Brauer groups of diagonal quartic surfaces

Martin Bright^{1}

Department of Mathematical Sciences University of Liverpool Liverpool L69 7ZL England

Abstract

We describe explicit methods of exhibiting elements of the Brauer groups of diagonal quartic surfaces. Using these methods, we compute the algebraic Brauer–Manin obstruction in two contrasting examples. In the second example, the obstruction is found to be trivial but a computer search reveals no points of small height on the surface.

1 Introduction

We are concerned with the solubility in rational numbers of the equation

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

where the a_{i} are non-zero rational numbers. Let V denote the smooth surface defined by this equation over ℚ. We will always suppose that V has points everywhere locally, that is, in the real numbers ℝ and in each p-adic field ℚ_{p}; for otherwise the equation (1) certainly has no rational solutions.

One reason that V might have no rational points is given by the Brauer–Manin obstruction, first described by Manin (1971, 1974). This obstruction is defined in terms of the Brauer group of V, which is the group of Azumaya algebras on V modulo equivalence. Our objective is to show how one part of the Brauer group, the so-called algebraic part, may be computed, and how the associated obstruction may thence be evaluated.

Email address: mjbright@liverpool.ac.uk (Martin Bright). 1 The author was supported by the Engineering and Physical Sciences Research Council, first through a studentship and then through Grant GR/R82975/01.

Preprint submitted to Journal of Symbolic Computation 5 December 2005

The surface V is geometrically a K3 surface; for certain choices of coefficients a_{i}, there is a fibration V → ℙ^{1} defined over ℚ, the fibres of which are curves of genus 1.

Colliot-Thélène et al. (1998) studied elliptic fibrations over ℙ^{1} satisfying certain criteria: that the singular fibres are all of type I_{2}; and a technical condition (“Condition D”) on the class of the fibration in a geometric Selmer group of its Jacobian. Conditionally on two well-known conjectures (the finiteness of the Tate–Shafarevich groups of elliptic curves, and Schinzel’s hypothesis), they were able to prove that the obstruction associated to the so-called vertical Brauer group is the only obstruction to the Hasse principle.

Swinnerton-Dyer (2000) proved a similar result for a particular family of diagonal quartic surfaces, those where a_{0}a_{1}a_{2}a_{3} is a square. In that case there is a fibration over ℙ^{1} with six fibres each of type I_{4}, consisting of four straight lines arranged in a skew quadrilateral. He defined a new obstruction to the existence of rational points on these surfaces, and showed under analogous conditions to those mentioned above that this obstruction is the only obstruction to the Hasse principle. He then showed that the new obstruction is in fact a Brauer–Manin condition.

The object of this article is to show how the algebraic Brauer group and associated obstruction may be computed for any diagonal quartic surface, whether or not it has the structure of an elliptic fibration over ℙ^{1}. By automating this procedure and calculating a large number of examples, we hope to gather evidence on the Brauer–Manin obstruction on these surfaces. In addition to describing the general procedures for carrying out the calculations, we present in detail two examples which exhibit contrasting results.

The first example computed in the present article is another example of a diagonal quartic surface with a fibration. We show that there is a non-trivial Brauer–Manin obstruction and therefore that the surface has no rational points.

On the other hand, a general diagonal quartic surface has no fibration defined over ℚ. In this case, the algebraic Brauer group has order 2 and it is relatively straightforward to compute the associated obstruction. Our second example has no algebraic Brauer–Manin obstruction and yet no rational points of small height; and numerical evidence suggests that this is a common occurrence.

1.1 Overview

The approach we adopt is as follows. Let Br_{1} V denote the algebraic Brauer group, and V̄ the extension of V to an algebraic closure ℚ̄ of ℚ. It is well known that Br_{1} V/ Br ℚ is isomorphic to the Galois cohomology group H^{1}(ℚ, Pic V̄). We can write down a set of generators for Pic V̄ and compute this cohomology group, which is finite; this process has been implemented in Magma (see Magma, 2003; Bosma et al., 1997), using Gavin Brown’s algebraic geometry machinery and the group cohomology algorithms implemented by Derek Holt. Code to do this is available from the author’s web page at Bright (2002). Unfortunately the isomorphism from H^{1}(ℚ, Pic V̄) to Br_{1} V/ Br ℚ is rather awkward to calculate. For certain special types of algebra, though, we can write down Azumaya algebras on V and easily compute the corresponding elements of H^{1}(ℚ, Pic V̄). In this way it is often possible to find a set of representatives for the whole algebraic Brauer group.

The first special class of algebras we consider are those which arise from a fibration of V. Let π be a surjective morphism from V to ℙ^{1}; then any element of the Brauer group of ℚ(ℙ^{1}) lifts to an element of the Brauer group of ℚ(V); and it is possible to write down simple criteria which determine when this resulting algebra is Azumaya. These algebras, which live in the so-called “vertical” Brauer group of V, are discussed in Section 2. We compute the algebraic Brauer–Manin obstruction for a particular diagonal quartic surface equipped with a fibration over ℙ^{1}.

The second special class of algebras are the cyclic algebras. Just as in the theory of Brauer groups of fields, it is more straightforward to compute with cyclic algebras than with general algebras. In Section 3 we describe the map from Br_{1} V/ Br ℚ to H^{1}(ℚ, Pic V̄) for cyclic algebras. We compute the algebraic Brauer–Manin obstruction for a surface which has no fibration over ℙ^{1}.

Although we are primarily concerned with surfaces as above defined over the rational numbers, the techniques described apply more generally. Throughout, k will denote a number field over which V is defined.

1.2 The Brauer group

For background information on Brauer groups of schemes, we refer the reader to Grothendieck (1968).

Let V be any smooth surface over k. We will write Br_{0} V for the image of Br k in Br V; when V has points everywhere locally, this map is injective. Let Br_{1} V denote the kernel of Br V → Br V̄, the “algebraic” part of the Brauer group.

As described by Skorobogatov (2001, pp. 116–117), there is an isomorphism between Br_{1} V/ Br k and H^{1}(k, Pic V̄). We now discuss computational aspects of this map.

Proposition 1 Let Prin V denote the group of principal divisors on the variety V. There are isomorphisms as follows:

Br_{1} V/ Br k easy ↿⇂ easy ker(H^{2}(k, k̄(V)^{×}) → H^{2}(k, Div V̄))/ Br k hard ↿⇂ easy ker(H^{2}(k, Prin V̄) → H^{2}(k, Div V̄)) easy ↿⇂ easy H^{1}(k, Pic V̄)

where the arrows are labelled to show ease of computation.

PROOF. The top isomorphism is the correspondence between an Azumaya algebra on V (specified as a vector space over k(V) together with a multiplication law) and the corresponding 2-cocycle (or factor set, in old terminology). This is well known and described explicitly in, say, Deuring (1935).

For the middle isomorphism, note that, as H^{3}(k, k̄^{×}) = 0 (see Tate, 1967, p. 199), there is an exact sequence

H^{2}(k, k̄^{×}) → H^{2}(k, k̄(V)^{×}) → H^{2}(k, k̄(V)^{×}/k̄^{×}) → 0

and so the map from H^{2}(k, k̄(V)^{×}) to H^{2}(k, Div V̄), being trivial on Br k, factors through that quotient. Now the group k̄(V)^{×}/k̄^{×} is isomorphic to the group Prin V̄ of principal divisors on V̄. There are two problems with computing this map: firstly, we must make effective the triviality of H^{3}(k, k̄^{×}); and secondly, we must have some procedure which, given a divisor known to be principal, finds a function having that divisor. Hence the upward arrow is labelled “hard”.

The lower isomorphism is the coboundary map from the long exact sequence in cohomology associated to

0 → Prin V̄ → Div V̄ → Pic V̄ → 0

where H^{1}(k, Div V̄) = 0 because it is a permutation module. Computing the upward arrow is easy: given a 1-cocycle with values in Pic V̄, we lift to Div V̄ and take the coboundary. For the downward arrow, we must compute the inverse of the coboundary map; splitting Div V̄ into its irreducible components makes this a simple problem in linear algebra. □

For more details on how these isomorphisms may be computed, see Bright and Swinnerton-Dyer (2004).

1.3 Computing H^{1}(k, Pic V̄)

There are two main reasons why the Picard group of V̄ is particularly amenable to computational methods. Firstly, the surface V̄ is a K3 surface, which means that it has trivial canonical sheaf and

H^{1}(V̄ , OV̄) = 0

and therefore Pic V̄ is finitely generated. Moreover, this means that divisors are linearly equivalent if and only if they are numerically equivalent, that is, their difference has intersection number zero with all divisors. The geometry of K3 surfaces is thoroughly covered by Barth et al. (1984).

Secondly, there is a very convenient set of generators. Let ϵ denote a fixed primitive eighth root of unity and set

α_{ij} = \sqrt[4]{a_{i}/a_{j}}

where the a_{i} are the coefficients from (1) and the fourth roots are chosen such that

α_{ij}α_{jk} = α_{ik}

for all i, j, k. Then we can write down the equations of 48 straight lines contained in V̄:

L_{mn}^{123} : X_{0} = ϵ^{m}α_{10}X_{1}, X_{2} = ϵ^{n}α_{32}X_{3};
L_{mn}^{231} : X_{0} = ϵ^{m}α_{20}X_{2}, X_{3} = ϵ^{n}α_{13}X_{1};
L_{mn}^{312} : X_{0} = ϵ^{m}α_{30}X_{3}, X_{1} = ϵ^{n}α_{21}X_{2}.

Here in each case m and n take the values 1, 3, 5, 7.

Let Λ denote the (free) subgroup of Div V̄ generated by the 48 lines. The intersection numbers of straight lines are easy to compute; let Λ^{0} denote the subgroup of Λ consisting of those linear combinations of lines which are numerically equivalent to 0, in other words, principal.

The following is well known; a proof may be found in Pinch and Swinnerton-Dyer (1991, Lemma 1).

Proposition 2 The Picard group Pic V̄ is free of rank 20, and is generated by the classes of the 48 lines L_{mn}^{pqr}. □

Let K denote the field

K = k(ϵ, α_{10}, α_{20}, α_{30}).

Then all 48 lines are clearly defined over K, and in fact K is the smallest common field of definition of the 48 lines. The Galois action on the lines therefore factors through

G = Gal(K/k)

and the inflation-restriction sequence gives us H^{1}(k, Pic V̄) ≅ H^{1}(G, Λ/Λ^{0}).

Let m be an integer known to annihilate H^{1}(G, Pic V̄), such as |G|. By considering the long exact sequence in cohomology associated to the exact sequence

0 → Λ/Λ^{0} ⟶^{×m} Λ/Λ^{0} → Λ/Λ^{0} ⊗ ℤ/mℤ → 0

(remember that Λ/Λ^{0} is torsion-free), we see that there is an isomorphism

\frac{H^{0}(G, Λ/Λ^{0} ⊗ ℤ/mℤ)}{H^{0}(G, Λ/Λ^{0}) ⊗ ℤ/mℤ} ⟶^{inf ∘d} H^{1}(k, Pic V̄)  (2)

where the map is the coboundary map followed by inflation.

The Galois action on the 48 lines is straightforward. Using the above isomorphism it is easy to compute H^{1}(k, Pic V̄) for any diagonal quartic surface. In fact the author has computed this group for all diagonal quartic surfaces. These results, together with some Magma and Pari scripts for producing them, are available electronically (Bright, 2002).

2 The vertical Brauer group of a fibration

2.1 The vertical Brauer group

In this section we suppose that the surface V has a surjective morphism π : V → ℙ^{1}, whose fibres must be curves of genus 1. This gives a useful way of producing Azumaya algebras on V. We choose once and for all a coordinate t on ℙ^{1}, such that the fibre of π over t = ∞ is non-degenerate. Whenever we write 𝔸^{1}, we mean that open subscheme of ℙ^{1} where t ≠ ∞.

Definition 3 We define the following groups associated to the fibration π:

B := {A ∈ Br k(ℙ^{1}) | π^{∗}A ∈ Br V }/ Br k
Br_{vert} := π^{∗}(Br k(ℙ^{1})) ∩ Br V
Pic_{vert} := ker(Pic V̄ → Pic V̄_{η})

where η is the generic point of ℙ^{1}, and V_{η} the generic fibre of π.

Tsen’s theorem (Milne, 1998, 13.5(b)) shows that Br_{vert} is contained in Br_{1} V. A simple piece of diagram-chasing gives the following result.

Proposition 4 There is an isomorphism B ≅ H^{1}(k, Pic_{vert}), such that the diagram

H^{1}(k, Pic_{vert}) ⟶^{∼}  B
↓  ↓
H^{1}(k, Pic V̄) ⟶^{∼} Br_{1} V/ Br k

commutes. □

If we can compute Pic_{vert}, then Proposition 4 allows us to find out how much of Br_{1} V/ Br k comes from the Br_{vert} without having to compute with elements of the Brauer group. In general, finding a generating set for Pic_{vert} is not totally straightforward. Such a generating set is given by the irreducible components of the singular fibres of π, as described by Shioda (1972). The procedure of Section 1.3 gives us H^{1}(k, Pic V̄) in terms of the straight lines L_{mn}^{pqr}. In order to compute H^{1}(k, Pic_{vert}) compatibly with this, we need to know a sum of classes of straight lines which is equal to the class in Pic V̄ of each irreducible component of each singular fibre. However, it is possible to find these without ever knowing explicitly what the singular fibres are, as follows.

• Each component of a singular fibre is a rational curve, of self-intersection −2; and the degree (that is, intersection number with a plane section) of a component is bounded by the degree of a general fibre of π, which is easily found from any explicit formula for π.

• There exist only finitely many divisor classes of a given self-intersection and bounded degree, and when the degree is reasonably small they may easily be listed.

• Any component of a fibre has intersection number 0 with the class of a fibre; this condition rules out any rational curves which are transverse to the fibration.

• Each fibre is connected; this means that, by considering the intersection numbers of the rational curves we find, we can partition them into possible fibres.

In this way it is possible to find a generating set for Pic_{vert}.

2.2 Structure of the vertical Brauer group

To compute the Brauer–Manin obstruction coming from Br_{vert}, we want to write down elements of B.

Lemma 5 For any closed point P of 𝔸^{1}, identify the residue field k(P) with k[t]/F (t), where F = N_{k(P)/k}(t − t_{P}) is the monic irreducible polynomial generating the maximal ideal associated to P. When c ∈ H^{1}(ℓ, ℚ/ℤ) and f ∈ ℓ(t) for some field ℓ, let (c, f) denote the algebra in Br ℓ(t) associated to the 2cocycle f ∪ dc.

Any element A of Br k(ℙ^{1}) may be written uniquely as

A = ∑_{P} cores_{k(P)/k}(c_{P} , t − t_{P}) + α  (3)

where the sum is over finitely many distinct closed points P of ℙ^{1}; c_{P} lies in H^{1}(k(P), ℚ/ℤ); and α lies in Br k.

PROOF. Colliot-Thélène and Swinnerton-Dyer (1994, Proposition 1.2.1). □

The following description of which of these algebras lift to Azumaya algebras on V is stated by Swinnerton-Dyer (2000).

Proposition 6 The algebra A described in (3) lies in B if and only if the following two conditions hold:

(1) ∑_{P} cores_{k(P)/k}(c_{P}) = 0 in H^{1}(k, ℚ/ℤ);

(2) for each P, res_{ℓ/k(P)}(c_{P}) = 0 whenever ℓ is the least field of definition of an irreducible component of the fibre π^{−1}P. □

Note that condition 2 of the proposition shows that non-trivial Azumaya algebras are only obtained by taking the points P to lie below geometrically reducible fibres of π. Therefore there are, up to elements of Br k, only finitely many classes of Azumaya algebras arising by this method: at each of the finitely many geometrically reducible fibres of π, the group

ker (H^{1}(k(P), ℚ/ℤ) → H^{1}(ℓ, ℚ/ℤ)) = H^{1}(ℓ/k(P), ℚ/ℤ)

is finite.

2.3 Example

2.3.1 A surface and a fibration

We consider the diagonal quartic surface V defined by

X_{0}^{4} + X_{1}^{4} = 6X_{2}^{4} + 12X_{3}^{4}.  (4)

It is easily checked with Magma, using routines by Nils Bruin, that the surface has points everywhere locally. A search using an algorithm described by Bernstein (2001) finds that there are no global solutions with the |X_{i}| less than 10^{4}. The 48 straight lines on this surface are defined over the field

L = ℚ(ϵ, \sqrt[4]{2}, \sqrt[4]{3})

of degree 32 over ℚ. (Here ϵ denotes, as always, a primitive eighth root of unity.) Looking at the Galois action on the lines shows that Pic V is free of rank 2, generated by the classes of a plane section and of the divisor

F = L_{11}^{123} + L_{15}^{123} + L_{31}^{123} + L_{35}^{123}.

While the class of F is defined over ℚ and contains a divisor defined over ℚ, the divisor F itself is defined only over K = ℚ(\sqrt{−2}). We will write σ for the non-trivial automorphism of K/ℚ. The conjugate over ℚ of F is another divisor

F^{σ} = L_{53}^{123} + L_{57}^{123} + L_{73}^{123} + L_{77}^{123}

linearly equivalent to F. The individual straight lines involved are defined over ℚ(i, \sqrt[4]{2}).

The divisor F has self-intersection 0 and hence arithmetic genus 1; therefore the linear system |F | defines a fibration π : V → ℙ^{1} whose generic fibre is a curve of genus 1. Although it would be possible to compute an equation, defined over ℚ, for this fibration, it will not be necessary. The morphism π is defined only up to an automorphism over ℚ of ℙ^{1}; we will later find a fibre of π defined over ℚ and choose this to be the fibre at infinity.

As described in Section 1.3, we can compute H^{1}(ℚ, Pic V̄). Using Magma and the routines available at Bright (2002), we find that it is of order 2:

  Magma V2.10-8     Tue Oct 14 2003 15:29:43 on kiwi
  Type ? for help.  Type <Ctrl>-D to quit.
  > load "quartic";
  Loading "quartic"
  > V := DiagonalQuarticSurface([1,1,-6,-12]);
  > V;
  Scheme over Rational Field defined by
  X0^4 + X1^4 - 6*X2^4 - 12*X3^4
  > H1Pic(V);
  Full Quotient RSpace of degree 1 over Integer Ring
  Column moduli:
  [ 2 ]

Thus Br_{1} V/ Br k is also of order 2. It would be possible to work out the singular fibres of the fibration and hence the associated Pic_{vert}, and then to show that the non-trivial element of Br_{1} V/ Br k lies in the image of Br_{vert}; but this is unnecessary, as any non-trivial Azumaya algebra which we produce with the help of Proposition 6 must lie in the unique non-trivial class of Azumaya algebras.

2.3.2 An Azumaya algebra

We will now use Proposition 6 to write down an Azumaya algebra on V arising from the fibration described above. As we already know that F is one geometrically reducible fibre of the fibration, we will try to find an Azumaya algebra without having to compute all the singular fibres.

Lemma 7 Let g be a function on V, defined over K, with divisor (g) = F − F_{∞}, where F_{∞} is a geometrically irreducible closed curve defined over ℚ. Then the quaternion algebra A = (−1, N_{K/ℚ}g) is Azumaya on V.

PROOF. Choose the fibration π so that F_{∞} lies above the point at infinity, that is, π^{∗}∞ = F_{∞}. Let P be the closed point of 𝔸_{ℚ}^{1} over which the two fibres F and F^{σ} lie; so π^{∗}P = F + F^{σ} and k(P) = K. Then g is, up to a constant multiple, equal to π^{∗}(t − t_{P}). We have

cores_{k(P)/ℚ}(−1, g) = (−1, N_{K/ℚ}g) = A

as the quadratic character on K^{×} associated to −1 comes by restriction from ℚ^{×}. All we must check are the conditions of Proposition 6. Condition 1 is satisfied because N_{K/ℚ}(−1) is a square. Condition 2 is satisfied because the least field of definition of each irreducible component of F or F^{σ} is ℚ(i, \sqrt[4]{−2}), in which −1 is a square. □

The function

f(X_{0}, X_{1}, X_{2}, X_{3}) = \frac{X_{0}^{2} − \sqrt{−2}X_{0}X_{1} − X_{1}^{2}}{X_{2}^{2} + \sqrt{−2}X_{3}^{2}}

is found to have divisor F − F^{σ}. Therefore the map from V to ℙ^{1}, defined over K by f, differs from π by an automorphism of ℙ^{1} over K. Note also the following property of f: that f^{σ} has a zero at F^{σ} and a pole at F; and therefore N_{K/ℚ}f = ff^{σ} is a constant function. In fact,

N_{K/ℚ}f = \frac{X_{0}^{4} + X_{1}^{4}}{X_{2}^{4} + 2X_{3}^{4}} = 6.  (5)

Suppose we can find a c ∈ K such that the fibre {f = c} is defined over ℚ and geometrically irreducible; then we can take

g = \frac{f}{f − c}

to satisfy the requirements of Lemma 7.

Lemma 8 The fibre {f = c} is defined over ℚ if and only if N_{K/ℚ}(c) = 6.

PROOF. The conjugate fibre to {f = c} is defined by the equation f^{σ} = c^{σ}; by (5) this is equivalent to f = 6/c^{σ}. The fibre is defined over ℚ if and only if the two equations define the same fibre, that is, c = 6/c^{σ}. □

We will take

c = 2 − \sqrt{−2}.

The fibre over c is defined over ℚ, and is easily checked to be non-singular. Following Lemma 7, we define

G = N_{K/ℚ} (\frac{f}{f − c})

= \frac{X_{0}^{4} + X_{1}^{4}}{2(X_{0}^{4} + X_{1}^{4}) − 4(X_{2}^{2} + X_{3}^{2})(X_{0}^{2} − X_{1}^{2}) − 4(X_{2}^{2} − 2X_{3}^{2})X_{0}X_{1}}

and then A = (−1, G) is an Azumaya algebra on V.

2.3.3 Computing the obstruction

To compute the Brauer–Manin obstruction on V, we must calculate the value of

∑_{v} inv_{v} A(x_{v})

for each adèlic point (x_{v}) in V (𝔸_{ℚ}). This task is made easier by several facts:

• The function inv_{v} A(x_{v}) is continuous, hence locally constant, on V (ℚ_{v}) for each v.

• The support of (G) (and, indeed, of any divisor) is closed in each V (ℚ_{v}), under the appropriate complete (real or p-adic) topology (see Colliot-Thélène et al., 1980, Lemma 3.1.2). It is therefore unnecessary to evaluate the local invariants along zeros or poles of G.

• At finite primes v where both the variety V and the algebra A extend to ℤ_{v}, the function inv_{v} A(x_{v}) is zero (see Skorobogatov, 2001, p. 101).

We must therefore evaluate inv_{v} A(x_{v}) at v = 2, 3 and at the infinite prime.

On ℝ, the algebra A = (−1, G) is trivial where G is positive, and non-trivial where G is negative. Now G has no zeros in V (ℝ), and a double pole along a curve of genus 1; therefore G is either everywhere positive or everywhere negative. Evaluating G at any one point of V (ℝ) gives a positive value; hence the local invariant is everywhere zero on V (ℝ).

Over ℚ_{3}, we ask Magma to compute the kernel of the local restriction map H^{1}(ℚ, Pic V̄) → H^{1}(ℚ_{3}, Pic V̄):

  > RestrictionKernel(V,3);
  Full Quotient RSpace of degree 1 over Integer Ring
  Column moduli:
  [ 2 ]

This computation involves finding the decomposition group at 3, and then using Derek Holt’s group cohomology functions to realise the restriction map. The necessary code can be found at Bright (2002). It turns out that the restriction map is the zero map; so any Azumaya algebra on V over ℚ_{3} is equivalent to a constant algebra over ℚ_{3}. We therefore only need to evaluate (−1, G) at one point of V (ℚ_{3}) to discover that the invariant is everywhere equal to 0.

On ℚ_{2}, we must work a little harder. Looking modulo 16 shows that every primitive solution of (4) in ℤ_{2} must have all the X_{i} units. Hensel’s Lemma then shows that any such solution is equivalent, modulo 2^{5}, to an integer point with

X_{0}^{4} + X_{1}^{4} ≡ 6X_{2}^{4} + 12X_{3}^{4} (mod 2^{7}).  (6)

Modulo squares, G is equal to

(X_{0}^{4} + X_{1}^{4}) (2(X_{0}^{4} + X_{1}^{4}) − 4(X_{2}^{2} + X_{3}^{2})(X_{0}^{2} − X_{1}^{2}) − 4(X_{2}^{2} − 2X_{3}^{2})X_{0}X_{1}) (7)

and knowing the X_{i} modulo 2^{5} gives the value of (7) modulo 2^{8}. We list the odd (X_{i}) modulo 2^{5} satisfying (6) and evaluate (7) at each such point; it turns out that the value of G is always either 96 or 176 modulo 2^{8}. In either case, the formula for the Hilbert symbol given in Serre (1973, Chapter III, 1.2, Theorem 1) shows that the local invariant is \frac{1}{2}.

Combining these calculations shows that

∑_{v} inv_{v} A(x_{v}) = \frac{1}{2}

for all points (x_{v}) ∈ V (𝔸_{ℚ}). Therefore there is a Brauer–Manin obstruction to the existence of rational points on V, and so there are no rational solutions to (4).

3 Cyclic algebras

In this section we look at another class of Azumaya algebras on a surface V which are often straightforward to find, that is, cyclic Azumaya algebras.

3.1 Background

The following well-known proposition is a straightforward generalisation of the Lemma 11 stated by Swinnerton-Dyer (2000).

Proposition 9 Let A = (K/k, f) be a cyclic algebra on V, where K is a cyclic extension of the base field k and f is some rational function in k(V). Then A is Azumaya if and only if the divisor (f) is the norm of some divisor D defined over K. If moreover we assume that V has points in every completion of k, then A is equivalent to a constant algebra if and only if we can take D to be principal. □

In the examples which follow, we will never need to compute the element of H^{1}(k, Pic V̄) corresponding to a cyclic algebra: the mere fact of being nontrivial gives it away. In more complex examples, though, this may be useful and we include it here for completeness.

Proposition 10 Let A = (K/k, f) be a cyclic Azumaya algebra on V, with (f) = N_{K/k}D for some divisor D defined over K. Let α be the 1-cocycle with values in Pic V_{K} defined by

α(1) = 0
α(σ^{r}) = ∑_{i=0}^{r−1} σ^{i}D

where σ is an (implicitly fixed) generator of Gal(K/k). Then the class inf_{k̄/K} α in H^{1}(k, Pic V̄) corresponds to the class of A in Br_{1} V/ Br k, under the isomorphism descibed in Proposition 1.

PROOF. Using the convention that an empty sum is 0, we work through the isomorphisms described in Proposition 1. Considering α as a 1-cocycle with values in Div V_{K}, its coboundary is given by

dα(σ^{r}, σ^{s}) = ∑_{i=0}^{r+s−1} σ^{i}D − ∑_{i=0}^{((r+s) mod n)−1} σ^{i}D

= \begin{cases}0 & r + s < n \\ N_{K/k}D = (f) & r + s ≥ n. \end{cases}

This is indeed the image of (K/k, f) in H^{2}(k, Prin V_{K}). □

3.2 Example

The example of Section 2.3 describes a cyclic algebra: there, the algebra produced was split by the extension ℚ(i). The function G had divisor F + F^{σ} − 2F_{∞}, which is easily shown to be the norm from ℚ(i) to ℚ of the divisor

L_{11}^{123} + L_{15}^{123} + L_{53}^{123} + L_{57}^{123} − F_{∞}

and so (−1, G) is Azumaya.

3.3 Example

In this section we will describe another example, where the surface has no fibrations in curves of genus 1 defined over ℚ. The approach used is a general one which extends to other similar cases. In this example, we will show that there is no algebraic Brauer–Manin obstruction on the chosen surface. The surface V this time is given by

7X_{0}^{4} + 15X_{1}^{4} = 2X_{2}^{4} + 6X_{3}^{4}.  (8)

This diagonal quartic surface shows the “most general” possible Galois module structure on the 48 straight lines: that is, the least common field of definition of the lines,

ℚ(ϵ, \sqrt[4]{15 × 7}, \sqrt[4]{2 × 7}, \sqrt[4]{6 × 7}),

is as large as it can be, of degree 256 over ℚ. The cohomology computations described earlier show that Pic V = H^{0}(ℚ, Pic V̄) is of rank 1, generated by the class Π of a plane section; and Br_{1} V/ Br k is again of order 2. A computer search reveals that there are no integer solutions with the |X_{i}| less than 10^{4}.

We will use a fibration of V in curves of genus 1, defined over a quadratic extension K of ℚ, to find a divisor F whose norm is linearly equivalent to twice a plane section. This will allow us to construct a function f satisfying the conditions described in Proposition 9, and hence an Azumaya algebra.

3.3.1 A useful fibration

There are no fibrations from V to ℙ^{1} defined over ℚ, for there are no divisors over ℚ which could be fibres of such a fibration. However, Swinnerton-Dyer (2000) describes two conjugate fibrations which are defined over

K = ℚ(\sqrt{7 × 15 × 2 × 6}) = ℚ(\sqrt{35}).

We will not reproduce the details of the geometry of these fibrations here, but will describe as much as is necessary for this example. The recipe given for finding such fibrations is as follows. Let

(a_{0}, a_{1}, a_{2}, a_{3}) = (7, 15, −2, −6)

be the coefficients in the equation of the surface V.

Lemma 11 Let θ, r_{1}, r_{2}, r_{3} be elements of K such that

a_{1}r_{1}^{2} + a_{2}r_{2}^{2} + a_{3}r_{3}^{2} = 0, θ^{2} = a_{0}a_{1}a_{2}a_{3}.

Define polynomials A, B, C, D by

A = θr_{2}X_{0}^{2} + a_{1}a_{3}(r_{3}X_{1}^{2} − r_{1}X_{3}^{2})
B = θr_{3}X_{0}^{2} − a_{1}a_{2}(r_{2}X_{1}^{2} + r_{1}X_{2}^{2})
C = a_{3}θr_{3}X_{0}^{2} − a_{1}a_{2}a_{3}(r_{2}X_{1}^{2} − r_{1}X_{2}^{2})
D = −a_{2}θr_{2}X_{0}^{2} − a_{1}a_{2}a_{3}(r_{3}X_{1}^{2} + r_{1}X_{3}^{2});

then the original equation (8) factorises over K as

A(X_{0}, X_{1}, X_{2}, X_{3})D(X_{0}, X_{1}, X_{2}, X_{3}) = B(X_{0}, X_{1}, X_{2}, X_{3})C(X_{0}, X_{1}, X_{2}, X_{3}).

The two morphisms π_{i} : V → ℙ^{1} given by

π_{1} = (A : B) = (C : D)
π_{2} = (A : C) = (B : D)

are fibrations whose fibres are curves of genus 1. The classes in Pic V_{K} of the fibres of π_{1} and π_{2} respectively are conjugate over ℚ, and the sum of these two classes is equal to twice the class of a plane section.

PROOF. See Swinnerton-Dyer (2000). □

If we write (A) for the divisor of zeros of the polynomial A, and so on, then

\begin{matrix}(A) = F_{1} + F_{2} (B) = F′_{1} + F_{2} \\ (C) = F_{1} + F′_{2} (D) = F′_{1} + F′_{2} \end{matrix}  (9)

where F_{1} and F′_{1} are two fibres of π_{1}, each defined over K, and similarly F_{2} and F′_{2} are two fibres of π_{2}.

Now consider the divisor F_{1}^{σ}, where σ is the non-trivial element of Gal(K/ℚ). This is a smooth curve lying in the class in Pic V_{K} of the fibres of π_{2}, and therefore is a fibre of π_{2}, defined by an equation B/D = c for some c ∈ K. To find c, it suffices to evaluate B/D at any point of F_{1}^{σ}. We first find a ℚ̄-valued point on F_{1}: such a point is a common zero of A and C. Rearranging the equation A = 0 gives us X_{3}^{2} in terms of X_{0} and X_{1}; similarly C = 0 yields an expression for X_{2}^{2}. Setting X_{0} = 0 and X_{1} = 1 gives

X_{2}^{2} = \frac{r_{2}}{r_{1}} X_{3}^{2} = \frac{r_{3}}{r_{1}}

and, taking conjugates and substituting into B/D, we obtain

c = \frac{B}{D}(F_{1}^{σ}) = \frac{r_{1}r_{2}^{σ} + r_{2}r_{1}^{σ}}{a_{3}(r_{1}r_{3}^{σ} + r_{3}r_{1}^{σ})}.  (10)

Lemma 12 The function

f = \frac{C}{X_{0}^{2}} (\frac{B}{D} − c) = \frac{A − cC}{X_{0}^{2}}

has divisor N_{K/ℚ}(F_{1} + D_{0}), where D_{0} is the plane section divisor defined by {X_{0} = 0}. The algebra

A = (K/ℚ, f)

is a non-trivial Azumaya algebra on V.

PROOF. The expression for (f) comes straight from (9) and (10). Proposition 9 then shows that A is Azumaya. Proposition 10 and the isomorphism (2) show that it is non-trivial. □

3.3.2 Computing the obstruction

To compute the algebra described in Lemma 12, we follow Lemma 11 and find r_{1}, r_{2} and r_{3} in ℚ(\sqrt{35}) satisfying

15r_{1}^{2} − 2r_{2}^{2} − 6r_{3}^{2} = 0.  (11)

This is not entirely straightforward. Such conic equations over the rational numbers have been well studied: Legendre famously described an algorithm for solving quadratic forms in three variables, and more recent work on efficient solution is described in Cremona and Rusin (2003). However, these methods run into problems when applied over a number field, as described in Simon’s thesis (Simon, 1998). In the present case, several steps of Legendre’s descent technique reduce the equation (11) to

x^{2} − ωy^{2} = 3z^{2}

where ω = 6+\sqrt{35} is a fundamental unit in K. Fortunately, this new equation succumbs to a computer search, and we deduce the solution

(r_{1}, r_{2}, r_{3}) = (20 + 2\sqrt{35}, 15, 20 + 5\sqrt{35})

to (11). Note the similarity to the example in Section 2.3, where finding the rational fibres of a fibration also involved solving a norm equation. Substituting these values into the expression in (10) gives c = −1. Lemma 12 produces, after removing a constant factor,

f = \frac{7X_{0}^{2} + 5X_{1}^{2} − 4X_{2}^{2} − 2X_{3}^{2}}{X_{0}^{2}}  (12)

such that (35, f) is an Azumaya algebra on V.

We now show that there is no Brauer–Manin obstruction on V associated to the algebra A.

Lemma 13 Let A be an element of the 2-torsion subgroup of Br V. Suppose that, for some place w of ℚ, the local invariant function

inv_{w} A(x_{w})

takes both values 0 and \frac{1}{2} for x_{w} in V (ℚ_{w}). Then there is no Brauer–Manin obstruction associated to A.

PROOF. Let (x_{v}) be a point of V (𝔸_{ℚ}). Then ∑_{v} inv_{v} A(x_{v}) is equal to either 0 or \frac{1}{2}. In the first case, this shows that there is no Brauer–Manin obstruction associated to A. In the second case, we replace x_{w} by x′_{w} so as to change the value of inv_{w} A(x_{w}) and make the sum equal 0. □

In our example, looking at ℚ_{5} shows that there is no Brauer–Manin obstruction. Using the formula in Serre (1973, Chapter III, 1.2, Theorem 1), we have

(35, b)_{5} = (\frac{7}{5})^{β}(\frac{u}{5}) = (−1)^{β}(\frac{u}{5})

where b = 5^{β}u with u a 5-adic unit. Now let

g = 7X_{0}^{2} + 5X_{1}^{2} − 4X_{2}^{2} − 2X_{3}^{2};

then (35, f) = (35, g). It is enough to show that g takes both square and non-square values in ℤ_{5}^{×} when evaluated at points of V (ℚ_{5}). For example, the point (2 : 1 : 1 : 5) satisfies

7X_{0}^{4} + 15X_{1}^{4} − 2X_{2}^{4} − 6X_{3}^{4} ≡ 0 (mod 5^{3})

and so by Hensel’s Lemma is equivalent modulo 5^{3} to a point of V (ℚ_{5}). But putting these values into g gives

g(2, 1, 1, 5) ≡ 4 (mod 5^{2})

which is a square in ℚ_{5}. On the other hand, the point (2 : 2 : 7 : 5) also lifts to a point of V (ℚ_{5}), this time modulo 5^{2}, but we have

g(2, 2, 7, 5) ≡ 2 (mod 5^{2})

which is not a square in ℚ_{5}. Therefore (35, f) takes values both +1 and −1 on V (ℚ_{5}), and so inv_{5} A(x) takes values 0 and \frac{1}{2}. By Lemma 13, there is no Brauer–Manin obstruction associated to A, and hence no algebraic Brauer– Manin obstruction on V.

That the quadratic form g should take both square and non-square values in ℤ_{5}^{×} on the surface V seems unsurprising; indeed, one might be surprised if the opposite were true. It seems likely that the absence of an algebraic Brauer– Manin obstruction on this particular surface is one instance of a more general trend. This is supported by the evidence described below.

On the other hand, it is possible to list surfaces which fall into this “most general” class, and to search for rational points of bounded height on them.

There are 424 everywhere locally soluble diagonal quartic surfaces of the form

aX_{0}^{4} + bX_{1}^{4} = cX_{2}^{4} + dX_{3}^{4}

with |a|, |b|, |c|, |d| less than 16, and “sufficiently general” in the sense described at the beginning of this section. Checking these conditions is simple with Magma: the only one not yet described is local solubility, for which the bounds given by the Weil conjectures show that we only need to check solubility at 2, 5 and the primes dividing the coefficients.

It is reasonably straightforward (see Bernstein, 2001) to search for solutions with the |X_{i}| less than 10^{4}. Let N(H) denote the number of such surfaces with no point satisfying |X_{i}| < H for all i; then we obtain the following data.

\begin{matrix}H & 100 & 1000 & 10000 \\ N(H) & 81 & 69 & 61 \end{matrix}

There appears to be a trend for surfaces in this family to have a point of reasonably small height if at all. Should this pattern continue, there would be some surfaces either with points of extremely large height, or with no points at all. Moreover, the same construction as used above allows us to compute the algebraic Brauer–Manin obstruction, which turns out to be trivial on all of these surfaces: in each case, there is some prime at which the local invariant takes both values 0 and \frac{1}{2}. Although very vague, this evidence does suggest that some diagonal quartic surfaces may have trivial algebraic Brauer–Manin obstruction but no rational points.

Acknowledgements

I would like to thank Sir Peter Swinnerton-Dyer for all his work as my research supervisor and Victor Flynn for many helpful comments. Part of this research was undertaken during a visit to the School of Mathematics, University of Sydney, with support provided by the Magma group.

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