

6 Martin Bright

\begin{matrix}H^{2}(K/ℚ_{p}, K^{×}) & ⟶^{v_{K}} & H^{2}(K/ℚ_{p}, ℤ) & ⟶ & H^{3}(K/ℚ_{p}, O_{K}^{×}) \\ ↓ & & ↓^{α} & & ↓^{β} \\ Br(X_{K}/X) & ⟶_{γ} & H^{2}(K/ℚ_{p}, Div′ X_{K}) & ⟶ & H^{3}(K/ℚ_{p}, O_{X_{K},Q}^{×}) \end{matrix}  (2·3)

Here α is the map 1 ↦ (𝔭_{K}), 𝔭_{K} being the prime in K. The top row is easily seen to be exact.

To prove the proposition, we must show that the image of αv_{K} contains that of γ. In fact it is enough to show that the image of α contains that of γ: for the evaluation map at the ℤ_{p}-point gives a left inverse to β and so β is injective. Some diagram-chasing shows that any element of H^{2}(K/ℚ_{p}, ℤ) whose image under α lies in the image of γ must lift to H^{2}(K/ℚ_{p}, K^{×}). The proof will be completed by Lemma 7.

Remark. When K/ℚ_{p} is cyclic and totally ramified, the proposition is trivial: the maps v_{K} and γ are zero.

In the statement and proof of the following lemma, the special fibre of X will no longer be required to be irreducible. Let Y be a component of the special fibre; then by Y_{L} we mean the fibre product Y ×_{X} X_{L}.

Lemma 7. Let L/K be a finite Galois extension of local fields. Let X be an integral scheme over O_{K} and Y an integral component of the special fibre of X. Suppose either that Y_{L} is irreducible, that X is regular, or that L/K is cyclic. Then the map

H^{2}(L/K, ℤ) → H^{2}(L/K, Div_{Y} X_{L})  (2·4)

given by 1 ↦ (𝔭_{L}) is surjective.

Proof. To save on notation, we will write Div_{Y} X_{L} for the group of Cartier divisors on X_{L} which are supported on the set-theoretic inverse image of Y.

Firstly, the lemma is true when Y_{L} is irreducible. In that case, we claim that the map

ℤ → Div_{Y} X_{L} : 1 ↦ (𝔭_{L})

is an isomorphism of Galois modules. It is easy to see that Div_{Y} X_{L} is free on one generator Y′, so that (𝔭_{L}) = nY′ for some integer n. The Galois action on both groups is trivial. We must show that n = 1, and it is enough to do this separately for L/K unramified and for L/K totally ramified. We will write π for the morphism X_{L} → X.

When L/K is unramified,

(𝔭_{L}) = (π^{∗}𝔭_{K}) = π^{∗}Y;

but, since π is étale and Y reduced, π^{∗}Y is reduced [9, I, 9.2] and so n = 1. When L/K is totally ramified of degree e, we have

e(𝔭_{L}) = (π^{∗}𝔭_{K}) = π^{∗}Y

and so π^{∗}Y = enY′. But π is a flat, finite map; thus en = [L : K] and so n = 1. This completes the proof when Y_{L} is irreducible.

We will now show that the lemma is true when the group Div_{Y} X_{L} of Cartier divisors is replaced by the group Z_{Y}^{1} X_{L} of Weil divisors supported on Y. In particular, this proves the lemma when X_{L} is regular: for then these two groups coincide.

Efficient evaluation of the Brauer–Manin obstruction 7

Write G for Gal(L/K). Let Y′ be an irreducible component of Y_{L}, and H ⊆ G the decomposition group of the generic point of Y′. Then the G-module Z_{Y}^{1} X_{L} is induced from the H-module Z_{Y′}^{1}X_{L}. For any i ≥ 0, we have the following commutative diagram.

\xymatrix{ & H^{i}(H, ℤ) \ar[r]^{∼} & H^{i}(H, Z_{Y′}^{1}X_{L}) \\ H^{i}(G, ℤ) \ar[r]^{δ} \ar[ur]^{res} & H^{i}(G, Ind_{H}^{G} ℤ) \ar[r]^{∼} \ar[u]^{∼} & H^{i}(G, Z_{Y}^{1} X_{L}) \ar[u]^{∼}}  (2·5)

Here the two vertical arrows are isomorphisms by Shapiro’s Lemma; and the top horizontal arrow is an isomorphism by the argument above. The arrow labelled δ comes from the map ℤ → Ind_{H}^{G} ℤ where, thinking of Ind_{H}^{G} ℤ as Hom_{H}(ℤ[G], ℤ), the image of 1 is the function which is identically 1. The composite map ℤ → Z_{Y}^{1} X_{L} is indeed the map (2·4), and moreover the composite map H^{i}(G, ℤ) → H^{i}(H, ℤ) is none other than the restriction: see Milne [14, II, 1.27].

Taking i = 2 in the diagram (2·5), the restriction map is surjective: for it is the map

Hom(G, ℚ/ℤ) → Hom(H, ℚ/ℤ)

and ℚ/ℤ is injective. It follows that the map (2·4) is surjective, completing this step of the proof.

Finally, suppose that G is cyclic. Then, for any G-module M, there is a surjective map

H^{0}(G, M) → H^{2}(G, M)

given by cup-product with a chosen generator of H^{2}(G, ℤ). There is a commutative diagram as follows.

\xymatrix{H^{0}(G, ℤ) \ar[rr]^{∼} \ar[dr]^{∼} \ar@{->>}[dd] &  & H^{0}(G, Z_{Y}^{1} X_{L}) \ar@{->>}[dd] \\  & H^{0}(G, Div_{Y} X_{L}) \ar[ur]^{∼} \ar@{->>}[dd] \\ H^{2}(G, ℤ) \ar@{->>}[rr] \ar[dr] &  & H^{2}(G, Z_{Y}^{1} X_{L}) \\  & H^{2}(G, Div_{Y} X_{L}) \ar[ur]}

In the top half, that the horizontal map is an isomorphism comes from the case i = 0 of the diagram (2·5). Since the two other maps are injective, it follows that they are isomorphisms.

Therefore the composition H^{0}(G, ℤ) → H^{2}(G, Div_{Y} X_{L}) is surjective, and hence so is the map (2·4).

2·3. Completion of the proof We can now finish the proof of Theorem 1.

Proof of Theorem 1 Recall that the scheme X was constructed from the given model of

8 Martin Bright

X by removing all of the special fibre apart from the open subset U. By construction, the geometric hypotheses of Proposition 6 are satisfied; and U has a ℤ_{p}-point, for otherwise the theorem is vacuous. The conclusion is that there exists an algebra B in Br(K/ℚ_{p}) such that ϕ_{K}(A − B) = 0.

Proposition 3 and its corollaries, applied to A − B, complete the proof.

3. Examples

3·1. Construction of an Azumaya algebra on some diagonal quartic surfaces Let V be the diagonal quartic surface in ℙ^{3} defined by the equation

a_{0}X_{0}^{4} + a_{1}X_{1}^{4} + a_{2}X_{2}^{4} + a_{3}X_{3}^{4} = 0  (3·1)

where the a_{i} are rational integers. This is a K3 surface, hence has finitely generated Picard group; so it is possible to apply the methods described by Bright and Swinnerton-Dyer [1] to compute the algebraic part of the Brauer group.

When a_{0}a_{1}a_{2}a_{3} is a square, Swinnerton-Dyer [17] describes a fibration of V over ℙ^{1} in curves of genus 1 and uses this to produce Azumaya algebras on V by lifting elements of Br k(ℙ^{1}). When a_{0}a_{1}a_{2}a_{3} is not a square, the same fibration itself defines an Azumaya algebra aplit by ℚ(\sqrt{a_{0}a_{1}a_{2}a_{3}}), as described by Bright [2]. This may be summarised as follows:

Proposition 8. Let V be the surface (3·1) and suppose that a_{0}a_{1}a_{2}a_{3} is not a square. Let θ denote a square root of a_{0}a_{1}a_{2}a_{3}. Then there exists a function f on V, defined over ℚ, such that

(f) = D + D′ − 2Π

where D is a curve of genus 1 defined over ℚ(θ), D′ is its conjugate, and Π is a plane section; and

A = (ℚ(θ)/ℚ, f)

defines a non-trivial Azumaya algebra on V.

We may assume that f can be written as a rational function on ℙ^{3}, such that neither the numerator nor the denominator is divisible by any constant prime factor; this can clearly be achieved by multiplying f by a constant factor.

Proposition 6 is very simple in this case of a quadratic extension. If ℚ(θ)/ℚ is unramified at p, then it shows that our Azumaya algebra A can be made to satisfy the conditions of Proposition 3 by maybe multiplying the function f by p. If, on the other hand, the extension is ramified, then Proposition 6 is trivial: at any component Y_{ℚ(θ)} of the special fibre, the function f has even valuation and so ϕ_{ℚ(θ)}(A) = 0.

We will now look at what happens at odd primes where V has bad reduction. We will assume that V has points everywhere locally, and that

0 ≤ v_{p}(a_{i}) ≤ 3 for all i, p.

Furthermore, we may assume that no prime p divides more than two of the a_{i}. For we can always multiply all the a_{i} by p and then remove fourth powers; in this way we will always reach the stage where at most two of the a_{i} are divisible by p, except when the v_{p}(a_{i}) are all different modulo 4. But in that case the surface is not locally soluble.

Efficient evaluation of the Brauer–Manin obstruction 9

3·2. Primes dividing one coefficient

Proposition 9. Let p be an odd prime dividing precisely one of the coefficients a_{i}. Then

(i) the reduction of V at p is a cone over a smooth plane quartic curve C; (ii) the reduction of f at p is the pullback of a function on C; (iii) the vertex of the cone does not lift to a point of V (ℚ_{p}).

Proof. Statement (i) is obvious. For statement (ii), consider the divisor of f, which is made up of curves of genus at most 1. The reduction of this modulo p will still be a union of curves of genus at most 1. By the Riemann-Hurwitz formula, the map from each of these components to the base curve C must be constant.

For statement (iii), suppose without loss of generality that p divides a_{0}. Then any lift of the vertex to ℚ_{p} is equivalent modulo p to (1 : 0 : 0 : 0). So

v_{p}(a_{1}X_{1}^{4} + z_{2}X_{2}^{4} + a_{3}X_{3}^{4}) ≥ 4

but v_{p}(a_{0}X_{0}^{4}) < 4, so the point does not lie on V (ℚ_{p}).

In particular, the reduction of V at p is irreducible. Proposition 6 takes a very simple form: having removed any factors of p from the numerator and denominator of the rational function f, we have ensured that Proposition 3 applies.

Corollary 10. If an odd prime p divides precisely one of the a_{i}, and does so to an even power, then inv_{p} A(x) = 0 for all x ∈ V (ℚ_{p}).

Proof. In this case, ℚ(θ)/ℚ is unramified at p. Combine Proposition 9 with Corollary 4.

Corollary 11. If an odd prime p divides precisely one of the a_{i}, and does so to an odd power, then to evaluate A on V (ℚ_{p}) it suffices to evaluate f at the points of C(𝔽_{p}).

Proof. In this case, ℚ(θ)/ℚ is tamely ramified at p. Combine Proposition 9 with Corollary 5.

Remark. In this case, the divisor cut out by f on the curve C is divisible by 2 in Div C. Before evaluating f at each point of C(𝔽_{p}), we should first check whether that divisor is in fact twice a principal divisor – if it is, then the reduction of f is a constant multiplied by a square and so inv_{p} A is constant on V (ℚ_{p}).

As an example, consider the surface

9X_{0}^{4} + 10X_{1}^{4} = 12X_{2}^{4} + 13X_{3}^{4}.  (3·2)

The construction described above gives the Azumaya algebra A = (ℚ(\sqrt{390})/ℚ, f) where

f = \frac{15X_{0}^{2} + 12X_{2}^{2} + 13X_{3}^{2}}{X_{0}^{2}} .

At p = 5, the equation reduces to that of a cone over a plane quartic curve. The extension is tamely ramified. To evaluate inv_{5} A at a point, we look to see whether the value of the function (12X_{2}^{2} + 13X_{3}^{2})/X_{0}^{2} is square or non-square modulo 5. It turns out that both square and non-square values occur, and therefore there is no Brauer–Manin obstruction on the variety (3·2). The same happens at p = 13.

10 Martin Bright

3·3. Primes dividing two coefficients

Let p be an odd prime dividing precisely two of the coefficients a_{i}. Then the reduction of V at p consists of four planes intersecting in a common line L. In this case, several different things can happen depending on the Galois structure of the four planes and whether f has zeros or poles on any of them.

On the surface (3·2) introduced above, the reduction at p = 3 is the union of four planes: two defined over 𝔽_{3}, and the other two quadratic conjugates. The extension ℚ(\sqrt{390})/ℚ is ramified at 3. The function defining the Azumaya algebra reduces to (X_{3}/X_{0})^{2}, which is invertible and square on each of those planes, and hence inv_{3} A is 0 except possibly on the line X_{1} = X_{3} = 0 where the planes meet.

On the surface

X_{0}^{4} + 13X_{1}^{4} = 3X_{2}^{4} + 13X_{3}^{4},  (3·3)

the construction gives the Azumaya algebra A = (ℚ(\sqrt{3})/ℚ, f) where

f = \frac{−3X_{0}^{2} − 26X_{1}^{2} + 12X_{2}^{2} + 13X_{3}^{2}}{X_{0}^{2}} .

At 13, the surface (3·3) reduces to four planes, each defined over 𝔽_{13}. The function f is identically zero (with multiplicity 1) on two of those planes but not on the other two. Since the extension is unramified at 13, we deduce that inv_{13} A is 0 at points of V (ℚ_{13}) lying above one of the planes where f is not zero. At points lying above the other two planes, we deduce that inv_{13} A is equal to inv_{13}(ℚ(\sqrt{3})/ℚ, 13) = 1/2. We obtain no information about any points of V (ℚ_{13}) lying above the line L.

Acknowledgements. The author was supported by the Engineering and Physical Sciences Research Council through Grant GR/R82975/01. I thank Professor Jean-Louis Colliot-Thélène for his comments on an earlier version of this article.

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