Computations on diagonal quartic surfaces

Martin James Bright Clare College

A dissertation submitted for the degree of Doctor of Philosophy at the University of Cambridge

September 2002

Preface

This dissertation is not substantially the same as any I have submitted for a degree or diploma or other qualification at any other university. Except where indicated in the text, it is the result of my own work and contains nothing which is the outcome of work done in collaboration.

I thank my research supervisor, Sir Peter Swinnerton-Dyer, for his expert guidance in the course of my studies. In particular, he has made me realise the great value of explicit calculations in understanding number theory. I would also like to thank the many people who have helped me with mathematical discussions, as well as those who have given moral support. I acknowledge the financial support of the Engineering and Physical Sciences Research Council. I am also very grateful to IBM Research for awarding me their first Special European Research Fellowship, including an IBM ThinkPad computer which has been invaluable both for computation and for producing this dissertation. Finally, I would like to thank the authors of PARI/GP for making the fruits of their work freely available.

Clare College September 2002

Summary

We are concerned with the solubility in integers of the equation

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

where a_{0}, a_{1}, a_{2} and a_{3} are fixed integral coefficients. As the equation is homogeneous, this is equivalent to studying rational solutions. This equation defines a surface in three-dimensional projective space ℙ^{3}, which we denote by V. Geometrically, V is a K3 surface.

One condition which is clearly necessary for V to have rational points is that there be points on V in each completion ℚ_{v} of ℚ. Varieties for which this condition is sufficient are said to satisfy the Hasse principle. In general, the Hasse principle does not hold, but many counterexamples are explained by the so-called Brauer–Manin obstruction. This was first put forward by Manin and is defined in terms of the Brauer group of the variety.

The aim of this dissertation is to classify all diagonal quartic surfaces, defined by the above equation, into finitely many cases. The cases are distinguished by the action of the absolute Galois group of ℚ on the Picard group of V, and we show that there are 546 such cases. They are described in terms of algebraic constraints on the coefficients a_{i}. In each case, we compute that part of the Brauer group of V which splits over the algebraic closure of ℚ; this is done using the Galois cohomology of the Picard group of V. We also investigate the elliptic fibrations which may exist on V: to such a fibration is associated a vertical Brauer group, and we show how these groups may also be computed. Finally, an indication is given of how these results may be used to look further into the Brauer–Manin obstruction on V.

Contents

List of notation vii

1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Geometry of the surface 11

2.1 K3 surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.1 Definition of K3 surfaces . . . . . . . . . . . . . . . . . 12 2.1.2 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Fibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.1 Divisors on fibrations . . . . . . . . . . . . . . . . . . . 18 2.2.2 The Picard group of the surface . . . . . . . . . . . . . 22 2.2.3 Picard groups over ℚ . . . . . . . . . . . . . . . . . . . 26

2.3 Fibrations on V . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.3.1 Finding fibrations . . . . . . . . . . . . . . . . . . . . . 28 2.3.2 Finding the singular fibres . . . . . . . . . . . . . . . . 32 2.3.3 An example . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Galois theory of the Picard group 38 3.1 The ‘generic’ Galois group and its action . . . . . . . . . . . . 38 3.2 Computing with Abelian groups . . . . . . . . . . . . . . . . . 42

3.2.1 Groups and matrices . . . . . . . . . . . . . . . . . . . 42 3.2.2 Normal forms for matrices . . . . . . . . . . . . . . . . 43 3.2.3 Subgroups and quotients . . . . . . . . . . . . . . . . . 45 3.3 Listing the cases . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3.1 Representation of subgroups . . . . . . . . . . . . . . . 47 3.3.2 Listing subgroups . . . . . . . . . . . . . . . . . . . . . 49 3.3.3 Equivalences between subgroups . . . . . . . . . . . . . 50 3.3.4 Finding a canonical representative . . . . . . . . . . . . 52

3.4 Finding the subgroup associated to a surface . . . . . . . . . . 53 3.5 Finding surfaces from subgroups . . . . . . . . . . . . . . . . . 59

3.5.1 Computing B^{H} . . . . . . . . . . . . . . . . . . . . . . 61 3.5.2 Giving examples . . . . . . . . . . . . . . . . . . . . . 65

3.6 Cohomology of the Picard group . . . . . . . . . . . . . . . . . 67 3.6.1 Computing with G-modules . . . . . . . . . . . . . . . 68 3.6.2 The action on the Picard group . . . . . . . . . . . . . 71

4 The Brauer group 76

4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.1.1 Cohomological theory . . . . . . . . . . . . . . . . . . . 77 4.1.2 Change of fields . . . . . . . . . . . . . . . . . . . . . . 81

4.2 Explicit descriptions . . . . . . . . . . . . . . . . . . . . . . . 82 4.2.1 Calculation of H^{1}(k, Pic V̄) . . . . . . . . . . . . . . . 83 4.2.2 Calculation of H_{Az}^{2}(V) . . . . . . . . . . . . . . . . . . 84 4.2.3 From H^{2} to algebras . . . . . . . . . . . . . . . . . . . 87

4.3 Cyclic algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.3.1 Cohomology of cyclic groups . . . . . . . . . . . . . . . 89 4.3.2 Cyclic algebras on V . . . . . . . . . . . . . . . . . . . 91

4.4 The vertical Brauer group . . . . . . . . . . . . . . . . . . . . 97

4.4.1 Identifying B . . . . . . . . . . . . . . . . . . . . . . . 99 4.5 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.5.2 Computing B . . . . . . . . . . . . . . . . . . . . . . . 105 4.5.3 The map to H^{1}(ℚ, Pic_{vert}) . . . . . . . . . . . . . . . . 106 4.5.4 Lifting to Br V . . . . . . . . . . . . . . . . . . . . . . 108 4.5.5 Other examples . . . . . . . . . . . . . . . . . . . . . . 109 5 Numerical methods and examples 110

5.1 Local solubility . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.2 Finding small solutions . . . . . . . . . . . . . . . . . . . . . . 114 5.3 Example with a fibration . . . . . . . . . . . . . . . . . . . . . 116 5.4 Example with no fibration . . . . . . . . . . . . . . . . . . . . 122 5.5 Trivial Brauer group . . . . . . . . . . . . . . . . . . . . . . . 126

A Tables of cases 128 A.1 k ∩ ℚ(ϵ) = ℚ . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 A.2 k ∩ ℚ(ϵ) = ℚ(i) . . . . . . . . . . . . . . . . . . . . . . . . . . 143 A.3 k ∩ ℚ(ϵ) = ℚ(\sqrt{2}) . . . . . . . . . . . . . . . . . . . . . . . . . 146 A.4 k ∩ ℚ(ϵ) = ℚ(\sqrt{−2}) . . . . . . . . . . . . . . . . . . . . . . . . 149 A.5 k ∩ ℚ(ϵ) = ℚ(ϵ) . . . . . . . . . . . . . . . . . . . . . . . . . . 152

B Locally soluble surfaces 154

C Surfaces with trivial Brauer group 159

D Implementation of algorithms 161

D.1 Subgroups of Γ . . . . . . . . . . . . . . . . . . . . . . . . . . 161 D.1.1 Algorithm 3.9 . . . . . . . . . . . . . . . . . . . . . . . 161 D.1.2 Algorithm 3.10 . . . . . . . . . . . . . . . . . . . . . . 162 D.1.3 Finding canonical subgroups . . . . . . . . . . . . . . . 162

D.2 Relating surfaces to subgroups . . . . . . . . . . . . . . . . . . 164 D.2.1 Algorithm 3.11 . . . . . . . . . . . . . . . . . . . . . . 164 D.2.2 Finding the constant field . . . . . . . . . . . . . . . . 166 D.2.3 Algorithm 3.17 . . . . . . . . . . . . . . . . . . . . . . 166 D.2.4 Algorithm 3.18 . . . . . . . . . . . . . . . . . . . . . . 169

D.3 Computing with G-modules . . . . . . . . . . . . . . . . . . . 170 D.3.1 Algorithm 3.19 . . . . . . . . . . . . . . . . . . . . . . 170 D.3.2 Working modulo m . . . . . . . . . . . . . . . . . . . . 170 D.3.3 Algorithm 3.20 . . . . . . . . . . . . . . . . . . . . . . 171 D.3.4 Algorithm 3.22 . . . . . . . . . . . . . . . . . . . . . . 171 D.3.5 Calculating a basis for Pic V̄ . . . . . . . . . . . . . . . 173 CONTENTS vi Bibliography 175

List of notation

Arithmetic

k_{v} Completion of the field k with respect to the val-

uation v

𝔽_{p}  Finite field with p elements
ϵ  Primitive eighth root of unity, ϵ = (1 + i)/\sqrt{2}
α_{ij}  Fourth root of a_{i}/a_{j}
H^{i}(L/K, M)  Galois cohomology of Gal(L/K) with values in M
H^{i}(k, M)  Galois cohomology of Gal(k̄/k) with values in M
Ĥ^{i}( )  Tate cohomology groups
∪  Cup product
Br L/K, Br k  Brauer group of L/K, of k̄/k
inv_{v}  Local invariant map Br k_{v} ⟶^{inv_{v}} ℚ/ℤ
( , )_{v}  Local Hilbert symbol at v

Linear algebra

M^{T}  Transpose of matrix M
M_{(i)}  ith column of matrix M
I_{n}  n × n identity matrix
(M_{1}|M_{2})  Matrix made by joining matrices M_{1} and M_{2}

Geometry

ℙ_{k}^{n}  n-dimensional projective space over the field k
Pic X  Picard group of X
NS X  Néron–Severi group of X
Pic^{0} X  Subgroup of Pic X algebraically equivalent to 0
Pic^{n} X  Subgroup of Pic X numerically equivalent to 0

Pic_{vert}  Vertical Picard group of a fibration
L(D)  Line bundle associated to divisor D

h^{i}(X, F) Dimension of cohomology group H^{i}(X, F) over

k(X)

χ(X, F) Euler characteristic of F, alternating sum of the

h^{i}(X, F)

ℓ(D)  h^{0}(L(D))
s(D)  h^{1}(L(D)), the superabundance of D

O_{X}, K_{X}, Ω_{X}, D_{X} Sheaves of regular functions, rational functions,

regular differentials and divisors on X

h^{i,j}(X)  Dimension of H^{i}(X, Ω_{X}^{j})
p_{a}, p_{g}  Arithmetic, geometric genus
Π  Divisor class of a plane section
D_{1}.D_{2}  Intersection number of divisors D_{1} and D_{2}
X_{t}  Fibre of map X → Y at the point t ∈ Y

Arithmetic and geometry

k(X) Function field of scheme X (nothing to do with

the field k)

X_{K}  Product scheme X ×_{Spec k} Spec K
X̄  Xk̄

𝔾_{m} Étale sheaf corresponding to the multiplicative group scheme; étale sheaf of rational functions on a variety

Br X  Brauer group of scheme X
Br_{0} X  Image of Br k in Br X
Br_{1} X  Kernel of Br X → Br X̄
H_{Az}^{2}(X)  Kernel of H^{2}(k, k(X̄)^{×}/k̄^{×}) → H^{2}(k, Div X̄)

Chapter 1

Introduction

We are concerned with the solubility in integers of the equation

a_{0}X_{0}^{4} + a_{1}X_{1}^{4} + a_{2}X_{2}^{4} + a_{3}X_{3}^{4} = 0  (1.1)

where a_{0}, a_{1}, a_{2} and a_{3} are fixed non-zero integral coefficients. As the equation is homogeneous, this is equivalent to studying rational solutions, and we allow the coefficients to be any rational numbers. The equation (1.1) defines a surface in three-dimensional projective space ℙ^{3}, which we denote throughout by V. Where we need to make a distinction between the variety defined over the rational numbers ℚ and that defined over the algebraic closure ℚ̄, we will always denote the former by V and the latter by V̄.

1.1 Background

One condition which is clearly necessary for rational solutions of (1.1) is that there be solutions in the real numbers ℝ and in each p-adic field ℚ_{p}, where p runs through all rational primes. In other words, we require that V have points everywhere locally. For certain classes of varieties, this condition is also a sufficient one: if points exist everywhere locally, then points also exist globally. This was first proved for quadratic forms, and in that case is the well-known theorem of Hasse–Minkowski [36, Chapter IV, Theorem 8]. A family of varieties for which local solubility is a sufficient condition for global solubility is said to satisfy the Hasse principle.

However, it has been known for a long time that not every variety satisfies the Hasse principle: in the case of curves, several famous examples of curves of genus 1 not satisfying the Hasse principle were known some time ago; see, for example, Selmer [34]. Such curves are now described by elements of the Tate– Shafarevich group of their Jacobians. Examples have also been exhibited of cubic surfaces not satisfying the Hasse principle: see Swinnerton-Dyer [42]; Mordell [30]; and Cassels and Guy [6]. To explain these counterexamples to the Hasse principle, Manin put forward in [25] and [24] a definition of an obstruction to the Hasse principle, based on the Brauer group of the variety, which accounted for all the examples listed above. This obstruction is known as the Brauer–Manin obstruction to the Hasse principle.

The Brauer group of a variety, defined in Chapter 4, is the group of Azumaya algebras on that variety modulo equivalence. An Azumaya algebra may be considered as a continuously varying family of central simple algebras over the base field k (in our case k = ℚ), parametrised by the points of the variety; alternatively, it is a central simple algebra over the function field of the variety which specialises at each point x to a central simple algebra over k(x). The equivalence relation defined on Azumaya algebras is a straightforward extension of that used to define the Brauer group of a field in terms of central simple algebras.

Manin’s obstruction relies on the fundamental theorem of Albert–Brauer– Hasse–Noether, which forms a basic part of class field theory:

Theorem 1.1 (Albert–Brauer–Hasse–Noether). Let k be an algebraic number field. Then there exist injective group homomorphisms

inv_{v} : Br k_{v} → ℚ/ℤ

for each place v of k, called local invariant maps, such that the sequence

0 → Br k → ⨁_{v} Br k_{v} \underrightarrow{∑_{v} inv_{v}} ℚ/ℤ

is exact.

Proof. See [46, Section 10.2, Theorem B].

Note that implicit in this statement is that, for any algebra in Br k, all but finitely many of the local invariants are zero.

Manin’s observation was that, for any rational point x on the variety and any Azumaya algebra A in the Brauer group, we must have

∑_{v} inv_{v} A(x) = 0.

Therefore, if it possible to show that, for any adèlic point (x_{v}) on the variety, the sum ∑_{v} inv_{v} A(x_{v}) is non-zero for some Azumaya algebra A, then no rational point can exist. This is Manin’s obstruction. A more formal definition follows.

Definition 1.2. Let X be a complete algebraic variety defined over a number field k. The set of adèlic points of X is simply

X(𝔸_{k}) = ∏_{v} X(k_{v}).

Let B be a subgroup of the Brauer group Br X. We define

X(𝔸_{k})^{B} = {(x_{v}) ∈ X(𝔸_{k}) | ∑_{v} inv_{v} A(x_{v}) = 0 for all A ∈ B}.

The sum exists because inv_{v} A(x_{v}) is zero for all but finitely many primes v. If X(𝔸_{k})^{B} is empty, we say that there is a Brauer–Manin obstruction to the Hasse principle attached to B.

The natural question to ask, given this definition, is the following: for a given family of varieties, is the Brauer–Manin obstruction the only obstruction to the Hasse principle? In other words, is it a sufficient condition for the existence of a rational point that the variety have points everywhere locally, and that there be no Brauer–Manin obstruction? This has been answered affirmatively for curves of genus 1, where it fits into the theory of the Tate–Shafarevich group (assuming this group to be finite; see [23, Chapter X, Theorem 2.1]); extensive work has also been done on rational varieties, which is described in the survey by Manin and Tsfasman [26]. However, the Brauer–Manin obstruction is known not to be the only obstruction to the Hasse principle for all varieties. Examples are known (see Skorobogatov’s paper [41]) of varieties for which the Brauer–Manin obstruction is not the only obstruction.

A good brief overview of this material is presented in Chapter X of Lang’s survey [23].

The geometry of the diagonal quartic surface V̄, that is, its structure as a variety over the complex numbers, does not depend on the coefficients a_{i} and is well understood. Any non-singular surface in ℙ^{3} defined by an equation of degree 4 falls into the class of K3 surfaces. The definition and properties of K3 surfaces, and how they fit into the more general classification of algebraic surfaces, are described in many books, such as those by Beauville [3], Iskovskikh and Shafarevich [22] and Barth, Peters and Van de Ven [2]. One property which makes K3 surfaces particularly suited to the computations performed here is that the Picard group is the same as the Néron–Severi group, which is finitely generated and torsion-free. The problem of finding rational points on a K3 surface defined over the rational numbers is a hard one, about which little is known; the study of the diagonal quartic surfaces (1.1) might be viewed as an attempt to gain some understanding of K3 surfaces more generally. The family of diagonal quartic surfaces is well suited to this, for it is broad enough to show some variation in characteristics of the surfaces, but narrow enough to allow much concrete computational experimentation.

One feature of K3 surfaces is that some have the structure of an elliptic fibration: that is, they possess a map to ℙ^{1}, the generic fibre of which is an elliptic curve. There are many such fibrations defined on the diagonal quartic surface V̄; depending on the coefficients a_{i}, these fibrations may or may not be defined over the rational numbers. Where there is a fibration defined over the rational numbers, this can be of use in trying to find rational points on V.

Some of the results in this dissertation overlap with those in [32] of Pinch and Swinnerton-Dyer. There, the authors were interested in verifying the Tate conjecture, which relates the order of a zero of a certain L-function to the rank of the Néron–Severi group of the surface. These ranks were therefore computed, using a method, similar to the one described here, of dividing the surfaces into cases. However, we are also concerned with the first Galois cohomology group of the Picard group, rather than just the invariant part; this means that our division into cases will be much finer than that in [32], making the calculations unreasonably large to perform by hand.

Approaches in the literature to studying the Brauer–Manin obstruction on K3 surfaces have arisen out of techniques using fibrations. These methods have previously been applied to various rational surfaces: see the survey by Colliot-Thélène [9] for a detailed account of this. In particular, a series of papers variously by Colliot-Thélène, Sansuc, Skorobogatov and Swinnerton-Dyer have proved that, under certain assumptions, the Brauer–Manin obstruction is the only obstruction to the Hasse principle for pencils of conics and more broadly for generalised Severi–Brauer varieties [13]. The same techniques have more recently been applied to pencils of curves of genus 1 [11, 12], of which the diagonal quartic surfaces studied in [44] are a specific example. The results obtained there make certain hypotheses on the elliptic fibrations which have an interpretation in terms of the relationship between the vertical Brauer group of the fibration and the whole Brauer group. It is hoped that the computations and methods put forward in this dissertation may be able to throw more light on the methods used there, and on the Brauer–Manin obstruction for K3 surfaces generally.

1.2 Overview

The aim of this dissertation is to show how computational methods may be used to investigate the Brauer–Manin obstruction on diagonal quartic surfaces. The algorithms and results presented here fall broadly into three related areas:

• the classification of all diagonal quartic surfaces into 546 cases, according to the Galois action on the Picard group; • the demonstration of how Galois cohomology can be used to compute,

and explicitly express, certain elements of the Brauer group of V;

• the finding of elliptic fibrations on V, where they exist, and the relating of computations in Galois cohomology to the theory of the vertical Brauer group attached to such a fibration.

The Brauer group of the variety V over ℚ has a natural filtration:

Br_{0} V ⊆ Br_{1} V ⊆ Br V

where Br_{0} V consists of constant algebras, that is, the image of Br ℚ in Br V; and Br_{1} V consists of those algebras which split over the algebraic closure ℚ̄ of ℚ. Now all the diagonal quartic surfaces are isomorphic over ℚ̄, so the part of the Brauer group which does not split over ℚ̄ (that is, the quotient Br V/ Br_{1} V) is fixed as the coefficients a_{i} vary. What concerns us in this dissertation is how to compute the part of the Brauer group which is dependent on the a_{i}, which is Br_{1} V. We will call this the arithmetic part of the Brauer group.

A constant algebra can give no Brauer–Manin obstruction: so we are interested in the quotient Br_{1} V/ Br_{0} V. Now this is known to be isomorphic to the Galois cohomology group H^{1}(ℚ, Pic V̄). We wish to calculate this cohomology group for all diagonal quartic surfaces.

One very useful observation about the surface V̄ is that it contains many straight lines. Indeed, writing

a_{0}X_{0}^{4} = −a_{1}X_{1}^{4} and a_{2}X_{2}^{4} = −a_{3}X_{3}^{4}

gives sixteen straight lines. These lines will be labelled L_{mn}^{123}, where m and n run through the values 1, 3, 5 and 7. In the equations of these lines, we write (as we will throughout) ϵ for a fixed primitive eighth root of unity, and α_{ij} for a fixed fourth root of a_{i}/a_{j}. We will choose the α_{ij} such that α_{ij}α_{jk} = α_{ik}. The lines are then given by

L_{mn}^{123} : X_{0} = ϵ^{m}α_{10}X_{1}, X_{2} = ϵ^{n}α_{32}X_{3}.  (1.2)

Similarly we define the lines

L_{mn}^{231} : X_{0} = ϵ^{m}α_{20}X_{2}, X_{3} = ϵ^{n}α_{13}X_{1}  (1.3)
L_{mn}^{312} : X_{0} = ϵ^{m}α_{30}X_{3}, X_{1} = ϵ^{n}α_{21}X_{2}  (1.4)

making 48 straight lines in all. In fact there are no other straight lines on the surface. It turns out that these 48 lines generate the Picard group of V̄, which is a free Abelian group of rank 20. Our problem of studying the action of Gal(ℚ̄/ℚ) on the Picard group is reduced to studying the action on these 48 straight lines, which is very straightforward to describe. For the lines are all defined over a finite extension of ℚ of simple structure, and thus the Galois action factors through a finite group which is very amenable to computation.

The first main result of this work is to show that the Galois action on the 48 lines, and hence on the Picard group of V̄, may take only finitely many forms, whatever the coefficients a_{i}. These ‘cases’ into which the diagonal quartic surfaces may fall correspond to subgroups of a certain Galois group Γ, up to various automorphisms which are made explicit. The cases are then described by putting various algebraic constraints on the a_{i}, and an algorithm is described for quickly finding the Galois action from the coefficients a_{i}.

A very simple example is useful to explain the concept of this approach. Suppose that we are studying the equation

X^{2} = a

over the rational numbers. Clearly, this equation has solutions when a is a rational square, and has no solutions otherwise. Thus there are two cases into which this equation may fall, and they are distinguished by the algebraic condition “a is a square”. The relevant Galois group here is Gal(ℚ(\sqrt{a})/ℚ), which is always a subgroup of the ‘generic’ group Gal(ℚ(\sqrt{x})/ℚ(x)), cyclic of order 2: the latter has two subgroups, being the whole group and the trivial subgroup, which correspond to the two cases. The classification of the diagonal quartic surfaces into cases follows the same lines, but the Galois group Γ is considerably larger (of order 256) and non-Abelian. There is a further complication: various different choices of roots and permutations of the coordinates can lead to different subgroups which we wish to consider as the same case.

In each case, the Galois action on the Picard group of V̄ allows us to compute both the Picard group of V, and the structure of the arithmetic part of the Brauer group of V. However, we are using the isomorphism between Br_{1} V/ Br_{0} V and H^{1}(ℚ, Pic V̄). This isomorphism must be made explicit if we are to write down elements of the Brauer group as Azumaya algebras. This is necessary in order to compute the Brauer–Manin obstruction. So the following part of this dissertation concerns the gap between the computational procedures expounded in the first part, and explicitly expressing elements of Brauer groups.

As described above, fibrations have been used to look into the Hasse principle and Brauer–Manin obstruction on various surfaces; in particular, one class of diagonal quartic surfaces (corresponding to several of our cases) was studied in depth by Swinnerton-Dyer [44]. The fibrations which may exist on K3 surfaces are morphisms to ℙ^{1} whose generic fibre is a curve of genus 1. We discuss the geometry of such fibrations, and in particular show that an elliptic fibration, defined over the rational numbers, exists on V if and only if the intersection quadratic form on Pic V represents 0. Later, after discussing Brauer groups, we show that the vertical Brauer group attached to a fibration may be computed in terms of a cohomology group H^{1}(ℚ, Pic_{vert}), where Pic_{vert} is the subgroup of Pic V̄ generated by the irreducible components of the fibres. Elements of the vertical Brauer group may be expressed in a comparatively simple form, making them easier to write down than arbitrary elements of the Brauer group.

The study of the geometry of the diagonal quartic surface V is covered in Chapter 2. There various geometric facts are reviewed, both about general K3 surfaces and about the diagonal quartic specifically. We discuss fibrations over the projective line, and how divisors on V interact with such fibrations; this will be important when we come to study the vertical Brauer group. Using a particular fibration, we prove in Proposition 2.15 that the 48 straight lines on V̄ do indeed generate the Picard group. Finally, we move away from pure geometry and look at fibrations defined over the rational numbers; in particular, we prove in Proposition 2.23 that such a fibration exists if and only if the intersection form represents zero.

Most of the computational work referred to above is described in Chapter 3. First, some general algorithms for computing with Abelian groups are recalled. The group Γ is not in fact Abelian, but is ‘almost’ so, having an Abelian subgroup of index 2. We must therefore demonstrate methods for representing, and computing with, elements and subgroups of Γ. Having done so, we define precisely the various automorphisms which lead to certain collections of subgroups being identified as one ‘case’ into which the diagonal quartic surface might fall. We then show how to list the 546 different cases, which are given in Appendix A.

After listing the various cases, certain useful computations are described: algorithms are given which implement the correspondence between subgroups of Γ and sets of coefficients (a_{i}). As the goal of these calculations is to study the Galois action on the Picard group of V̄, we then review some techniques for computing cohomology groups and apply them to the 48 lines on V̄.

Chapter 4 deals with the problem of moving between elements, computed in Chapter 3, of H^{1}(ℚ, Pic V̄) and elements of the Brauer group of V. While the correspondence between these objects is easy to write down in terms of maps of cohomology groups, performing such calculations explicitly is more difficult and we go into some detail. In particular, we describe how cyclic algebras appear under this correspondence. We then move onto studying the vertical Brauer group attached to a fibration, and again show the correspondence between Azumaya algebras and elements of a certain Galois cohomology group. An example is given showing how this material relates to the calculations of Swinnerton-Dyer [44].

Finally, in Chapter 5 we turn from algebra to numerical work. Useful algorithms are described for checking local solubility of the equations, and for searching for small solutions. We present some worked examples of finding the Brauer–Manin obstruction, and show some surfaces which have no arithmetic Brauer–Manin obstruction but no small solutions. This suggests that perhaps the Brauer–Manin obstruction is not the only obstruction to the Hasse principle on diagonal quartic surfaces, and thus on K3 surfaces more generally.

At several points we will implicitly assume that the particular diagonal quartic surface under investigation has points in every completion of ℚ. As this is an easily verified condition, and a necessary starting point for any search for rational points, this is a reasonable assumption and we will not always explicitly state it.

Most of the computations leading to this dissertation have been performed using the excellent number theory package PARI/GP, which is available freely under the GNU General Public License. Information on downloading and

using PARI/GP is at http://www.parigp-home.de.

Some of the algorithms and results described here may be downloaded

from http://www.boojum.org.uk/maths/quartic-surfaces/.

Geometry of the surface

In this chapter, we review various facts about the geometry of the surface (1.1). Much of the geometry is that which applies generally to the class of K3 surfaces, of which our surface is an example; so we begin in Section 2.1 by gathering together some results about these surfaces. Section 2.2 discusses the geometry of fibrations over the projective line, and also includes a proof of the well-known fact on which the techniques used in the following chapters rely: that the Picard group of our surface is generated by the 48 straight lines on it. In Section 2.3 we show how fibrations on V, defined over the rational numbers, may be found.

Note: In much of this chapter, we are concerned with complex surfaces. When we need to make a distinction between our surface (1.1) as a variety over ℚ and the corresponding variety over ℚ̄ or ℂ, we will refer clearly to the former variety as V and the latter as V̄.

2.1 K3 surfaces

In this section we gather some well-known results about K3 surfaces. The concepts used in this section may all be found in Hartshorne [21]; for more detailed information about the classification of surfaces see the book by Beauville [3]. Algebraic geometry on surfaces is covered very well in [22].

A more analytic approach, including a large chapter on K3 surfaces, is given in [2].

2.1.1 Definition of K3 surfaces

Recall that a surface X has two important birational invariants: the arithmetic genus p_{a}, defined by

p_{a}(X) = χ(X, O_{X}) − 1 = h^{2}(X, O_{X}) − h^{1}(X, O_{X});

and the geometric genus p_{g}, defined by

p_{g}(X) = h^{0}(X, ω_{X})

where ω_{X} = ⋀^{2} Ω_{X/k} is the canonical sheaf on X. In contrast to the situation for curves, these two quantities are not necessarily equal: we define the irregularity of X, written q(X), to be the difference

q(X) = p_{g}(X) − p_{a}(X) = h^{1}(X, O_{X})

since by Serre duality [21, III, 7] we know that p_{g} = h^{2}(X, O_{X}).

Definition 2.1. A K3 surface is a surface with trivial canonical sheaf and irregularity q = 0.

Proposition 2.2. The complex surface V̄ (1.1) is a K3 surface.

Proof. Any hypersurface of degree d in ℙ^{n} has ω_{X} ≅ O_{X}(d − n − 1) (see [21, II, 8.20.3]). Thus our particular surface has ωV̄ ≅ OV̄, that is, the canonical sheaf is trivial. To see that H^{1}(V̄ , OV̄) = 0, we use the exact sequence of sheaves on ℙ^{3}

0 → IV̄ → O_{ℙ^{3}} → OV̄ → 0

where IV̄ is the ideal sheaf of V̄. But IV̄ is isomorphic to O_{ℙ^{3}}(4), so has trivial cohomology in dimension 2; and O_{ℙ^{3}} has trivial cohomology in dimension 1 [21, III, 5.1]. Thus the long exact sequence in cohomology gives H^{1}(V̄ , OV̄) = 0, as the cohomology of OV̄ is the same taken on V̄ as on ℙ^{3}. (In fact, this approach works for any complete intersection in projective space: see [21, III, Ex. 5.5].)

2.1.2 Divisors

Here we state some properties of K3 surfaces which will be important in subsequent sections. In particular, we are interested in the Picard group of X; this turns out to be torsion-free and equal to the group of divisors modulo numerical equivalence, which makes computation in Pic X particularly simple.

Divisors are commonly defined in two different ways, as Weil divisors and as Cartier divisors; we will occasionally want to use both of these notions, so recall them now.

Definition 2.3. On a scheme X satisfying certain criteria (which certainly hold for a smooth projective variety), the group of Weil divisors is free group generated by the irreducible subschemes of codimension 1. Thus, on a surface, a prime Weil divisor is an irreducible curve, and a Weil divisor is a finite formal sum of irreducible curves. It is easy to define the valuation v_{D}(f) of a rational function f at an irreducible divisor D, as the local ring O_{X,D} is a discrete valuation ring; hence we can define the Weil divisor associated to a function f by

(f) = ∑_{D} v_{D}(f)D

where all but finitely many of the v_{D}(f) are non-zero. Two Weil divisors are said to be linearly equivalent if their difference is the divisor of a function, and the Weil divisor class group is defined to be the group of Weil divisors modulo linear equivalence.

Definition 2.4. On any scheme X, a Cartier divisor is a global section of the quotient sheaf D_{X} = K_{X}^{×}/O_{X}^{×}. Here K_{X} is the sheaf of rings formed locally from O_{X} by inverting all the elements which are not zero-divisors; so for an integral scheme it is simply the sheaf of rational functions.

Hence on a variety a Cartier divisor can be described by a covering of open sets {U_{α}}, together with rational functions f_{α} on each open set U_{α}, such that the restriction of f_{α}/f_{β} to U_{α} ∩ U_{β} lies in O_{X}^{×}(U^{α} ∩ U^{β}). Again we can define the divisor of a rational function f, this time simply by the quotient map K_{X}^{×} → K_{X}^{×}/O_{X}^{×}. For Cartier divisors on an integral scheme, we immediately see that the class group is isomorphic to H^{1}(X, O_{X}^{×}): this comes from the exact sequence in cohomology

⋯ → H^{0}(X, K_{X}^{×}) → H^{0}(X, K_{X}^{×}/O_{X}^{×}) → H^{1}(X, O_{X}^{×}) → H^{1}(X, K_{X}^{×})

where, since the sheaf of rational functions is flabby, its first cohomology group is trivial. There is a straightforward correspondence between Cartier divisors and invertible sheaves on X, which shows that (at least when X is integral) the Cartier divisor class group is isomorphic to the Picard group Pic X.

On a sufficiently nice scheme, such as a smooth projective variety, there is an isomorphism between the groups of Weil and Cartier divisors, and between the corresponding class groups [21, II, 6.11].

A notion of equivalence of divisors coarser than that of linear equivalence is given by algebraic equivalence.

Definition 2.5. Two divisors D_{1}, D_{2} on a variety X are said to be algebraically equivalent if there exists a smooth curve C and a divisor D on X × C such that D_{1}, D_{2} are the intersections of D with two fibres of the projection X × C → C.

It is clear that linearly equivalent divisors are algebraically equivalent (take C = ℙ^{1}). Algebraic equivalence is a geometric version of the topological concept of two cycles being homologous: in fact, on a variety over ℂ, the two notions coincide (see Griffiths & Harris [17, Lemma, page 462]).

Definition 2.6. The group of divisors modulo algebraic equivalence is called the Néron–Severi group of X and written NS(X). The subgroup of Pic X consisting of divisors algebraically equivalent to 0 is written Pic^{0} X; thus we have NS(X) = Pic X/ Pic^{0} X.

Now let X be a smooth projective surface. Given any two irreducible curves D_{1}, D_{2} in X, we can define their intersection number D_{1}.D_{2}. The construction is described in detail in [21, V, 1.1]. Briefly, if D_{1} and D_{2} intersect transversely, then their intersection number is the number of points of intersection; we then extend linearly to divisors which do not intersect transversely. This gives a symmetric bilinear form on Div X. It can be shown that, if D_{1} and D′_{1} are algebraically equivalent, then D_{1}.D_{2} = D′_{1}.D_{2} for all divisors D_{2}. Thus we have a symmetric bilinear form NS(X) × NS(X) → ℤ.

Definition 2.7. An element of the (left or right) kernel of the intersection form, that is, a divisor having intersection number zero with every other divisor, is said to be numerically equivalent to 0; the subgroup of Pic X consisting of such divisors is written Pic^{n} X.

By linearity, Pic^{n} X must contain all the torsion elements of NS(X); in fact, the converse is also true: Pic^{n} X consists of those divisors which are torsion in NS(X). (This is essentially due to the Riemann–Roch theorem: [29, V, Lemma 3.27] shows that Pic^{n} X/ Pic^{0} X is finite, hence torsion.)

Let X be any projective K3 surface. For any divisor D on X, let L(D) denote the invertible sheaf associated to D. The Riemann–Roch theorem [21, V, 1.6] gives

χ(X, L(D)) = \frac{1}{2}D.D + 2.  (2.1)

We will write ℓ(D) for dim H^{0}(X, L(D)) and s(D) for dim H^{1}(X, L(D)), the superabundance of D. It follows from Serre duality that

χ(X, L(D)) = ℓ(D) − s(D) + ℓ(−D).

If D is a non-zero effective divisor, then −D is certainly not effective so ℓ(−D) = 0. Let Π be the class of a plane section of X. If D is a non-zero effective divisor then D.Π ≥ 1. Conversely, if D.Π ≥ 1 then −D cannot be equivalent to an effective divisor, so again ℓ(−D) = 0.

If D is a non-singular curve, we can also control s(D):

Lemma 2.8. If D is a non-singular curve in X, then H^{1}(X, L(D)) = 0.

Proof. Let i : D → X be the inclusion map. We consider the exact sequence of sheaves on X

0 → L(−D) → O_{X} → i_{∗}O_{D} → 0

associated to the subscheme D. Tensoring this with L(D) and taking the long exact sequence in cohomology, we get

\begin{matrix}⋯ → H^{1}(X, O_{X}) → H^{1}(X, L(D)) → H^{1}(X, L(D) ⊗ i_{∗}O_{D}) → \\ → H^{2}(X, O_{X}) → H^{2}(X, L(D)) → · · ·. \end{matrix} (2.2)

Now X has irregularity 0, and so H^{1}(X, O_{X}) = 0. We also know that H^{2}(X, O_{X}) has dimension 1 and that H^{2}(X, L(D)) = 0, both by Serre duality. On the other hand, Proposition 8.20 of [21, II] states that

ω_{D} ≅ ω_{X} ⊗ L(D) ⊗ O_{D};

in our case this reduces to Ω_{D} ≅ L(D) ⊗ O_{D}. But now

H^{1}(X, L(D) ⊗ i_{∗}O_{D}) ≅ H^{1}(D, L(D) ⊗ O_{D})
≅ H^{1}(D, Ω_{D})
≅ H^{0}(D, O_{D}) by Serre duality

so H^{1}(X, L(D)⊗i_{∗}O_{D}) has dimension 1. Putting this into the exact sequence (2.2), we get the desired result.

For any effective divisor D, we know that the arithmetic genus p_{a}(D) satisfies 2p_{a}(D) − 2 = D.D and we may use this to define p_{a}(D) when D is not effective [21, V, Ex. 1.3].

We now put all these facts together into the following useful form of the Riemann–Roch theorem:

Proposition 2.9. Let X be any K3 surface, and let Π be a very ample divisor on X. If D is an effective divisor on X or, more generally, any divisor such that D.Π > 0, then the following inequality holds:

ℓ(D) ≥ p_{a}(D) + 1 = \frac{1}{2}D.D + 2.

If D is equivalent to a non-singular curve, then equality holds.

In particular, non-singular rational curves are isolated, and non-singular curves of genus 1 always lie in pencils.

We now show how the various notions of equivalence for divisors fit together on a K3 surface.

Proposition 2.10. On a K3 surface X over ℂ, the following notions of equivalence of divisors are the same:

1. linear equivalence;

2. algebraic equivalence;

3. homology;

4. numerical equivalence.

Proof. Firstly, we have already stated that algebraic equivalence and homology are the same. Now the exponential sequence of (analytic) sheaves

0 → ℤ → O_{X} ⟶^{exp} O_{X}^{×} → 0

gives rise to an exact sequence in cohomology

H^{1}(X, O_{X}) → H^{1}(X, O_{X}^{×}) ⟶^{α} H^{2}(X, ℤ)

where the map α associates to a divisor in Pic X ≅ H^{1}(X, O_{X}^{×}) its Poincaré dual cohomology class [22, 3.3]. So the kernel of α consists of the divisors homologous, and so algebraically equivalent, to 0; and the image of α is isomorphic to the Néron–Severi group of X. (In complex geometry, this is often used as the definition of the Néron–Severi group.) But for a K3 surface, H^{1}(X, O_{X}) = 0; thus α gives an isomorphism from Pic X to NS(X).

To show that algebraic and numerical equivalence coincide, we must establish that Pic X is torsion-free. Suppose that D is a divisor on X such that nD is principal, for some positive integer n. Then D.D = 0, and putting this into the Riemann–Roch formula (2.1) gives χ(X, L(D)) = 2, so either D or −D is equivalent to an effective divisor. But if D is a non-zero effective divisor, then nD cannot be principal. The same holds for −D. So D = 0, and hence Pic X is torsion-free.

Finally, the rank of Pic X is easily bounded by Noether’s formula (see [22], 1).

Proposition 2.11. Let X be a K3 surface over ℂ; then Pic X is free of rank no more than 20.

Proof. Using all the symmetries we get from Hodge theory and Serre duality, the definition of a K3 surface gives

h^{0,0} = h^{2,0} = h^{0,2} = h^{2,2} = 1
h^{1,0} = h^{0,1} = h^{1,2} = h^{2,1} = 0

and therefore Betti numbers b_{0} = b_{4} = 1, b_{1} = b_{3} = 0. Now Noether’s formula shows that the topological Euler characteristic of X is 24, so b_{2} = 22 and therefore h^{1,1} = 20. But the image of Pic X in H^{2}(X, ℤ) lies in H^{1,1}(X) [22, 3.3], so has rank at most h^{1,1} = 20.

2.2 Fibrations

Here we discuss fibrations of surfaces over curves. In particular, we are interested in fibrations of our surface V (1.1) over ℙ^{1}, where the fibres are curves of genus one. Some information on fibrations may be found in [2] and [22]; a more detailed treatment appears in Silverman’s book on elliptic curves [39], which is aimed at the study of elliptic surfaces with a section but contains much useful information for our situation.

2.2.1 Divisors on fibrations

It will be helpful to have a good understanding of the theory of divisors on fibrations, so here we gather some relevant facts. In this section, X is any smooth complete surface over a field k, C is a smooth complete curve over k, and π : X → C is a non-constant morphism. If t is a point of C, we write X_{t} for the fibre X ×_{C} t. We will be particularly interested in the generic fibre X_{η}, where η is the generic point of C.

Following [39], we make the following definitions.

Definition 2.12. A prime divisor on X is fibral if its image under π consists of a finite number of points, and horizontal otherwise, that is, when the image is the whole of C.

For each point t of C, there is a natural map of groups from Pic X to Pic X_{t}. This arises from the map of sheaves O_{X}^{×} → i_{∗}O_{X_{t}}^{×}, where i is the inclusion map X_{t} → X; thus we have

H^{1}(X, O_{X}^{×}) → H^{1}(X, i_{∗}O_{X_{t}}^{×})

where the left-hand group is Pic X; we claim that the right-hand group is naturally isomorphic to Pic X_{t}. If t is a closed point of C, then i is a closed immersion and hence i_{∗} an exact functor (see [28, Corollary 8.4]). If t is η, the generic point of C, then the claim follows from this lemma, when combined with the Leray spectral sequence:

Lemma 2.13. Let j : X_{η} → X be the inclusion of the generic fibre. Then R^{1}j_{∗}𝔾_{m} = 0.

Proof. See Lemma 4.4.1 of [11].

We study this map of Picard groups in more detail, and show that the kernel is the same in all cases.

First let t be a closed point of C. Then the fibre X_{t} is a curve over k. The map i is a closed immersion, so O_{X} → i_{∗}O_{X_{t}} is surjective, and hence so is O_{X}^{×} → i_{∗}O_{X_{t}}^{×}. We will describe the map of Picard groups first in terms of Cartier divisors.

A Cartier divisor D on X is a covering of X by pieces of rational functions; suppose that X_{t} is not contained in the support of D: that is, the rational functions have no zeros or poles along the curve X_{t}, or equivalently that the functions are invertible elements of O_{X,X_{t}}. Then restricting the functions to X_{t} gives a Cartier divisor on X_{t}. This gives a map from most of Div X into Div X_{t}. It is easy to see that this map respects linear equivalence. Moreover, every divisor is equivalent to one whose support does not contain X_{t}; so we have defined a natural map

Pic X → Pic X_{t}.

Of course, this map also admits a description in terms of Weil divisors: any Weil divisor whose support does not contain X_{t} intersects X_{t} is a finite number of points; the image of the divisor in Pic X_{t} is just those points, taken with appropriate intersection multiplicities.

Now consider the generic fibre X_{η}. This is a curve over k(C). The function field of X_{η} is the same as that of X. The sheaf of regular functions on X_{η} is obtained from O_{X} by inverting all the functions lying in the inverse image π^{∗}O_{C}; so, algebraically, it is just a localisation of O_{X}. Several useful properties follow immediately: the points of X_{η} are a subset of those of X, and in fact the closed points of X_{η} correspond precisely to the horizontal divisors on X. Note that X_{η} may well not have any k(C)-valued points: such a point would be a horizontal divisor D on which π restricts to an isomorphism D → C, that is, a section of π. In general, π gives a map D → C of some degree, often greater than 1; so D corresponds to a point of X_{η} whose residue field is some finite extension of k(C).

Let j denote the inclusion X_{η} → X, and let U be an open set in X. Then U is the complement of finitely many closed subschemes {Z_{i}} in X, which we can split into points, fibral divisors and horizontal divisors. The inverse image j^{−1}U should be an open set in X_{η}, that is, the complement of finitely many horizontal divisors; and following the definitions through shows that in fact j^{−1}U is simply the complement of those Z_{i} which are horizontal divisors. Knowing how j^{−1} acts on open sets allows us to describe the induced map on Picard groups.

A Cartier divisor on X is a covering of X by open sets U_{α}, together with rational functions f_{α} on each U_{α} such that f_{α}/f_{β} is a regular function on U_{α} ∩ U_{β} for all α, β. On the other hand, a Cartier divisor on X_{η} requires that the open sets U_{α} all be complements of finite sets of horizontal divisors (as opposed to complements of finite sets of arbitrary divisors); but now the f_{α}/f_{β} only have to be regular on X_{η}: that is, the divisor (f_{α}/f_{β}) may have fibral components but not horizontal ones. So, given a Cartier divisor on X, we can turn it into a Cartier divisor on X_{η} just by replacing each open set U_{α} by j^{−1}U_{α}; this process might mean that the f_{α}/f_{β} are no longer regular on the enlarged intersections U_{α}∩U_{β}, but we can only have introduced problems along fibral divisors, which as remarked does not affect whether the f_{α}/f_{β} is regular as a function on X_{η}.

On Weil divisors, the map is entirely obvious: a horizontal divisor maps to itself, and a fibral divisor maps to zero.

Proposition 2.14. The map Pic X → Pic X_{η} is surjective; its kernel is generated by the irreducible components of the fibres of π.

Proof. Let j denote the inclusion X_{η} → X. The sheaf j_{∗}O_{X_{η}} consists of those rational functions on X which are regular on X_{η}, that is, whose divisors contain only fibral components. So the map O_{X} → j_{∗}O_{X_{η}} is injective. As j_{∗}O_{X_{η}}^{×} is a subsheaf of the sheaf K_{X} of rational functions on X, so we may regard the quotient sheaf

F = j_{∗}O_{X_{η}}^{×}/O_{X}^{×}

as a subsheaf of K_{X}^{×}/O_{X}^{×}; but that is the sheaf of divisors, D_{X}. In fact F consists precisely of the divisors of functions regular on X_{η}, that is, the (locally) fibral divisors. We get an exact sequence in cohomology

H^{0}(X, F) → H^{1}(X, O_{X}^{×}) → H^{1}(X, j_{∗}O_{X_{η}}^{×}) → H^{1}(X, F)

where the first term is the group of fibral divisors, and the last term is zero because F is a direct summand of the sheaf of divisors, which has trivial 1dimensional cohomology (see Lemma 4.3). The middle two terms are Pic X and Pic X_{η} respectively, and the kernel of the map between them therefore is the subgroup of Pic X generated by the fibral divisors.

When the fibration π is fixed, we will refer to the kernel of the map Pic X → Pic X_{η} as the vertical Picard group, and write it as Pic_{vert}. This group will be further studied in Section 4.4, where we will be investigating the vertical Brauer group attached to a fibration.

2.2.2 The Picard group of the surface

The information about fibrations in the previous section allows us to prove the following well-known result, which describes precisely the Picard group of our surface V (1.1).

Proposition 2.15. The Picard group of the complex variety V̄ is a free Abelian group of rank 20, and is generated by the classes of the 48 straight lines on V̄.

As we are only concerned here with the structure of V̄ as a complex variety, the coefficients a_{i} make no difference and to prove Proposition 2.15 we may assume that the equation of the surface is

X_{0}^{4} − X_{1}^{4} = X_{2}^{4} − X_{3}^{4}.  (2.3)

Let Λ denote the subgroup of Div V̄ generated by the 48 lines. We must show that the image of Λ in Pic V̄ is the whole of Pic V̄.

Firstly, it is a simple matter to calculate the intersection numbers of the 48 lines: two distinct straight lines intersect with multiplicity 1 or not at all, and the self-intersection of each line is −2. This allows us to compute the subgroup of Λ^{0} of Λ consisting of those divisors numerically equivalent to 0, and hence (by Proposition 2.10) linearly equivalent to 0. The quotient Λ/Λ^{0} turns out to have rank 20, so by Proposition 2.11 the image of Λ is a subgroup of finite index in Pic V̄.

Lemma 2.16. 1. The determinant of the intersection form on the image of Λ is −64.

2. The index of the image of Λ in Pic V̄ is a power of 2.

Proof. The first statement comes from a simple computation, given that we already know the intersection numbers of the 48 lines. For the second statement, observe that the index of the image Λ in Pic V̄ must divide the determinant of the image of Λ.

We will use the fibration described in [44] and some standard results on elliptic curves to show that in fact the image of Λ is the whole of Pic V̄.

The fibration is defined as follows: we rewrite (2.3) as

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

which is then a covering in the obvious way of the quadric surface defined by the equation

(Y_{0} + Y_{1})(Y_{0} − Y_{1}) = (Y_{2} + Y_{3})(Y_{2} − Y_{3}).

Now this surface is isomorphic to ℙ^{1} ×ℙ^{1}, so has two pencils of straight lines: we will use the one defined by

y(Y_{0} + Y_{1}) = z(Y_{2} + Y_{3}) and z(Y_{0} − Y_{1}) = y(Y_{2} − Y_{3})

which, when lifted to V̄, gives us the pencil of curves of genus 1 with equation

y(X_{0}^{2} + X_{1}^{2}) = z(X_{2}^{2} + X_{3}^{2}) and z(X_{0}^{2} − X_{1}^{2}) = y(X_{2}^{2} − X_{3}^{2}). (2.4)

This is the fibration over ℙ^{1} which we will now use to investigate the structure of Pic V̄. As in the previous section, we write η for the generic point of ℙ^{1} and Pic_{vert} for the vertical Picard group, that is, the subgroup of Pic V̄ generated by the irreducible components of the fibres of π. The map from Pic V̄ to the Picard group of the generic fibre of this fibration gives us, by Proposition 2.14, an exact sequence

0 → Pic_{vert} → Pic V̄ → Pic V̄_{η} → 0

where Pic_{vert} is generated by the irreducible components of the fibres, and V̄_{η} is a curve of genus 1 over ℂ(t) (where t = y/z).

Lemma 2.17. Pic_{vert} is contained in the image of Λ.

Proof. Looking at the partial derivatives of the equations (2.4) reveals that the singular fibres occur when y/z is equal to 0, ±1, ±i and ∞. The fibres can then be found by substituting these values of y/z into (2.4), and we find that each of the singular fibres consists of four straight lines forming a skew quadrilateral.

We must now show that the image of Λ in Pic V̄_{η} is the whole of Pic V̄_{η}. We write Λ_{vert} for the subgroup of Λ generated by the irreducible components of the fibres. Further computations (using the methods described in Section 3.2) reveal the following facts.

Lemma 2.18. 1. the image in Pic V̄ of Λ_{vert}, that is, Pic_{vert}, has rank 19;

2. the torsion part of the quotient Λ/(Λ^{0}+Λ_{vert}) is isomorphic to (ℤ/4ℤ)^{2};

3. the intersection number of any element of Λ with a fibre is even.

The first of these facts means that the image of Λ in Pic V̄_{η} has rank 1; and, as the image of Λ is of finite index, therefore Pic V̄_{η} has rank 1. Now the degree map on Pic V̄_{η} gives an exact sequence

0 → Pic^{0} V̄_{η} → Pic V̄_{η} ⟶^{deg} ℤ

and we must show that Pic^{0} V̄_{η} is contained in the image of Λ, and further that the image of Λ → Pic V̄_{η} → ℤ is the same as the image of Pic V̄_{η} in ℤ.

It follows from the first of the above observations that the rank of Pic V̄_{η} is 1. The degree map on a curve is non-constant; therefore the degree 0 part Pic^{0} V̄_{η} is finite and is just the torsion subgroup of Pic V̄_{η}. It is at least as large as (ℤ/4ℤ)^{2}, since that is the part generated by Λ; we must show that this is the whole of Pic^{0} V̄_{η}. On the other hand, Pic^{0} V̄_{η} is isomorphic to the group of ℂ(t)-valued points on the Jacobian of V̄_{η}, which [44] shows may be described by the equation

E : Y^{2} = X(X + (t^{2} − 1)^{2})(X + (t^{2} + 1)^{2}).  (2.5)

Lemma 2.19. The Mordell–Weil group of the curve (2.5) over ℂ(t) is isomorphic to (ℤ/4ℤ)^{2}.

Proof. By the second statement of Lemma 2.18, the Mordell–Weil group of E contains a subgroup isomorphic to (ℤ/4ℤ)^{2}. By the second statement of Lemma 2.16, the index of this subgroup in the whole group is a power of 2; so it suffices to find the 4-division points of E which are not 2-division points, and show that they are not divisible by 2.

Hand calculation quickly gives the non-trivial 4-division points to be

• four points with X = ±(t^{4} − 1), which double to give (0, 0);

• four points given by X = −(t^{2} − 1)(t ± i)^{2}, which when doubled give (−(t^{2} − 1)^{2}, 0);

• four points given by X = −(t^{2} + 1)(t ± 1)^{2}, which when doubled give (−(t^{2} + 1)^{2}, 0).

A point P on an elliptic curve having rational 2-division points P_{1}, P_{2}, P_{3} is divisible by 2 if and only if the three quantities X(P) − X(P_{i}) are all squares [38, X, §1]; but the points listed clearly do not satisfy this condition.

To complete the proof of Proposition 2.15, we must show that the image of Λ under the degree map Pic V̄_{η} → ℤ is the whole of the image of Pic V̄_{η}. The degree map is the same as the intersection number with a fibre; as stated above, calculation shows that the smallest non-zero intersection number of an element of Λ with a fibre is 2. Therefore the image of Pic V̄_{η} under the degree map may only be either ℤ or 2ℤ.

Lemma 2.20. The image of the degree map Pic V̄_{η} → ℤ is 2ℤ.

Proof. Suppose that V̄_{η} contained a divisor of degree 1; the Riemann–Roch theorem shows that such a divisor is equivalent to an effective divisor of degree 1: that is, a ℂ(t)-valued point of V̄_{η}. Then V̄_{η} would be isomorphic to its Jacobian E (2.5). To show that this is not the case, we use the wellknown theory of 2-coverings (see [38, Chapter X]). The group H^{1}(ℂ(t), E[2]) classifies the 2-coverings of E; as all the 2-division points are defined over ℂ(t), we have E[2] ≅ μ_{2}^{2} and hence H^{1}(ℂ(t), E[2]) ≅ (ℂ(t)^{×}/(ℂ(t)^{×})^{2})^{2}. There is a coboundary map from E(ℂ(t))/2E(ℂ(t)) to (ℂ(t)^{×}/(ℂ(t)^{×})^{2})^{2} given by

P ↦ (X(P), X(P) + (t^{2} − 1)^{2})

and those 2-coverings which have a ℂ(t)-valued point are those lying in the image of this map. But, as stated in [44], equation (23), V̄_{η} is a 2-covering of E associated to the element (t^{4} − 1, t(t^{2} − 1)). On the other hand, the points of E are listed above and their images are

(1, 1), (t^{4} − 1, t^{2} − 1), (t^{2} − 1, t(t^{2} − 1)), (t^{2} + 1, t)

which form a 4-group not containing (t^{4} − 1, t(t^{2} − 1)). Hence V̄_{η} has no ℂ(t)-valued point.

Therefore Pic V̄_{η} is indeed generated by the image of Λ, and this completes the proof of Proposition 2.15.

2.2.3 Picard groups over ℚ

If X is an arbitrary variety over a number field k, then we can define Pic X to be the group of divisors on X modulo principal divisors. Here a divisor on X is a closed subscheme of codimension 1 which is defined over k. This is subtly different from H^{0}(k, Pic X̄), which consists of those divisor classes fixed by the Galois action of k̄/k: for it is quite possible for a divisor class to be fixed by the Galois action but contain no divisors which are themselves fixed. This distinction is rather tiresome, but fortunately a standard result shows that in the interesting cases, that is, where X has a point in each completion k_{v} of k, the two groups Pic X and H^{0}(k, Pic X̄) are the same.

Proposition 2.21. Let X be a smooth variety over a number field k, and suppose that X(k_{v}) ≠ ∅ for each completion k_{v} of k. Then Pic X = H^{0}(k, Pic X̄).

Proof. Let D be a divisor on X̄ in a class fixed by the action of Gal(k̄/k). If Π denotes a very ample divisor on X, then there exists an integer m such that D + mΠ is effective; and D + mΠ is equivalent to a k-rational divisor if and only if D is; thus to prove the proposition we may assume that D is effective. We may also assume that the linear system |D| has no fixed part: for if it does, then the fixed part is Galois-invariant and we may subtract it. Finally, if ℓ(D) = 1 then D is the only effective divisor in its equivalence class, and is therefore defined over k; so we now assume that ℓ(D) > 1.

Write L for |D|. The linear system L is parametrised by the scheme ℙ(L(D)), making it by definition a Severi–Brauer variety. Such varieties were shown by Châtelet [7] to satisfy the Hasse principle. We will show that L contains a divisor defined over k_{v} for each v, and deduce that L contains a divisor defined over k.

Fix a place v of k. By the hypothesis, there is a point in X(k_{v}) for each v. In fact, in a neighbourhood of such a point a local parametrisation shows that there are infinitely many more. We choose x_{v} to be a point of X(k_{v}) which is not one of the finitely many base points of L. As dim L ≥ 1, there is some divisor D′ in L, defined over a finite extension of k_{v}, on which x_{v} lies. If D′ is defined over k_{v}, we are finished; otherwise, note that x_{v} also lies on each conjugate of D′ over k_{v}; we conclude that these conjugates do not span the whole linear system L, for if they did then x_{v} would be a base point of L. We therefore replace L by the linear system spanned by the conjugates of D′, which has dimension at least 1 but strictly less than that of L, and start again. As the dimension of L decreases each time, this process must eventually find a D′ in the linear system |D| and defined over k_{v}.

In this dissertation we are always primarily concerned with surfaces V which are everywhere locally soluble: for otherwise there is no reason to study the Brauer–Manin obstruction. Therefore we will sometimes assume that Pic V and H^{0}(ℚ, Pic V̄) are the same without explicitly stating this.

2.3 Fibrations on V

Some K3 surfaces, in particular our surface V̄ (1.1), may be given the structure of a fibration over ℙ^{1}, with fibres being curves of genus 1. When such a fibration is defined over the rational numbers, it may be used as in [44] to obtain information about the Brauer group of V. In this section we study such fibrations. Here V will always denote the surface considered as a variety over the rational numbers, and V̄ the same surface as a variety over ℚ̄.

2.3.1 Finding fibrations

We are interested in finding fibrations V → ℙ^{1} defined over ℚ. The calculations described in Chapter 3 allow the computation of H^{0}(ℚ, Pic V̄), the part of the Picard group which is defined over ℚ; when V has a point in each completion ℚ_{v} of ℚ, this is equal to Pic V. The existence of fibrations on K3 surfaces over ℂ has been much studied; in particular, an account is given in [33], where it is proved that a K3 surface has a fibration in curves of genus 1 if and only if the intersection form on the Picard group represents zero. The version of the Riemann–Roch theorem stated above in Proposition 2.9 already shows that any nonsingular curve of genus 1 occurs as part of a linear system of dimension 1; and, as such curves have self-intersection zero, the linear system has no base points; thus any nonsingular curve of genus 1 defines a fibration over ℙ^{1}. The first step in [33] is to turn this into a more useful criterion:

Lemma 2.22. Let D be a divisor class on V such that D^{2} = 0 and D has non-negative^{1} intersection number with all effective divisors. Then D is equal to mF, where m is a positive integer and F a divisor such that the linear system |F | defines a fibration V → ℙ^{1}.

Proof. See [33].

In fact the second condition is simpler than it might appear. In the language of [2, VIII], we require that D lie in the closure of the Kähler cone of divisors on V; but Corollary 3.8 there shows that the Kähler cone is characterised by the fact that D.C is positive for every rational curve C on V.

1In fact, the statement in [33] requires that D have strictly positive intersection number with all effective divisors. This is clearly wrong: for a start, D.C = 0 where C is any component of a fibre. The proof, though, assumes that the intersection number is merely non-negative.

The second step is to show that, by using certain automorphisms of the Picard lattice called reflections, any divisor D satisfying D^{2} = 0 may be transformed into one satisfying the conditions of the lemma. If C is a rational curve on V, then the reflection s_{C} is defined by

s_{C}(D) = D + (D.C)C.

One checks that this defines an involution of Pic V̄ which preserves intersection numbers. This part of the proof does not immediately work over ℚ, but can be adapted to do so. I am indebted to Professor Shepherd-Barron for indicating this method of proof.

Proposition 2.23. The surface V has a fibration V → ℙ^{1} in curves of genus 1, defined over ℚ, if and only if the intersection form on H^{0}(ℚ, Pic V̄) represents zero non-trivially.

Proof. We follow the proof of [33, §6, Theorem 1]. Let D be a non-trivial element of H^{0}(ℚ, Pic V̄) such that D^{2} = 0. Then D.Π cannot equal zero, by the Hodge index theorem, so we assume (replacing D by −D if necessary) that D.Π > 0. Then D is effective, by the Riemann–Roch theorem (Proposition 2.9).

If D does not satisfy the conditions of Lemma 2.22, then there is a rational curve C on V such that D.C < 0. Let C_{i,j} be the conjugates of C over ℚ, numbered such that {C_{1,j}} form one connected component, {C_{2,j}} another, and so on. Each connected component is defined over the same finite extension K/ℚ. We will now use the theory of the fundamental cycle, as introduced in [1]. Consider first the connected component {C_{1,j}}. As D is defined over ℚ, we have that D.C_{1,j} < 0 for all j. Therefore, by Proposition 2 of [1], the intersection form on the components {C_{1,j}} is negative definite; in other words, this connected component is an A-D-E curve, as defined in [2, III.3]. The fundamental cycle Z_{1} of this connected component is defined to be the unique minimal effective divisor

Z_{1} = ∑_{j} a_{j}C_{1,j}

such that Z_{1} has non-positive intersection number with each C_{1,j}. It turns out that Z_{1}^{2} = −2 (see [2, III, Proposition 3.6]). Also, Z_{1} is defined over K: for otherwise its conjugates over K would have the same properties, contradicting uniqueness. Similarly we define Z_{i} for each connected component {C_{i,j}}; it is clear that the Z_{i} are the complete set of conjugates over ℚ of Z_{1}.

Note that D.Z_{1} < 0: otherwise, we would have ∑_{j} a_{j} < 0, so Z_{1}.Π < 0, contradicting that Z_{1} is effective.

Now the proof proceeds as in [33]. We replace D by

D′ = D + (D.Z_{1}) ∑_{i} Z_{i}

which is the result of applying the reflections s_{Z_{i}} in turn: as the Z_{i} do not intersect, these reflections commute. Then D′ is defined over ℚ. Since reflections preserve intersection number and the positive cone, we still have (D′)^{2} = 0 and D′.Π > 0. Moreover,

D′.Π = D.Π + (D.Z_{1}) ∑_{i} Z_{i}.Π < D.Π

as each Z_{i} is effective. Repeating this procedure, we must eventually reach a D which satisfies the conditions of Lemma 2.22.

As the lattice Pic V is of a very specific type, we can give more detail about exactly when a pencil of curves of genus 1 exists.

We begin with a simple lemma about lattices. Definition 2.24. A lattice is a free Abelian group equipped with a nondegenerate symmetric bilinear form, written ( , ). The determinant of a lattice Λ, written det Λ, is the determinant of the matrix (e_{i}, e_{j}) where (e_{i}) is any basis for Λ. Lemma 2.25. Let Λ be a lattice. Suppose that a finite group G acts on Λ preserving the bilinear form. Then the determinant of H^{0}(G, Λ) divides |G|^{dim Λ} det Λ. Proof. We write A for H^{0}(G, Λ). Denote by A^{⊥} the orthogonal complement of A, that is, the group of elements x of Λ such that (x, y) = 0 for all y in A.

Since the bilinear form is non-degenerate, A and A^{⊥} have trivial intersection. Now, as vector spaces, we have

Λ ⊗ ℚ = (A ⊗ ℚ) ⊕ (A^{⊥} ⊗ ℚ)

essentially by the Gram–Schmidt algorithm. We will show that |G|Λ is contained in A ⊕ A^{⊥}, and hence of finite index in A ⊕ A^{⊥}.

Let x be any element of Λ; then, as the action of G preserves the bilinear form, x − gx lies in A^{⊥} for all g ∈ G. Summing over all g gives

|G|x − Nx ∈ A^{⊥}

where Nx denotes ∑_{g∈G} gx. But Nx lies in A, so therefore |G|x is in A+A^{⊥}. This shows that |G|Λ is of finite index in A ⊕ A^{⊥}. Therefore

det(A ⊕ A^{⊥}) divides det |G|Λ
i.e. det A × det A^{⊥} divides |G|^{dim Λ} det Λ
so det A divides |G|^{dim Λ} det Λ.

We use this lemma to find the discriminant of the intersection form, which by definition is the determinant of the lattice H^{0}(ℚ, Pic V̄). Corollary 2.26. Let ρ denote the rank of H^{0}(ℚ, Pic V̄). If ρ = 1, then the determinant of H^{0}(ℚ, Pic V̄) is 4. If ρ > 1, then the determinant of H^{0}(ℚ, Pic V̄) is of the form (−1)^{ρ+1}2^{k} for some integer k. Proof. If ρ = 1 then H^{0}(ℚ, Pic V̄) is generated by the class Π of a plane section, which has self-intersection 4.

Suppose that ρ > 1. We shall see in Chapter 3 that the Galois action on Pic V̄ factors through a finite group G which is a subgroup of a certain group Γ of order 2^{8}. The determinant of Pic V̄ is −2^{6}, as is easily found by writing down a set of generators. Lemma 2.25 shows that the determinant of H^{0}(G, Pic V̄) divides −64 × 2^{8×20}; and the Hodge index theorem gives the sign.

Corollary 2.27. Let V be a diagonal quartic surface, and let ρ denote the rank of Pic V.

• If ρ = 1, then V does not have a fibration in curves of genus 1.

• If ρ = 2, then V has a fibration in curves of genus 1 if and only if − det H^{0}(ℚ, Pic V̄) is a square.

• If ρ ≥ 3, then V always has a fibration in curves of genus 1.

Proof. By Proposition 2.23, we must decide when the intersection form on H^{0}(ℚ, Pic V̄) represents 0.

The cases ρ = 1, 2 are trivial. For ρ ≥ 5, by [36, Chapter IV, 3.2, Corollary 2], all we need show is that the form is indefinite; but this is true, as the discriminant is negative. This leaves ρ = 3, 4.

If ρ = 3, then [36, Chapter IV, 3.2, Corollary 3] shows that it is enough to show that the intersection form represents 0 in ℚ_{v} for all v except one. But the discriminant is a unit at all odd primes, so the form does represent 0 in all ℚ_{p} for p ≠ 2, and also in ℝ. Therefore it represents 0 in ℚ.

If ρ = 4, then we must show that the intersection form represents 0 in every ℚ_{v}. In ℚ_{p}, p an odd prime, and ℝ this is trivial. In ℚ_{2}, by [36, Chapter IV, 2.2, Theorem 6] it is enough to show that the discriminant is not a square in ℚ_{2}. But the determinant is −1 × 2^{k} for some k; and −1 and 2 are independent elements of ℚ_{2}^{×}/(ℚ_{2}^{×})^{2} (see, for example, [36, Chapter II, 3.3, Corollary to Theorem 4]). Hence the discriminant is not a square in ℚ_{2}, and the result follows.

2.3.2 Finding the singular fibres

Suppose that we are given an element F ∈ Pic V̄ which is the class of the fibres of a fibration V → ℙ^{1}. In order to study the vertical Brauer group attached to this fibration (see Section 4.4), we need to know generators for the group Pic_{vert}, that is, we must find the irreducible components of each singular fibre. It is possible, though rather messy, to write down an explicit formula for the morphism from V to ℙ^{1}; and given such a formula, one can solve certain equations in partial derivatives to find the singular fibres. However, there is a more efficient way to find these fibres, which we now describe.

The possible structures which can be taken by singular fibres of an elliptic fibration are described by the Kodaira–Néron classification [39, Theorem 8.2]. A rather more elementary observation is that the singular fibres may only contain rational curves. Moreover, the degree of the rational curves which may occur in singular fibres is bounded, since any fibre may be described by a set of equations of fixed degree.

Lemma 2.28. There are only finitely many rational curves on V̄ of a given degree.

Proof. The set of rational curves on V̄ of degree d corresponds to the set of divisor classes in Pic V̄ satisfying

D^{2} = −2 and D.Π = d  (2.6)

where Π is the class of a plane section. We work in Pic V̄ ⊗ ℝ, and let D_{0} = (d/Π^{2})Π; then D_{0} satisfies the second equation, and writing D = D_{0} + D′ we must solve

(D′ − D_{0})^{2} = −2 and D′.Π = 0.  (2.7)

The second equation defines a plane through the origin, and rearranging the first gives

(D′)^{2} = −2 − (d^{2}/Π^{2}).

Now the Hodge Index Theorem [21, V, Theorem 1.9] states that the intersection form is negative definite on the set of D′ satisfying D′.Π = 0; thus the solution set of (2.7) is compact, so is that of (2.6), and the set of points of Pic V̄ lying in the solution set is discrete and hence finite.

This proof indicates how all the rational curves of a given degree may be enumerated: the problem is to find all D′ in Pic V̄ ⊗ ℝ such that (D′)^{2} is equal to a certain constant, and D′ + D_{0} lies in Pic V̄.

More generally, we consider the following problem: given a positive definite quadratic form Q on ℝ^{n}, a vector α ∈ ℝ^{n}/ℤ^{n} and a bound C, find all vectors v ∈ ℤ^{n} + α such that Q(v) ≤ C. When α = 0, this is the problem solved by the algorithm of Fincke and Pohst described in [16] and [8]. The algorithm is easily modified to cope with the offset α.

To apply such an ‘inhomogeneous’ Fincke–Pohst algorithm to the problem of finding divisors with prescribed degree and self-intersection, we construct the following algorithm. In this form it can also be used when working over ℚ instead of over ℚ̄. We assume that there is a standard basis chosen for Pic V̄, and that we already have a matrix N whose columns represent, in this standard basis, a basis for Pic V. Such a matrix is computed using the algorithms of Chapter 3.

Note: For the purposes of this algorithm, we assume that the first column of N, that is, the first generator of Pic V, represents the class of a plane section.

Algorithm 2.29. Given a 20 × ρ matrix N whose columns describe a basis for Pic V, a 20×20 matrix Q representing the intersection form on Pic V̄, and integers d and n, this algorithm produces a matrix whose columns describe all elements of Pic V with degree d and self-intersection at least n.

1. If d^{2}/4 < n then return an empty matrix.

2. If ρ = 1, then the plane section Π generates Pic V. In this case, return a 1-column matrix representing (d/4)Π if d is divisible by 4, and an empty matrix otherwise.

3. Let d_{0} ← (d/4)π, where π is the column vector representing the class of Π.

4. Let a ← π^{T} QN.

5. Using a standard algorithm, find: an integer vector d_{1} such that

a^{T} d_{1} = d;

and an integer matrix A whose columns generate the solution set of a^{T} x = 0. ^{2} If no such d_{1} exists, return an empty matrix.

6. Let B be any left inverse matrix to NA.

7. Let Q′ ← A^{T} N^{T} QNA.

8. Let α ← B(Nd_{1} − d_{0}).

9. Apply the inhomogeneous Fincke–Pohst algorithm to find all vectors x ∈ ℤ^{ρ} + α such that x^{T} Q′x ≤ d^{2}/4 − n.

10. Return the matrix whose columns are NAx + d_{0} for each such vector x.

Being able to list all rational curves of a given degree makes it straightforward to find the irreducible components of the fibres of a given fibration. If we know the class F in Pic V of a fibre, then we start by listing all the rational curves of degree 1, say, which have intersection number 0 with F. Each fibre must be connected (Zariski’s Main Theorem, [21, III, Corollary 11.4]) and disjoint from each other fibre; using this it is easy to search for connected sets of rational curves which together add to the class of F. If curves of degree 1 do not provide sufficient singular fibres, we continue by looking at rational curves of degree 2, and so on; the process is guaranteed to terminate because the total degree of the components of a singular fibre is simply F.Π.

There is a formula described in [2, III, Proposition 11.4] which shows that the (complex) Euler characteristics of the singular fibres must sum to 24. It is a simple exercise in topology to compute the Euler characteristic of each of the possible fibre types in the Kodaira–Néron classification, and so this formula provides an easy way to determine when all the singular fibres of a fibration have been found.

The rank of the group Pic_{vert} and thence of the Jacobian of the generic fibre may be found without computing a set of generators for Pic_{vert}.

2In PARI, this is accomplished by using the function matsolvemod to solve the equation ∑_{i=2}^{ρ} a_{i}x_{i} ≡ d (mod a_{1}).

Proposition 2.30. Suppose that all the singular fibres have type I_{n}, where n ≥ 2. Then the rank of Pic_{vert} is

24 − (number of singular fibres) + 1

and the rank of the Jacobian of the generic fibre is

(number of singular fibres) − 5.

Proof. Note that the Euler characteristic of a fibre of type I_{n} is n, and these must therefore sum to 24. We then apply the results of Shioda [37, §1].

This algorithm may also be used to list divisor classes of given degree and self-intersection zero, and therefore to list the fibrations on V. In general, there is a better way to do this, but this algorithm is nonetheless useful.

2.3.3 An example

As an example, we consider those cases where Pic V has rank 2; here V has a fibration if and only if the determinant of the lattice Pic V is minus a square. The fibration is easy to find, as solving a quadratic form in two variables is trivial.

The fibrations which arise on V when Pic V has rank 2 fall into three distinct types, up to various Galois actions and permutations of variables. These three types may be summarised as follows:

• Fibrations having a fibre which looks like

L_{11}^{123} + L_{15}^{123} + L_{51}^{123} + L_{55}^{123}

defined over ℚ(\sqrt{a_{0}a_{1}a_{2}a_{3}}). These are the fibrations studied in [44]. Such a fibration exists in case A1 of Appendix A and all its subcases. The six singular fibres each consist of four straight lines, arranged as a fibre of type I_{4}. The Pic_{vert} has rank 19 and, as shown in the proof of Proposition 2.15, the generic fibre is a 2-covering of its Jacobian, which has finite Mordell–Weil group.

• Fibrations with a fibre which looks like

L_{11}^{123} + L_{17}^{123} + L_{51}^{123} + L_{57}^{123}.

This fibre and a similar one are the two fibres of type I_{4}, consisting each of four straight lines. There are eight other singular fibres, each of which is of type I_{2} made up of two rational curves of degree 2. This fibration does have a section: in fact, any line L_{mn}^{231} or L_{mn}^{312} has intersection number 1 with the fibre. The Pic_{vert} is of rank 15, and therefore the generic fibre is an elliptic curve of rank 4. (This fits with the results given in [37].) This particular fibration is defined, for example, in case A212 of our classification.

• Fibrations with a fibre which looks like

L_{11}^{123} + L_{17}^{123} + L_{31}^{123} + L_{33}^{123} + L_{53}^{123} + L_{55}^{123} + L_{75}^{123} + L_{77}^{123}.

This fibre and another similar one are of type I_{8}, consisting each of eight straight lines arranged in an octagon. There are four more singular fibres, each of type I_{2} consisting of two rational curves of degree 4. The fibration does not have a section; the Pic_{vert} is of rank 19, indicating that the Jacobian of the generic fibre is an elliptic curve of rank zero. The particular fibration given here is defined, for example, in case A176 of our classification.

In fact, these fibrations which are defined for surfaces V of Picard rank 2 between them cover almost all surfaces which have a fibration. That is, there are only a small number of cases in which V has a fibration, but not one of the types listed above. Further calculation reveals that every surface possessing a fibration has one of degree at most 8.

Having found some fibrations defined on V, we may go on to investigate the associated vertical Brauer group. An example of how this can be computed is given in Section 4.5.

Galois theory of the Picard group

This chapter describes techniques for investigating the structure of Pic V̄ as a Galois module. By Proposition 2.15 this depends only on knowing the Galois action on the 48 lines on V̄. We show in Section 3.1 that all possible Galois actions for varying parameters a_{0}, a_{1}, a_{2}, a_{3} may be studied by considering subgroups of a ‘generic’ Galois group Γ. We then, in Section 3.2, review some basic algorithms for computing with finitely generated Abelian groups. Sections 3.3 to 3.6 demonstrate algorithms for performing various computations using the Galois action on the 48 lines. The results of these computations are given in the tables in Appendix A.

The main reference for this chapter is [8]. We will make use of some general results on the Galois theory of rings and schemes, which may be found in [5] (Chapter 5) and in [19] respectively.

3.1 The ‘generic’ Galois group and its action

The 48 lines on the surface V̄ were defined in equations (1.2)–(1.4). Firstly, we note that the 48 lines are all defined over a finite extension of ℚ, so that the action of Gal(ℚ̄/ℚ) on them factors through a finite group. Recall that α_{ij} denotes a fourth root of a_{i}/a_{j}, chosen such that α_{ij}α_{jk} = α_{ik}.

Lemma 3.1. The field ℚ(α_{10}, α_{20}, α_{30}, ϵ) is the least field of definition for the 48 lines L_{mn}^{pqr}.

Proof. All 48 lines are clearly defined over this field. Conversely, let K be the least field of definition of the 48 lines. Then K contains ϵ^{m}α_{pq} for all p ≠ q and all odd m. In particular, K contains ϵα_{12} and ϵα_{02}; dividing these, we see that K contains α_{10}. Similarly, K contains α_{20} and α_{30}. But now K contains both α_{10} and ϵα_{10}, so must also contain ϵ.

How does the Galois action on the 48 lines vary as the coefficients a_{0}, a_{1}, a_{2}, a_{3} are changed? To answer this we will construct a ‘generic’ diagonal quartic surface.

Let U be the (affine) subscheme of ℙ_{ℚ}^{3}, with homogenous coordinates S_{i}, defined by S_{0}S_{1}S_{2}S_{3} ≠ 0. This will be the space in which the parameter a = (a_{0} : a_{1} : a_{2} : a_{3}) lives. Let X be the scheme in ℙ_{U}^{3} defined by

X : S_{0}X_{0}^{4} + S_{1}X_{1}^{4} + S_{2}X_{2}^{4} + S_{3}X_{3}^{4} = 0

which is our ‘generic’ surface. For any a ∈ U(ℚ), the fibre X_{a} is the diagonal quartic surface over ℚ given by (1.1). The 48 lines are given by the closed subscheme L ⊂ X defined by

L : {S_{0}X_{0}^{4} = −S_{1}X_{1}^{4}, S_{2}X_{2}^{4} = −S_{3}X_{3}^{4}}

∪ {S_{0}X_{0}^{4} = −S_{2}X_{2}^{4}, S_{3}X_{3}^{4} = −S_{1}X_{1}^{4}} ∪ {S_{0}X_{0}^{4} = −S_{3}X_{3}^{4}, S_{1}X_{1}^{4} = −S_{2}X_{2}^{4}}

which has 3 irreducible components as a scheme over ℚ (and, indeed, over ℚ̄).

To split L into its 48 components, we consider a scheme Ũ → U constructed as follows. Let Ũ be the subscheme of ℙ_{ℚ(ϵ)}^{3}, with homogeneous coordinates T_{i}, defined by T_{0}T_{1}T_{2}T_{3} ≠ 0. Define a map Ũ → U by T_{i}^{4} = S_{i}. Then Ũ is an étale covering of U, and in fact the covering is Galois of degree 256 (degree 4 for the base field extension, and degree 64 for the geometric covering). We will call the group of this covering Γ, and it will play a central rôle in the rest of this chapter.

We can explicitly write down a set of generators for Γ, for it is the same as the Galois group of the extension of function fields k(Ũ)/k(U). Writing s_{i} for S_{i}/S_{0} and t_{i} for T_{i}/T_{0}, this extension is

ℚ(ϵ, t_{1}, t_{2}, t_{3})/ℚ(s_{1}, s_{2}, s_{3})  (3.1)

where again t_{i}^{4} = s_{i}. We note here that ℚ(ϵ) = ℚ(i, \sqrt{2}), which will be useful for calculations. Kummer theory then shows that the Galois group Γ of (3.1) fits into an exact sequence

0 → Γ′ → Γ → Gal(ℚ(i)/ℚ) → 0

and that Γ′ is Abelian. We see that a system of generators for Γ is given by

\begin{aligned}σ_{1} & : t_{1} ↦ it_{1} \\ σ_{2} & : t_{2} ↦ it_{2} \\ σ_{3} & : t_{3} ↦ it_{3} \\ σ_{4} & : \sqrt{2} ↦ −\sqrt{2} \\ τ & : i ↦ −i \end{aligned}  (3.2)

where each generator fixes those of \{t_{1}, t_{2}, t_{3}, \sqrt{2}, i\} not mentioned. The multiplication in Γ is as follows: all the σ_{i} commute with each other; and τσ_{i} = σ_{i}^{−1}τ.

What effect does the base change Ũ → U have on the 48 lines L? Let L̃ denote the fibre product L ×_{U} Ũ. Then L̃ is the scheme defined over ℚ(ϵ) by the equations

L̃ : {T_{0}^{4}X_{0}^{4} = −T_{1}^{4}X_{1}^{4}, T_{2}^{4}X_{2}^{4} = −T_{3}^{4}X_{3}^{4}}

∪ {T_{0}^{4}X_{0}^{4} = −T_{2}^{4}X_{2}^{4}, T_{3}^{4}X_{3}^{4} = −T_{1}^{4}X_{1}^{4}} ∪ {T_{0}^{4}X_{0}^{4} = −T_{3}^{4}X_{3}^{4}, T_{1}^{4}X_{1}^{4} = −T_{2}^{4}X_{2}^{4}} (3.3)

and each quartic equation now splits into four linear factors: L̃ is the union of 48 straight lines. The group Γ acts on L̃ in the obvious way, permuting the lines within their three sets.

Now let a = (a_{0} : a_{1} : a_{2} : a_{3}) be a rational point of U. The inverse image of a in Ũ is étale over a, and so consists of finitely many points; and, as the covering is Galois, these points are permuted transitively by Γ and so are isomorphic. We let b be one of these points, and denote by Γ_{b} the decomposition group of b, that is, the subgroup of Γ which fixes b ([19], Section 2). The residue field k(b) is then generated over ℚ by ϵ and fourth roots of the a_{i}/a_{0}, and has Galois group Γ_{b}. In fact, k(b)/ℚ is easily checked to be the extension described in Lemma 3.1. Note that different choices of b will give rise to conjugate groups Γ_{b}; in terms of the extension k(b)/ℚ this simply means a different choice of the various fourth roots and square roots. For an example of how this works, see the beginning of Section 3.4.

Considering the 48 lines on the surface X_{a}, we now have a Galois covering L̃_{b} → L_{a} which is simply the covering we obtain by extending the base field of L_{a} from ℚ to k(b): in other words, by the smallest extension such that the 48 lines are all individually defined. The corresponding Galois action on the 48 lines is that induced by Γ_{b} acting on the generic lines L̃.

We may summarise all this discussion in the following statement:

Proposition 3.2. For any choice of a_{0}, a_{1}, a_{2}, a_{3}, the action of Gal(ℚ̄/ℚ) on the 48 lines of V factors through a finite quotient G. This group G, and its action on the 48 lines, are isomorphic to a subgroup of Γ and its action on the generic 48 lines L̃.

All the groups Γ_{b} which arise in this way will have the property that ℚ(ϵ)^{Γ_{b}} = ℚ, for the residue field at a is ℚ. Although we are primarily interested in the case where the base field is ℚ, the above discussion makes sense when the base field is any number field k; and the Galois structure of Pic V̄ depends on k only in that it depends on k ∩ ℚ(ϵ). It is no extra effort for us to study such cases, and indeed will be useful. The tables in Appendix A are divided into one section for each possibility for k ∩ ℚ(ϵ).

It follows from Proposition 3.2 that to study all possible actions of the Galois group Gal(ℚ̄/ℚ) on the 48 lines on V̄, it is necessary to study the subgroups of Γ. Specifically, this chapter will address the following problems:

• to list all the subgroups of Γ;

• to find equivalences between the subgroups of Γ which correspond to various trivial rearrangements of the coefficients a_{i}, and list the different equivalence classes;

• given non-zero rational numbers a_{i}, to find the subgroup of Γ associated to the surface V;

• given a subgroup H of Γ, to find conditions on the coefficients a_{i} which imply that the group G of Proposition 3.2 is contained in H;

• given a subgroup of Γ, to find the part of Pic V̄ which is defined over ℚ;

• given a subgroup of Γ, to compute the corresponding cohomology group H^{1}(ℚ, Pic V̄).

Before we embark on these computations in Section 3.3, we review some theory of finitely generated Abelian groups in the following section.

3.2 Computing with Abelian groups

In this section we gather some results and algorithms for dealing with finitely generated Abelian groups. Of course our group Γ is not Abelian, but it is “almost” Abelian, and we will be able to use the techniques described in this chapter to deal with it. A standard reference for the algorithms used is Cohen’s book [8].

All the matrices we will consider are integer matrices, and a matrix is invertible if it has an inverse which is also an integer matrix.

3.2.1 Groups and matrices

We will always think of Abelian groups in a concrete way as ℤ-modules. Every finitely generated ℤ-module has a resolution by free modules: let G be a finitely generated Abelian group; then we can find an exact sequence

ℤ^{m} ⟶^{α} ℤ^{n} → G → 0  (3.4)

of ℤ-modules. We may always assume either that m = n or that α is of maximal rank; if G is finite, we may assume both. What we are doing here is expressing G as a quotient of ℤ^{n} by a lattice, the image of α. Any set of generators of ℤ^{n} also generates G, though the converse is not true.

By choosing bases for the ℤ^{m} and ℤ^{n} in (3.4), we obtain a matrix A for the linear map α.

Definition 3.3. Any matrix for the linear map α with respect to some choice of bases is a realisation of the group G.

Given a choice of basis for ℤ^{n} in (3.4), we may represent a subgroup of G by giving its generators as elements of ℤ^{n}. We will often represent such a subgroup by a matrix whose columns form a set of generators. Note that there are thus two very different meanings we can attach to a matrix: it may be a realisation of a finitely generated Abelian group, or it may be a set of generators for a subgroup of such a group.

3.2.2 Normal forms for matrices

There are two well-known normal forms for matrices, the Hermite normal form and the Smith normal form, which have relevance to the use of matrices to describe Abelian groups. We describe both of these normal forms and explain how they will be used in our algorithms. Both the Hermite and Smith normal forms relate to choosing bases for the ℤ^{m} and ℤ^{n} in (3.4).

The Hermite normal form

Here we think of the basis for ℤ^{n} as fixed in some way: perhaps it has some special significance in the problem being studied. We are interested in properties of the lattice Λ = im α and the quotient ℤ^{n}/Λ. In particular, we would like to be able to tell whether two lattices, specified by generating sets, are equal. The Hermite normal form allows us to do this.

As before, A is the matrix representing the map α with respect to our chosen basis of ℤ^{n}, and with respect to some basis of ℤ^{m}. The columns of A form a generating set for Λ. What we would like to have is a canonical ordered set of generators for Λ, and the Hermite normal form provides just that: every lattice has precisely one ordered set of generators which form a matrix in Hermite normal form. The Hermite normal form is discussed in greater detail in section 2.4.2 of [8], and we do not prove its existence or uniqueness here; we simply recall some properties.

Proposition 3.4. 1. Given an n × m integer matrix A, there exists a unique m × m invertible integer matrix U such that AU is in Hermite normal form.

2. A square matrix M in Hermite normal form satisfies the following properties:

(a) M is upper triangular;

(b) All the elements of M are non-negative;

(c) Each element on the diagonal of M is the largest element in its row.

An n × m matrix, where m > n, in Hermite normal form has the first m − n columns equal to zero, and the remaining n columns form a square matrix in Hermite normal form.

The columns of the matrix U give the new basis for Λ in terms of the old generating set. Choosing a unique ordered basis for a lattice will be useful in several ways in our calculations.

The Smith normal form

When discussing the Hermite normal form, we assumed that the basis chosen for ℤ^{n} in (3.4) was special in some way. We now relax that assumption: all that interests us is the structure of the quotient G as an abstract ℤmodule. We assume here that m = n. The structure theorem for finitely generated modules over a principal ideal domain states that G is isomorphic to a product of a free group by some cyclic groups, and moreover that the orders of these cyclic groups may be chosen to satisfy certain properties; the Smith normal form of a matrix provides a concrete interpretation of that theorem.

An integer matrix M is said to be in Smith normal form if it is of the form

M = (\begin{matrix}d_{1} & 0 & ⋯ & 0 \\ 0 & d_{2} & ... & ⋮ \\ ⋮ & ... & ... & 0 \\ 0 & ⋯ & 0 & d_{n} \end{matrix})  (3.5)

where d_{i} | d_{i+1} for i = 1, …, n − 1. It is then immediately clear that the group realised by the matrix M is isomorphic to

ℤ/d_{1}ℤ × ⋯ × ℤ/d_{r}ℤ × ℤ^{n−r}  (3.6)

where d_{1}, …, d_{r} are the non-zero diagonal entries of M.

A simple algorithm (given in [8]) proves:

Proposition 3.5. Given a square integer matrix A, there exist invertible matrices U, V such that D = UAV is in Smith normal form. The resulting diagonal matrix D is unique.

The columns of U^{−1} give generators for G, in terms of the old basis, which are generators of the cyclic groups in (3.6). Such a system of generators for G will be called a Smith normal form basis for G.

3.2.3 Subgroups and quotients

Here we apply the Hermite and Smith normal forms to some problems which we will have to solve in future sections. We are always working in the context of a finitely generated Abelian group G, and assume we have been given a matrix A which is a realisation of G. Let Λ denote the lattice which is the image of A in ℤ^{n}, so that G ≅ ℤ^{n}/Λ. We are interested in certain subgroups and subquotients of G. Both the algorithms we describe here are very simple, almost trivial, but deserve proper explanation.

The first problem is this: given a set of generators for a subgroup H of G, we wish to find a canonical set of generators for H. Now the subgroups H are in 1-1 correspondence with the lattices Λ′ of ℤ^{n} containing Λ: given a set of generators for H, we adjoin a basis for Λ to obtain a set of generators for Λ′. Conversely, any generating set for Λ′ is also a generating set for H. This leads to the following short algorithm:

Algorithm 3.6. Given a matrix A which is a realisation of a group G, and a matrix B whose columns are generators for a subgroup H of G, find a unique canonical matrix whose columns generate H.

1. Let B ← (B | A), the concatenation of B and A.

2. Put B into Hermite normal form.

This description of subgroups also allows us to make a further observation. Given that Λ ⊆ Λ′, and given a choice of basis for Λ′ as a matrix B, we may write down a matrix C whose columns are the coordinates, in this basis, of a basis for Λ. Then the the columns of the product matrix A = BC are the coordinates of the basis for Λ, but now given in the standard basis of ℤ^{n}; so A is a realisation of G. Conversely, any such factorisation A = BC arises in this way from a subgroup H of G. This means that to list the subgroups of G, it is enough to list the matrices in Hermite normal form which divide A on the left: and this is precisely the method adopted by PARI.

The second problem which we will need to solve is the following. Given two subgroups K ⊂ H of G, we wish to know the structure of the quotient H/K. The Smith normal form is how one discovers the abstract structure of an Abelian group, so we must apply it here.

Let Λ′, Λ′′ be the lattices which are the liftings of H and K, respectively, to ℤ^{n}. Then Λ′′ ⊂ Λ′, and

\frac{H}{K} ≅ \frac{Λ′/Λ}{Λ′′/Λ} ≅ \frac{Λ′}{Λ′′}

If we let B and C be matrices whose columns are bases for Λ′ and Λ′′ respectively, then C = BE for some matrix E; and the columns of E are the coordinates of the basis for Λ′′ with respect to the chosen basis of Λ′. So the matrix E is a realisation of the group Λ′/Λ′′, and its Smith normal form shows the structure of this group, which is isomorphic to H/K.

Algorithm 3.7. Given matrices B, C whose columns are generators for H and K respectively, subgroups of a group G, find: a matrix B′ whose columns are generators for H; and a matrix D in Smith normal form, such that the columns of B′D generate K. Then D contains the structure of the quotient H/K, and the columns of B′ form a Smith normal form basis for H/K.

1. Let E ← B^{−1}C.

2. Use the Smith normal form algorithm to find matrices U, V such that D = UEV is in Smith normal form.

3. Let B′ ← BU^{−1}.

3.3 Listing the cases

We now turn to the problem of finding all the different cases into which our surface (1.1) might fall. As described in Section 3.1, this involves listing all of the subgroups of the generic Galois group Γ, up to certain equivalences which we must describe exactly.

3.3.1 Representation of subgroups

We choose how a subgroup of Γ will be described in a canonical way. We fix the generators of Γ listed in (3.2), and denote by Γ′ the Abelian subgroup generated by σ_{1}, …, σ_{4}. The matrix for this realisation of Γ′ is

(\begin{matrix}4 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 2 \end{matrix}).

Firstly we consider subgroups of Γ′. Any element g of Γ′ may be represented uniquely as an integer column vector using this basis: the first three coordinates, corresponding to σ_{1}, σ_{2} and σ_{3}, are considered modulo 4; and the fourth coordinate, corresponding to σ_{4}, is considered modulo 2. Now if G is a subgroup of Γ′, let M be a matrix whose columns represent generators of G. Algorithm 3.6 gives a way of turning M into a canonical 4 × 4 matrix whose columns generate G.

The quotient Γ/Γ′ has order 2 and is generated by the element τ, so any element of Γ is either an element of Γ′ or is τ multiplied by an element of Γ′. Let H be a subgroup of Γ, and write H′ for the intersection of H with Γ′. Either H is entirely contained in Γ′, in which case H = H′; or else H/H′ is of order 2, and H is generated by a set of generators for H′, together with any element of H which is not in H′. In other words, we have shown

Proposition 3.8. Any subgroup H of Γ can be expressed in one of the two forms

• H = ⟨h_{1}, h_{2}, h_{3}, h_{4}⟩

• H = ⟨h_{1}, h_{2}, h_{3}, h_{4}, τh_{5}⟩

where the h_{i} are elements of Γ′.

In calculations, we will represent H by a 4 × 4 matrix M of generators for H′, together with another optional element v of ℤ^{4} representing h_{5}. This pair (M, v) will be called a matrix-vector representation of H. If there is no fifth generator h_{5}, then v will be an “empty” vector (represented in PARI as

[]).

This representation is far from unique: when choosing h_{5}, we could have picked any element of H not in H′. But we can take advantage of the upper triangular shape of the Hermite normal form to come up with the following algorithm:

Algorithm 3.9. Given a set of generators for a subgroup H of the Galois group Γ, this algorithm produces a uniquely defined canonical set of generators for H. Let (M, v) be a set of generators for H, where M is a 4×4 matrix whose columns generate H′ over the basis σ_{1}, …, σ_{4} and v is a column vector representing an element h_{5} of Γ′ (given in the same basis) such that τh_{5} is a fifth generator of H.

1. Apply Algorithm 3.6 to M.

2. Set i ← 4.

3. Let v ← v − ⌊v_{i}/M_{ii}⌋ × M_{(i)}.

4. Let i ← i − 1.

5. If i > 0, go to step 3.

Here M_{(i)} denotes the ith column of M. This algorithm is listed in Section D.1.1.

It is clear that the resulting representation does indeed represent the same subgroup H: we have simply chosen a canonical basis for H′, and then changed h_{5} only by subtracting elements of H′. And the representation is unique: for the ith element of v lies in the interval [0, M_{ii}); adding any multiple of the ith column of M to v would violate this condition. This, together with the fact that M is upper triangular, proves the uniqueness.

We note that it is easy to find the order of H:

|H| = \begin{cases}128/ det M & if v is empty; \\ 256/ det M & otherwise. \end{cases}

As M will usually be upper triangular, calculating its determinant is particularly simple.

3.3.2 Listing subgroups

Having fixed a canonical way to represent subgroups of Γ, we are now in a position to list them.

There exists a function in PARI which lists the subgroups of an Abelian group given in Smith normal form (or, equivalently, lists the Hermite normal form left divisors of a diagonal integer matrix). This gives a simple way of listing all the subgroups of Γ:

1. List all the subgroups of Γ′.

2. For each subgroup H′ of Γ′ and each element h_{5} of Γ′, reduce the subgroup generated by H′ and τh_{5} to canonical form using Algorithm 3.9.

3. Remove the many repeated entries from the list of subgroups.

This method of course counts large subgroups many times; but this was not a problem.

There are 10,095 distinct subgroups of Γ.

3.3.3 Equivalences between subgroups

We have been listing subgroups of Γ, but what we are interested in is listing the different possible Galois actions on the 48 lines of our surface (1.1). The relation between subgroups and cases is described in Section 3.1, but we should decide which cases we will consider to be “the same” as each other. This will allow us to reduce the 10,095 subgroups to a more manageable number of cases.

Firstly, the subgroup of Γ associated to a surface depends on choices of roots, and so is defined only up to conjugacy class. Therefore any two subgroups which are conjugate to each other will belong to the same case.

Secondly, permuting the coefficients a_{0}, a_{1}, a_{2}, a_{3} results in a new surface isomorphic over ℚ to the original one, and we will consider any two such surfaces to belong to the same case.

Choice of roots

In Section 3.1 we saw that the subgroup of Γ associated to a given surface is only determined up to conjugacy class, because we have choices of ϵ and of the α_{ij}. Accordingly, any two subgroups of Γ which differ only by such choices will be considered equivalent.

Let H be a subgroup of Γ, with a set of generators labelled as in Proposition 3.8. Replacing t_{1} by it_{1} has the effect of conjugating H by σ_{1}. This has no effect on the subgroup H′ of H (it is Abelian), but the ‘other’ generator τh_{5} becomes τσ_{1}^{2}h_{5}. Similarly, conjugating by σ_{2} or σ_{3} has the effect of multiplying h_{5} by σ_{2}^{2} or σ_{3}^{2} respectively. If (M, v) is a matrix-vector representation of H, we are adding 2 to the appropriate elements of v.

We must also consider choice of ϵ. Replacing \sqrt{2} by −\sqrt{2} is conjugation by σ_{4}, which has no effect at all. Replacing i by −i is conjugation by τ. The element τh_{5} becomes τh_{5}^{−1}; in other words, our vector v is negated. But we are already only considering it modulo 2, so that makes no difference. Conjugation by τ also replaces each σ_{i} by σ_{i}^{−1}; but that is the same as replacing H by H^{−1}, which is the same thing. So there is nothing more to consider to cope with the choice of ϵ.

Permuting the coefficients

In terms of the generic Galois group, we want any two subgroups of Γ which are related only by permuting the parameters T_{i} to be equivalent. To do this, it is useful to consider firstly those permutations which fix T_{0}, and then the transposition of T_{0} and T_{1}. Between them, these generate the whole permutation group (of order 24).

Any permutation fixing T_{0} simply permutes the generators σ_{1}, σ_{2}, σ_{3} of Γ. If (M, v) is a matrix-vector representation of a subgroup H, the effect of the permutation is to permute the first three rows of M and of v.

Now consider the effect of interchanging T_{0} and T_{1}. The inhomogeneous coordinates t_{i}, from which our generators σ_{i} are defined, are changed as follows:

(t_{1}, t_{2}, t_{3}) ↦ (1/t_{1}, t_{2}/t_{1}, t_{3}/t_{1})

and the effect on the Galois group is to replace (σ_{1}, σ_{2}, σ_{3}) with

(σ_{1}^{−1}, σ_{2}σ_{1}^{−1}, σ_{3}σ_{1}^{−1}).

Let H be a subgroup of Γ, and let (M, v) be a matrix-vector representation of H. The effect of this substitution is to multiply both M and v on the left by the matrix

(\begin{matrix}−1 & 0 & 0 & 0 \\ −1 & 1 & 0 & 0 \\ −1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}).  (3.7)

It is readily verified that the matrix (3.7), together with the matrices representing permutation of the first three rows, generate a group isomorphic to S_{4} which acts on our subgroups H. We will consider any two subgroups lying in the same orbit under this action to be equivalent.

3.3.4 Finding a canonical representative

Having explained when two subgroups are to be equivalent to each other, we must now produce an algorithm for finding a canonical representative of each equivalence class. This will be approached in two steps: firstly, finding a canonical subgroup in each conjugacy class; and secondly, finding a canonical subgroup in each equivalence class. Note that we may do this, because permutation of the a_{i} followed by conjugation is equal to (some other) conjugation followed by permutation.

We already have Algorithm 3.9 to produce a canonical representation of a subgroup H of Γ. Let (M, v) be a matrix-vector representation of H; in Section 3.3.3 we saw that conjugation has the effect of changing the elements of v by a multiple of 2. To get a canonical representative for a conjugacy class of subgroups, it might appear that all we have to do is take a canonical representation (M, v) of a subgroup H (which already has v “as small as possible”) and then reduce v modulo 2. Unfortunately this does not quite work, for it may be possible to change v by adding or subtracting columns of M to produce a “smaller” v than we have already. However, we can change Algorithm 3.9 to do as we want, to get the following:

Algorithm 3.10. Given a set of generators for a subgroup H of the Galois group Γ, this algorithm produces a uniquely defined canonical set of generators for a canonical subgroup conjugate to H. Let (M, v) be a matrix-vector representation of H.

1. Apply Algorithm 3.6 to M.

2. Let N ← (M | 2I_{4}).

3. Put N into Hermite normal form.

4. Set i ← 4.

5. Let v ← v − ⌊v_{i}/N_{ii}⌋ × N_{(i)}.

6. Let i ← i − 1.

7. If i > 0, go to step 5.

Here I_{4} is the 4 × 4 identity matrix, and N_{(i)} denotes the ith column of N. This algorithm is listed in Section D.1.2.

Now we must choose a unique representative modulo permutation of the a_{i}. We do this in the following, unsophisticated, way: by listing representatives of all 24 conjugacy classes which are equivalent to the given one, and by choosing the one which comes first in some arbitrary ordering (we use the lexicographic order). The implementation of this is listed in Section D.1.3. There are 546 different equivalence classes of subgroups, which are listed individually in the tables in Appendix A.

3.4 Finding the subgroup associated to a sur-

face

We have now listed all possible cases into which the Galois action on the 48 lines on V might fall. However, this information is still not in a very helpful format: all we have is a list of subgroups of a large group. This section and the following one describe algorithms to perform two useful functions: given a surface, to find the subgroup of Γ which is associated to it; and given a subgroup, to describe in general the coefficients a_{i} which give rise to that subgroup.

In these two sections, we illustrate the algorithms with a running example. This will be the same surface as in the worked example of Section 5.3. It is defined by the equation

X_{0}^{4} + X_{1}^{4} = 6X_{2}^{4} + 12X_{3}^{4}  (3.8)

and so (a_{0}, a_{1}, a_{2}, a_{3}) = (1, 1, −6, −12). In this section we will compute the subgroup of Γ associated to this surface, and in the following section we will derive a general formula for surfaces with that subgroup.

Suppose we are given non-zero rational numbers a_{0}, a_{1}, a_{2} and a_{3}, or in other words a point a of the scheme U described in Section 3.1. We wish to find the subgroup of Γ associated to the diagonal quartic surface defined by the a_{i}; that is, to find a point b of Ũ lying above a, and the subgroup Γ_{b} of Γ which fixes b. In more concrete terms, we fix fourth roots α_{i0} of a_{i}/a_{0} for i = 1, 2, 3 and ask what subgroup of Γ, acting via t_{i} ↦ α_{i0}, takes each α_{i0} to a conjugate of itself over ℚ.

In our running example, the inverse image of a in Ũ is given by the ideal

𝔞 = (t_{1}^{4} − 1, t_{2}^{4} + 6, t_{3}^{4} + 12)

in ℚ(ϵ)[t_{1}, t_{2}, t_{3}]. This decomposes into eight maximal ideals, of which a typical one is

𝔟 = (t_{1} − 1, \sqrt{2}t_{2}^{2} − t_{3}^{2}, t_{3}^{4} + 12)

and the others are obtained by replacing 1 with another fourth root of unity, and maybe replacing \sqrt{2} with −\sqrt{2}. We choose b to be the point associated to the maximal ideal 𝔟. Each element of Γ either fixes 𝔟 or takes it to one its conjugate maximal ideals. An example of an element of Γ which does not fix 𝔟 is σ_{1}, which takes t_{1} ↦ it_{1}. Examples of elements which do fix 𝔟 are σ_{2}^{2}, taking t_{2} ↦ −t_{2}; and σ_{2}σ_{3}, taking t_{2} ↦ it_{2} and t_{3} ↦ it_{3}. Those elements which do fix 𝔟 induce automorphisms on the quotient

ℚ(ϵ)[t_{1}, t_{2}, t_{3}]/𝔟 = ℚ(ϵ, \sqrt[4]{−6}, \sqrt[4]{−12})

fixing ℚ, and so are identified with elements of the Galois group of this field over ℚ.

The object is to find a matrix M and vector v which represent the desired subgroup H of Γ. We will produce M in an upper triangular form, which comes from building up our field extension piece by piece.

Consider the tower of field extensions

K_{1} = ℚ(α_{10}, α_{20}, α_{30}, ϵ)
|
K_{2} = ℚ(α_{20}, α_{30}, ϵ)
|
K_{3} = ℚ(α_{30}, ϵ)
|
K_{4} = ℚ(ϵ) = ℚ(\sqrt{2}, i)
|
K_{5} = ℚ(i)

Each extension is cyclic over the previous one; a generator for Gal(K_{i}/K_{i+1}) is given by the image of a suitable power (1, 2 or 4) of σ_{i}. In what follows, we will identify H with Gal(K_{1}/ℚ) in the obvious manner. Let h_{1}, …, h_{5} be the generators for H as in Proposition 3.8, represented by M and v. We will choose them as follows:

• h_{1} is a generator for Gal(K_{1}/K_{2}). In our example, K_{1} = K_{2} and so h_{1} is the identity.

• h_{2} is a lifting of a generator for Gal(K_{2}/K_{3}). In our example, K_{2} is of degree 2 over K_{3} and so h_{2} is of the form σ_{2}^{2}σ_{1}^{a}.

• h_{3} is a lifting of a generator for Gal(K_{3}/K_{4}). On the example, K_{3} is of degree 4 over K_{4} and h_{3} is therefore of the form σ_{3}σ_{2}^{b}σ_{1}^{c}.

• h_{4} is a lifting of the generator for Gal(K_{4}/K_{5}); this is always of the form σ_{4}σ_{3}^{d}σ_{2}^{e}σ_{1}^{f}.

• τh_{5} is a lifting of the generator for Gal(ℚ(i)/ℚ). Here h_{5} is an element of Γ′ which we must find.

It is the lifting of each generator to Gal(K_{1}/ℚ) which requires some thought; the rest is simple. In other words, finding the diagonal elements of M is straightforward, but finding the elements above the diagonal is less so.

We may immediately fill in two elements of M and v, as K_{4} and K_{5} are fixed. We get

(M, v) = ((\begin{matrix}∗ & ∗ & ∗ & ∗ \\ 0 & ∗ & ∗ & ∗ \\ 0 & 0 & ∗ & ∗ \\ 0 & 0 & 0 & 1 \end{matrix}) , (\begin{matrix}∗ \\ ∗ \\ ∗ \\ 0 \end{matrix}))

Suppose that K is a number field containing a primitive fourth root of unity. It is well-known that the cyclic extensions of K of degree dividing 4 are given by K(\sqrt[4]{d}), where d is an element of K, and that they are classified by the class of d in K^{×}/(K^{×})^{4}. The degree of the extension is the order of d in the quotient group. If in addition we know that d lies in ℚ, we may work in ℚ^{×}/(ℚ^{×} ∩ (K^{×})^{4}): this has the advantage that, as ℚ has unique factorisation, ℚ^{×} is the free group generated by −1 and all the primes. By abuse of language, we will call −1 a prime.

For calculations we work with the part of ℚ^{×} generated by −1, 2 and all the primes dividing any of the a_{i}. Write F for this free group, and write F_{i} for the intersection of (K_{i}^{×})^{4} with F. We then have

M_{ii} = 4/[F_{i} : F_{i+1}] for i = 1, …, 3.

After applying this, we have

(M, v) = ((\begin{matrix}□ & ∗ & ∗ & ∗ \\ 0 & □ & ∗ & ∗ \\ 0 & 0 & □ & ∗ \\ 0 & 0 & 0 & 1 \end{matrix}) , (\begin{matrix}∗ \\ ∗ \\ ∗ \\ 0 \end{matrix}))

where □ denotes either 1, 2 or 4 in each case.

Consider our running example. The primes we need to consider are −1, 2 and 3, so let F be the free group generated by these three elements. In ℚ(i), the non-trivial fourth powers are 1 and −4; so F_{5} is generated by (−1)^{2} and (−1) × 2^{2} together with 2^{4} and 3^{4}. Continuing with the other F_{i} gives

F_{5} = ⟨(−1)^{2}, (−1) × 2^{2}, 2^{4}, 3^{4}⟩
F_{4} = ⟨−1, 2^{2}, 3^{4}⟩
F_{3} = ⟨−1, 2^{2}, 3⟩
F_{2} = ⟨−1, 2, 3⟩
F_{1} = F_{2}.

This gives the diagonal elements of M to be 4, 2, 1, 1.

Next we come to the off-diagonal elements. Consider first M_{12}. The second column of M represents an element h_{2} of Γ which, when mapped to Gal(K_{2}/ℚ), gives σ_{2}^{M_{22}}. That is, h_{2} = σ_{1}^{n}σ_{2}^{M_{22}} for some integer n. To find n, we consider the action of h_{2} on α_{10}. Let m be the least power (1, 2 or 4) of α_{10} which lies in K_{2}. We know the action of h_{2} on K_{2}, so get

h_{2}α_{10}^{m} = (i^{n}α_{10})^{m} = i^{r}α_{10}^{m}

whence n = r/m. This will be an integer, as α_{10}^{4} always lies in ℚ. A similar process allows us to discover M_{23}, and thence M_{13}. The same argument gives the fourth column of M.

The running example works as follows:

1. We already know that h_{2} = σ_{2}^{2}σ_{1}^{a} for some a, and wish to compute a. Now α_{10} = 1, which must be fixed by any Galois action; so a = 0 and h_{2} = σ_{2}.

2. Now we consider h_{3} = σ_{3}σ_{2}^{b}σ_{1}^{c}. The degree computations we have already done show that α_{20}^{2} lies in K_{3}, and linear algebra in F gives

(−6)^{2} = (−12)^{2}/4
and so α_{20}^{2} = α_{30}^{2}/\sqrt{2}.

We know the effect of h_{3} on the right hand side of this expression: \sqrt{2} is fixed and α_{30}^{2} negated. Therefore h_{3} negates α_{20}^{2} and so b is either 1 or 3 (it does not matter which). As before, we find c = 0.

3. For h_{4}, we know that the smallest power of α_{30} lying in K_{4} is the fourth power; so h_{4} may have any effect on α_{30}, and we choose that α_{30} should be fixed by h_{4}. The above expression for α_{20}^{2} then implies that h_{4} should change the sign of α_{20}^{2}, and we end up with h_{4} = σ_{4}σ_{2}.

Calculating the entries of v is rather easier, as ℚ(i)/ℚ only has degree 2, and we are only interested in the values modulo 2 (see Section 3.3.3). We get

v_{i} = \begin{cases}0 & if a_{i}/a_{0} is positive; \\ 1 & if a_{i}/a_{0} is negative. \end{cases}

Combining all these calculations, the example we are studying ends up with

(M, v) = ((\begin{matrix}4 & 0 & 0 & 0 \\ 0 & 2 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) , (\begin{matrix}0 \\ 1 \\ 1 \\ 0 \end{matrix})).

Putting these steps all together results in the following algorithm. We assume the existence of a function which, given a finite ordered set p_{1}, …, p_{n} of primes and a rational number a = p_{1}^{r_{1}} ⋯ p_{n}^{r_{n}}, returns the vector (r_{1}, …, r_{n}). The matrices K_{i} in this algorithm are bases for the groups (K_{i}^{×})^{4} ∩ F, where F as above is the free group on the primes in S.

Algorithm 3.11. Given rational numbers a_{0}, a_{1}, a_{2} and a_{3}, this algorithm calculates a matrix M and a vector v which are a matrix-vector representation of the subgroup H of Γ corresponding to the diagonal quartic surface defined by the a_{i}.

1. Let S be the ordered set of primes occurring in the prime factorisations of the a_{i}, together with −1 and 2. Let n ← |S|.

CHAPTER 3. GALOIS THEORY OF THE PICARD GROUP 59

2. Let

K_{4} ← (\begin{matrix}0 & 1 & 0 \\ 2 & 2 & 0 \\ 0 & 0 & 4I_{n−2} \end{matrix}).

3. Let vectors f_{i} (i = 1, 2, 3) be the factorisations of a_{i}/a_{0} in terms of the primes in S.

4. Let K_{3} ← (f_{3} | K_{4}), K_{2} ← (f_{2} | K_{3}) and K_{1} ← (f_{1} | K_{2}).

5. Let d_{i} (i = 1, …, 4) be the determinant of the Hermite normal form of K_{i}.

6. Let M_{ii} ← d_{i}/d_{i+1} (i = 1, 2, 3). Let M_{44} ← 1.

7. Let w_{i} (i = 1, 2, 3) be solutions of K_{i}w_{i} ≡ f_{i} (mod 4).

8. Let i ← 2.

9. Let j ← i − 1.

10. Let M_{ij} ← ∑_{k=j+1}^{i} M_{ik}(w_{i})_{k} × M_{ii}/4.

11. Let j ← j − 1. If j ≥ 1, go to step 9.

12. Let i ← i + 1. If i ≤ 4, go to step 8.

13. Reduce M to Hermite normal form.

14. For each i from 1 to 3, let v_{i} be either 0 if a_{i}/a_{0} is positive, or 1 if a_{i}/a_{0} is negative. Let v_{4} ← 0.

The implementation of this algorithm is listed in Section D.2.1.

3.5 Finding surfaces from subgroups

In this section we address the following problem: given a subgroup H of Γ, find conditions on the coefficients a_{i} which imply that the subgroup attached to the surface defined by the a_{i} is contained in H. What we will find is that, for example, a certain subgroup H might give the condition “a_{1}/a_{0} is a square”; a smaller subgroup H′ ⊂ H might then give the condition “a_{1}/a_{0} is 4 times a fourth power”, which is clearly a narrowing of the first condition.

Whenever the a_{i} are rational numbers, the subgroup H associated to the surface defined by the a_{i} must satisfy ℚ(ϵ)^{H} = ℚ. The arguments and algorithms developed here will be equally applicable to other base fields, but for the sake of clarity we will assume that the base field is ℚ. A short function to find the constant field fixed by H is listed in Section D.2.2.

To set out the problem, we return to the situation described in Section 3.1. A set of coefficients a_{i} is described by a rational point a of the open subscheme U of 3-dimensional projective space; and this point has several points lying above it in Ũ; we pick one of them and call it b. The question now is: what conditions must we impose on a to ensure that the decomposition group Γ_{b} of b be contained in H?

Let

A = ℚ[s_{1}^{±}, s_{2}^{±}, s_{3}^{±}]  (3.9)

be the coordinate ring of U, and let

B = ℚ(ϵ)[t_{1}^{±}, t_{2}^{±}, t_{3}^{±}]  (3.10)

be the coordinate ring of Ũ. Then Γ acts on B, and we write B^{H} for the subring of B fixed by H. The spectrum of B^{H} is the quotient scheme Ũ/H ([19], 1.1), and we let b′ be the image of b in this quotient.

Lemma 3.12. The decomposition group Γ_{b} of b is contained in H if and only if b′ is a rational point of Ũ/H.

Proof. The Galois group of k(b)/k(a) is Γ_{b}, and the Galois group of k(b)/k(b′) is Γ_{b} ∩ H, both by ([19], 1.1). If b′ is a rational point of Ũ/H, then k(b′) is ℚ and the two Galois groups are the same; that is, Γ_{b} is contained in H. Conversely, if Γ_{b} is contained in H then by ([19], 2.2) the completed local rings at b′ and a are isomorphic, and hence so are the residue fields, giving k(b′) = ℚ.

We have reduced the question to the following problem: which points of U are the images of rational points on Ũ/H?

A rational point of Ũ/H is given by a homomorphism B^{H} → ℚ: that is, by a point in projective space (1 : α_{10} : α_{20} : α_{30}), with the α_{i0} elements of ℚ̄^{×}, such that the image of B^{H} under the map t_{i} ↦ α_{i0} lies in ℚ. The image of this point in U is a point (a_{0} : a_{1} : a_{2} : a_{3}), with the a_{i} non-zero and rational, such that the image of B^{H} under the map t_{i} ↦ a_{i}/a_{0} is contained in the set (ℚ^{×})^{4}. Thus our conditions on the a_{i} will be of the form “certain monomials in the a_{i} lie in (ℚ^{×})^{4}”.

3.5.1 Computing B^{H}

Let H be a subgroup of Γ. The coordinate rings of the affine schemes U and Ũ are called A and B and were defined in (3.9) and (3.10) above. The group Γ acts on B by automorphisms; the subring fixed by this action is precisely equal to A. We now wish to find a set of elements which generate B^{H}, the subring of B fixed by H, as an algebra over A.

While B is conceptually a simple object, it can be regarded in several different ways: as an A-module; as an A-algebra; as a Γ-module; or as an A[Γ]-module.

First consider B as a module over A. A set of generators is given by the 256 elements of the form

x = t_{1}^{m_{1}}t_{2}^{m_{2}}t_{3}^{m_{3}} \sqrt{2}^{m_{4}}i^{m_{5}}  (3.11)

where m_{1}, m_{2} and m_{3} take values from 0 to 3; and m_{4} and m_{5} take values 0 or 1. Now the action of any of the generators of Γ given in (3.2) is to take x to either x, −x, ix or −ix. Thus the A-submodule of B generated by x and ix is fixed under the action of Γ, which we may write as the following lemma.

Lemma 3.13. As an A[Γ]-module, B is the direct sum of 128 submodules; these are each generated by an element x, of the form (3.11) with m_{5} = 0.

This means that to find B^{H}, it is enough to find the fixed part of each submodule separately.

Let x be an element of the form (3.11) with m_{5} = 0, and let H be a subgroup of Γ given by a set of generators as in Proposition 3.8. Let B_{x} denote the A[Γ]-module generated by x, that is, the free A-module generated by x and ix. An element of B_{x} is fixed by H if and only if it is fixed by each generator of H.

First, we consider the four generators h_{1}, …, h_{4} which fix i. The subgroup of H they generate is called H′ and is Abelian.

Lemma 3.14. The submodule of B_{x} fixed by H′ is

B_{x}^{H′} = \begin{cases}B_{x} & if H′ fixes x \\ \{0\} & otherwise \end{cases}

Proof. Since H′ commutes with multiplication by i, we can (compatibly with the action of H′) think of B_{x} as the rank-1 free module over A[i] generated by x. A rank-1 free module is fixed by a group action if and only if its single generator is fixed.

We return to the running example of the previous section, where the group H is given by (3.4). Let x be a general element of the form (3.11). We find:

• Every x is fixed by h_{1}, which equals the identity.

• x is fixed by h_{2} = σ_{2}^{2} if and only if m_{2} is even.

• x is fixed by h_{3} = σ_{2}σ_{3} if and only if m_{2} + m_{3} = 0 (mod 4).

• x is fixed by h_{4} = σ_{2}σ_{4} if and only if 2m_{2} + m_{4} = 0 (mod 4).

Combining these results, those x which are fixed by the whole of H′ have either m_{2} = m_{3} = m_{4} = 0; or m_{2} = m_{3} = 2 and m_{4} = 1. That is,

x = \begin{cases}t_{1}^{m_{1}}i^{m_{5}} \\ t_{1}^{m_{1}}t_{2}^{2}t_{3}^{2}\sqrt{2}i^{m_{5}} \end{cases} .  (3.12)

Next we must find the part of B_{x} which is fixed by τh_{5}, the remaining generator of H. Enumerating the possible cases gives the following:

Lemma 3.15. The submodule of B_{x} fixed by τh_{5} is

B_{x}^{τh_{5}} = \begin{cases}Ax & if h_{5}x = x \\ Aix & if h_{5}x = −x \\ A(1 − i)x & if h_{5}x = ix \\ A(1 + i)x & if h_{5}x = −ix \end{cases}

We note, for computational use, that an alternative statement is this: if h_{5}x = i^{k}x, then the part of B_{x} fixed by τh_{5} is generated over A by (1+i)^{−k}x.

In the example, the elements (3.12) are fixed by the automorphism h_{5} = τσ_{2}σ_{3} if and only if m_{5} = 0. Therefore B^{H} is generated as a module over A by the eight elements \{t_{1}^{m_{1}} | m_{1} = 0, …, 3\} and \{\sqrt{2}t_{1}^{m_{1}}t_{2}^{2}t_{3}^{2} | m_{1} = 0, …, 3\}.

These two lemmas together allow us to find a generating set for B^{H} as a module over A. With very little extra effort, we can turn this into a generating set for B^{H} as an algebra over A.

Lemma 3.16. Let X be the finite subgroup of B^{×}/A^{×} generated by the five elements t_{1}, t_{2}, t_{3}, \sqrt{2} and (1 + i). Any generating set for X^{H} also generates B^{H} as an algebra over A.

Proof. From the previous two lemmas, we may find a generating set for B^{H} as a module over A with all generators contained in X: that is, X^{H} generates B^{H} as a module over A. So any multiplicative generating set for X^{H} must therefore generate B^{H} as an algebra over A.

In the example, B^{H} is generated as an algebra over A by the two elements t_{1} and \sqrt{2}t_{2}^{2}t_{3}^{2}.

To write down an algorithm for computing a generating set for B^{H}, let x be as in (3.11) with m_{5} = 0, and suppose that σ is an element of Γ′ given by

σ = σ_{1}^{n_{1}}σ_{2}^{n_{2}}σ_{3}^{n_{3}}σ_{4}^{n_{4}}.

CHAPTER 3. GALOIS THEORY OF THE PICARD GROUP 64

Then the action of σ on x is to multiply x by exp(k(σ, x)π/2), where

k(σ, x) = n_{1}m_{1} + n_{2}m_{2} + n_{3}m_{3} + 2n_{4}m_{4}.

The following algorithm accomplishes these computations. Algorithm 3.17. Given a matrix-vector representation (M, v) of a subgroup H of Γ, this algorithm finds a matrix N whose columns are the coordinates, on the ordered basis (1 + i, \sqrt{2}, t_{1}, t_{2}, t_{3}), of a set of generators for B^{H} as an algebra over A.

1. Let N be the 5 × 5 diagonal matrix with diagonal entries 4, 2, 4, 4, 4.

2. Let m_{i} ← 0 (i = 1, …, 4).

3. If

m_{1}M_{j1} + m_{2}M_{j2} + m_{3}M_{j3} + 2m_{4}M_{j4} ≡ 0 (mod 4)

for all j = 1, …, 4, then go to step 4; else go to step 6.

4. If v is empty, let k ← 0. Otherwise, let k ← m_{1}v_{1} + m_{2}v_{2} + m_{3}v_{3} + 2m_{4}v_{4}

5. (Apply Lemma 3.15.) Let

N ← (\begin{matrix}& −k \\ & m_{4} \\ N & m_{1} \\ & m_{2} \\ & m_{3} \end{matrix})

6. Repeat steps 3–5 for m_{1}, m_{2},m_{3} taking all values between 0 and 3, and for m_{4} taking values 0 and 1.

7. If v is empty, let

N ← (\begin{matrix}& 2 \\ & 0 \\ N & 0 \\ & 0 \\ & 0 \end{matrix})

8. Put N into Hermite normal form. The implementation of this algorithm is listed in Section D.2.3.

3.5.2 Giving examples

Once a set of conditions on the coefficients a_{i} is given, a convenient and simple extension is to give ‘generic’ coefficients which satisfy the conditions. For example, the condition “a_{1}/a_{0} is a square, and a_{3}/a_{2} is 4 times a fourth power” is satisfied by any numbers of the form (1, c_{1}^{2}, c_{2}, 4c_{2}), where c_{1} and c_{2} are arbitrary rational numbers; conversely, up to projective equivalence and multiplication by fourth powers, any set of coefficients satisfying the conditions must be of this form. Of course, taking c_{1} and c_{2} to be insufficiently ‘general’ may give coefficients whose surface falls into a more specific case than the general one under consideration: thus the surface (1, 1, 1, 4) has rather different properties from the general surface satisfying these conditions.

From Algorithm 3.17 we obtain a matrix in Hermite normal form, whose columns represent monomials in the α_{i0} which must lie in ℚ. Assuming that the constant field for the subgroup under consideration is ℚ, the first two columns will always represent (1 + i)^{4} and \sqrt{2}^{2}; the remaining three columns contain the information we need. As remarked above, we may take fourth powers of these monomials to obtain monomials in the a_{i} which we will require to lie in (ℚ^{×})^{4}.

The last three columns of the matrix give us conditions of this form:

\begin{aligned}(−1)^{n_{13}}4^{n_{23}}a_{1}^{n_{33}} & ∈ (ℚ^{×})^{4} \\ (−1)^{n_{14}}4^{n_{24}}a_{1}^{n_{34}}a_{2}^{n_{44}} & ∈ (ℚ^{×})^{4} \\ (−1)^{n_{15}}4^{n_{25}}a_{1}^{n_{35}}a_{2}^{n_{45}}a_{3}^{n_{55}} & ∈ (ℚ^{×})^{4} \end{aligned}.  (3.13)

To turn these conditions into general forms for the a_{i}, we simply:

• set each expression in (3.13) equal to a rational fourth power c_{i}^{4};

• solve each equation in turn to express each a_{i} in terms of the c_{i};

• cast out any fourth powers.

In our example, the previous algorithm produces three expressions which generate B^{H}: t_{1}, t_{2}^{4} (a trivial generator) and \sqrt{2}t_{2}^{2}t_{3}^{2}. This means that, for the surface with coefficients (1, a_{1}, a_{2}, a_{3}) to give rise to a subgroup contained in H, the following conditions must be satisfied:

α_{10} ∈ ℚ
α_{20}^{4} ∈ ℚ
\sqrt{2}α_{20}^{2}α_{30}^{2} ∈ ℚ.

Setting these expressions equal to rational numbers c_{1}, c_{2}, c_{3}, setting a_{0} = 1 and taking appropriate powers, we get

a_{1} = c_{1}^{4}
a_{2} = c_{2}
2a_{2}a_{3} = c_{3}^{2}.

We can now rearrange these equations and cast out fourth powers to get

(a_{0}, a_{1}, a_{2}, a_{3}) = (1, 1, c_{2}, 2c_{3}^{2}c_{2}^{3}).

Substituting any sufficiently general values for c_{2} and c_{3} will give a surface whose associated subgroup is H. For example, we can recover the surface (3.8) by taking c_{2} = −6 and c_{3} = 1/6.

The algorithm is as follows.

Algorithm 3.18. Given a subgroup H of Γ as a matrix-vector representation (M, v), such that ℚ(ϵ)^{H} = ℚ, this algorithm computes monomials P_{1}(X), CHAPTER 3. GALOIS THEORY OF THE PICARD GROUP 67

P_{2}(X, Y), P_{3}(X, Y, Z) such that, for sufficiently general rational constants c_{i}, the surface defined by the coefficients

(1, P_{1}(c_{1}), P_{2}(c_{1}, c_{2}), P_{3}(c_{1}, c_{2}, c_{3}))

gives rise to the subgroup H.

1. Compute N using Algorithm 3.17.

2. Let

N ← (\begin{matrix}0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ & & N \end{matrix})

3. Let i ← 3.

4. Let N_{(i)} ← N_{(i)} × 4/N_{i+3,i}.

5. For each 3 ≤ j < i, let N_{(i)} ← N_{(i)} − (N_{j+3,i}/N_{j+3,j})N_{(j)}.

6. Let i ← i + 1. If i ≤ 5, go to step 3.

7. For each i = 1, …, 3, let P_{i} ← (1 + i)^{N^{4i}} \sqrt{2}^{N^{5i}}X^{N^{1i}}Y^{N^{2i}}Z^{N^{3i}}.

8. Cast out any powers of 4 from each P_{i}. An implementation of this algorithm is shown in Section D.2.4.

3.6 Cohomology of the Picard group

Our goal in studying the Picard group of the surface V is to be able to compute certain cohomological invariants of it. Specifically, we are interested in H^{0}(ℚ, Pic V̄), which is the part of the Picard group defined over ℚ; and in H^{1}(ℚ, Pic V̄), which is related to the Brauer group of V and will be discussed further in the following chapter. Here we present the algorithms for calculating these invariants.

3.6.1 Computing with G-modules

In this section we present some general routines for performing calculations on free Abelian groups with a group action. Here G will be a finitely generated group, and M a free ℤ-module of finite rank on which G acts by automorphisms. We will be concerned with calculating the first two cohomology groups H^{0}(G, M) and H^{1}(G, M).

The concrete representation of a G-action used is as follows: first pick generators g_{1}, …, g_{n} for G; then let M_{i} be the matrix for the action of g_{i} on M, with respect to some previously chosen basis. The vector (M_{i})_{i=1}^{n} is the object which will be used in algorithms to represent the G-module M.

The calculation of H^{0}(G, M), the part of M fixed by the action of G, is extremely simple, and is given here only for completeness. We assume that we already have the basic algorithms to perform basic matrix calculations, as provided for example by PARI.

Algorithm 3.19. Given a vector (M_{i})_{i=1}^{n} of matrices representing a group action on a free ℤ-module M of rank r, this algorithm calculates a matrix U whose columns form a basis for H^{0}(G, M).

1. Let U be a matrix whose columns form a basis for the kernel of M_{1}−I_{r}.

2. If n = 1, terminate. Otherwise, let i ← 2.

3. Let V be a matrix whose columns form a basis for the kernel of M_{i}−I_{r}.

4. Let U be a matrix whose columns form a basis for the intersection of the two lattices generated by the columns of U and V, respectively.

5. Let i ← i + 1. If i ≤ n, go to step 3.

This algorithm is listed in Section D.3.1.

We now turn to calculating the H^{1}(G, M). We do this by dimensionshifting. The sequence

0 → ℤ → ℚ → ℚ/ℤ → 0

of ℤ-modules is exact. M is torsion-free, hence flat, and so tensoring the previous sequence with M gives another exact sequence

0 → M → M ⊗ ℚ → M ⊗ ℚ/ℤ → 0.  (3.14)

The long exact sequence in cohomology associated to (3.14) is

⋯ H^{0}(G, M ⊗ ℚ) → H^{0}(G, M ⊗ ℚ/ℤ) ⟶^{∂} H^{1}(G, M) → 0  (3.15)

that is, every element of H^{1}(G, M) is of the form

σ ↦ σx − x

for some x in M ⊗ ℚ which is fixed modulo M by the action of G; such an element is a coboundary if and only if we may take the corresponding x to lie in M + H^{0}(G, M ⊗ ℚ). This is exactly the same description of H^{1}(G, M) as that given in [43], and may be written as

H^{1}(G, M) ≅ \frac{H^{0}(G, M ⊗ ℚ/ℤ)}{im H^{0}(G, M ⊗ ℚ)}.  (3.16)

The inverse map, from H^{1}(G, M) to H^{0}(G, M ⊗ ℚ/ℤ), is given by

α ↦ \frac{1}{|G|} ∑_{σ∈G} α(σ)  (3.17)

which is clearly a left inverse to the coboundary.

For computation, we use the well-known fact that H^{1}(G, M) is killed by the order of G, which we will write as m. Thus H^{0}(G, M ⊗ ℚ/ℤ) is actually contained in the m-torsion subgroup, and (multiplying through by m) we can rewrite (3.16) as

H^{1}(G, M) = H^{1}(G, M)[m] ≅ \frac{H^{0}(G, M ⊗ ℤ/mℤ)}{im H^{0}(G, M)}.  (3.18)

This isomorphism can equally well be derived from the exact sequence

0 → M ⟶^{×m} M → M ⊗ ℤ/mℤ → 0

and is valid for any G-module M with no m-torsion.

We will calculate the top and bottom of the quotient (3.18), and then apply Algorithm 3.7 to find generators for the quotient. The bottom expression is essentially already found by Algorithm 3.19; for the top we will adapt that algorithm slightly to work modulo m. The only change necessary is to calculate the kernel of a matrix modulo m, and this is performed by the PARI function matsolvemod. The modified algorithm is listed in Section D.3.2.

The whole algorithm for calculating H^{1}(G, M) is described below. We work in M ⊗ ℤ/mℤ; all matrices represent subgroups of this group.

Algorithm 3.20. Given a vector (M_{i})_{i=1}^{n} of matrices representing a group action on a free ℤ-module M of rank r, this algorithm calculates a matrix U whose columns represent elements of M ⊗ℚ, the coboundaries of which form a Smith normal form basis for the m-torsion of H^{1}(G, M); and a diagonal matrix D in Smith normal form describing the orders of those generators.

1. Let V be a matrix whose columns form a basis for H^{0}(G, M), using Algorithm 3.19.

2. Let V ← (V | mI_{r}).

3. Put V into Hermite normal form. This is in fact Algorithm 3.6.

4. Let W be a matrix whose columns form a basis for H^{0}(G, M ⊗ℤ/mℤ), using the modified version of Algorithm 3.19.

5. Apply Algorithm 3.7 to V and W to obtain a matrix U, whose columns form a Smith normal form basis for the quotient in (3.18), and a matrix D in Smith normal form describing the orders of these generators.

6. Let U ← U/m.

This algorithm is listed in Section D.3.3.

3.6.2 The action on the Picard group

In order to apply the general results of the previous section to the group Pic V̄, we must find the matrices representing generators of a subgroup H of Γ. This section describes that simple procedure.

The Picard group is generated by the 48 straight lines L_{mn}^{pqr} on V, as defined in (1.2)–(1.4). The action of Gal(ℚ̄/ℚ) on the lines is the same as the action on the generic lines L̃, defined in (3.3), of the subgroup H of Γ associated to the surface V. We will use the same names to refer to the generic lines as the specific ones:

L_{mn}^{pqr} : {X_{0} = ϵ^{m}t_{p}X_{p}, X_{q} = ϵ^{n}(t_{r}/t_{q})X_{r}}.  (3.19)

The action of Γ is simply that induced by its action on B, the coordinate ring of Ũ:

σL_{mn}^{pqr} : {X_{0} = σ(ϵ^{m}t_{p})X_{p}, X_{q} = σ(ϵ^{n}(t_{r}/t_{q}))X_{r}}

which is another line, of the form L_{m′n′}^{pqr}. Let σ be the element of Γ′ defined by

σ = σ_{1}^{n_{1}}σ_{2}^{n_{2}}σ_{3}^{n_{3}}σ_{4}^{n_{4}} ;  (3.20)

then, remembering that ϵ = (1 + i)/\sqrt{2}, we see that

\begin{aligned}m′ & ≡ m + 2(2n_{4} + n_{p}) (mod 8) \\ n′ & ≡ n + 2(2n_{4} − n_{q} + n_{r}) (mod 8). \end{aligned}  (3.21)

Also, τL_{mn}^{pqr} = L_{−m,−n}^{pqr}. Let Λ denote the free ℤ-module of rank 48 generated by the lines, which is a subgroup of Div V̄. We have shown that Λ, as a Γmodule, is the direct sum of three Γ-modules Λ^{1}, Λ^{2}, Λ^{3}: here Λ^{1} is generated by the 16 lines L_{mn}^{123}, and similarly for the other two cases. Now Λ^{1} is isomorphic to the free ℤ-module generated by the 16 elements (ϵ^{m}t_{1}, ϵ^{n}(t_{3}/t_{2})), with the obvious Γ-action; and this in turn is the tensor product, over ℤ, of two free modules of rank 4, the first generated by the ϵ^{m}t_{1} and the second by the ϵ^{n}(t_{3}/t_{2}). The representation of the element σ, defined by (3.20), on this module is then given easily by (3.21).

Lemma 3.21. Let M be the free ℤ-module generated by the four elements ϵ^{m}(t_{i}/t_{j}), where i and j are fixed and m takes the values 1, 3, 5, 7; and let Γ act on M by permuting these generators in the natural way. Let σ be defined by (3.20). Then the matrix for the action of σ on M is given by

(\begin{matrix}0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix})^{2n_{4}+n_{i}−n_{j}}

and the matrix for the action of τ on M is given by

(\begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}).

It is then straightforward to calculate the matrix for an element of Γ acting on, say, Λ^{1} as follows: take the two 4 × 4 matrices for the action on the modules generated by the ϵ^{m}t_{1} and the ϵ^{n}(t_{3}/t_{2}) respectively, and then compute the tensor product of the two matrices. The matrix for that element of Γ acting on the whole of Λ is formed simply by combining the matrices for Λ^{1}, Λ^{2} and Λ^{3} diagonally into one 48 × 48 matrix.

When implementing the algorithm, we may order the 48 lines L_{mn}^{pqr} first by p, then by m, and finally by n: thus the basis is

L_{11}^{123}, L_{13}^{123}, …, L_{75}^{312}, L_{77}^{312}.  (3.22)

Algorithm 3.22. Given an element τ^{n_{5}}σ of Γ, where n_{5} is either 0 or 1 and σ is given by (3.20) and so lies in Γ′, this algorithm computes a matrix M which represents the action of this element of Γ on the module Λ, where the basis for Λ consists of the 48 lines ordered as in (3.22).

1. Let p ← 1.

2. Let

A ← (\begin{matrix}0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix})^{2n_{4}+n_{p}}.

3. Let

B ← (\begin{matrix}0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix})^{2n_{4}−n_{p+1}+n_{p+2}}

where the indices p + 1, p + 2 are taken modulo 3, to lie in the set {1, 2, 3}.

4. If n_{5} = 1, let

A ← (\begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}) A , B ← (\begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}) B.

5. Let M_{p} be the 16 × 16 matrix such that

(M_{p})_{i+4(k−1),j+4(l−1)} = A_{ij}B_{kl}.

6. Let p ← p + 1. If p ≤ 3, go to step 2.

7. Let

M ← (\begin{matrix}M_{1} & 0 & 0 \\ 0 & M_{2} & 0 \\ 0 & 0 & M_{3} \end{matrix}).

The implementation of this algorithm is listed in Section D.3.4.

We now must choose a basis for Pic V̄ and a surjective map α from Λ to Pic V̄, and thence represent elements of Γ acting on Pic V̄. As every surjective map of vector spaces has an injective left inverse, we may take Pic V̄ to be embedded in Λ ⊗ ℚ.

We recall from Proposition 2.10 that linear and algebraic equivalence of divisors on K3 surfaces are both the same as numerical equivalence: thus if Λ^{0} denotes the kernel of the intersection form on Λ, then Pic V̄ is isomorphic to the quotient Λ/Λ^{0}. The intersection pairing is easily calculated from the explicit equations of the 48 lines:

(L_{mn}^{pqr}, L_{ij}^{pqr}) = \begin{cases}−2 & if m = i and n = j \\ 1 & if m = i or n = j, but not both \\ 0 & otherwise; \end{cases}

(L_{mn}^{pqr}, L_{ij}^{qrp}) = \begin{cases}1 & if n − m ≡ i + j (mod 8) \\ 0 & otherwise. \end{cases}

Let Q be the 48 × 48 matrix representing this quadratic form; then Λ^{0} is the (left or right) kernel of Q. We now consider the properties we would like our basis of Pic V̄ to have. It is helpful to take the class of a plane section, denoted by Π, to be the first basis vector. Now ideally the remaining basis vectors would be orthogonal to Π with respect to the intersection pairing; but this is impossible, as the resulting lattice is of finite index in Pic V̄ but is not the whole of it. Instead, we do the following:

1. Choose a ℚ-basis for Λ^{0}.

2. Extend this to a ℚ-basis for ⟨Π⟩^{⊥}; call the vector space spanned by the new basis vectors A.

3. The chosen basis for Pic V̄ will be the Hermite normal form basis for the image of Λ when projected along Λ^{0} onto Π ⊕ A.

We will consider the lattice spanned by this basis to be our concrete representation of Pic V̄. The implementation of the algorithm for calculating this basis is listed in Section D.3.5.

Finally, we can represent an element of Γ on this basis of Pic V̄. Let D be the matrix whose columns are our chosen basis for Pic V̄, expressed as elements of Λ ⊗ ℚ; and let E be the matrix representing the projection map along Λ^{0} onto Pic V̄, with respect to this basis. If σ is an element of Γ, and M is the matrix produced by Algorithm 3.22 corresponding to σ, then the matrix representing the action of σ on our basis of Pic V̄ is given by EMD. This allows us to represent any subgroup H of Γ in the form needed to apply the algorithms of Section 3.6.1 to H acting on Pic V̄.

The Brauer group

Here we investigate the Brauer group of the surface V. After recalling the definition and properties of the Brauer group of a scheme, we relate it in Section 4.1 to the cohomology of the Picard group of V̄. This enables us in Section 4.2 to apply the computational machinery of the previous chapter to calculate the Brauer group of V according to the parameters a_{0}, a_{1}, a_{2}, a_{3}. The case of cyclic algebras is studied in greater detail in Section 4.3. In Section 4.4, we look at the so-called vertical Brauer group attached to a fibration V → ℙ^{1}. The chapter concludes with a worked example of the case studied by Swinnerton-Dyer in [44], showing how the theory described in the chapter relates to the calculations given there.

Background on the Brauer groups of fields may be found in Serre [35] or, in an earlier but more explicit form, in Deuring’s book [15]. The canonical reference for Brauer groups of schemes is the set of three articles by Grothendieck [18]. An account is also given in Milne’s book [29]. Section 1 of [13] recounts much of the theory which we will use here.

4.1 Background

The Brauer group of a field k, written Br k, is a well-known object, the group of equivalence classes of central simple algebras over that field. It features heavily in the study of the arithmetic of the field; in particular, much of class field theory is concerned with Brauer groups. It is easily shown that Br k is isomorphic to the cohomology group H^{2}(k, k̄^{×}).

The definition of the Brauer group may be extended to an arbitrary scheme, as described in detail in [18]. The idea is as follows.

Definition 4.1. An Azumaya algebra on a scheme X is a coherent O_{X}- module A, with the structure of an algebra over O_{X}, such that the specialisation of A at each point x of X is a central simple algebra over the residue field at x. Equivalently, A is locally (for the étale topology) isomorphic to a matrix algebra. An Azumaya algebra is trivial if it is isomorphic to the sheaf of endomorphisms of some locally free O_{X}-module. The Brauer group of X, written Br X, is the group of equivalence classes of Azumaya algebras modulo trivial ones.

When X is Spec K for some field K, this definition agrees with the usual definition of the Brauer group of a field.

There is a natural map Br X → Br X̄. We will be entirely concerned with the part of Br X which vanishes under this map, which we will call the arithmetic part of the Brauer group of X, and denote by Br_{1}(X). This consists of those Azumaya algebras on X which are split by some finite extension of the base field.

There is always an injection Br X → H^{2}(X, 𝔾_{m}); under certain conditions this is known to be an isomorphism. In particular, this is true when X is a smooth surface. We will use this cohomological description of the Brauer group to produce alternative descriptions of Azumaya algebras.

4.1.1 Cohomological theory

In this section, X is a connected smooth projective variety over a field k of characteristic 0. The algebraic closure of k will be denoted by k̄, and X̄ will mean X ×_{k} k̄. The field of rational functions on X will be denoted by k(X).

The machinery used in this section is described in many texts. The canonical reference is [14], and other useful accounts may be found in the books by Milne [29] and Tamme [45], and also the lecture notes by Milne [28].

We will study the cohomology of the sheaf 𝔾_{m} on X and also of the sheaf of divisors on X. In this section and later, a useful tool will be the following form of the Hochschild–Serre spectral sequence:

Lemma 4.2. Let k′ be a (possibly infinite) Galois extension of k. Write G for Gal(k′/k), and X′ for X ×_{k} k′. Let F be a sheaf on X. Then there is a spectral sequence

E_{2}^{pq} = H^{p}(G, H^{q}(X′, F)) ⟹ H^{p+q}(X, F).

Proof. See [29] or any of several other references.

Applying this to k′ = k̄ and F = 𝔾_{m}, we get a filtration on H^{2}(X, 𝔾_{m}):

0 ⊆ im Br k ⊆ ker(H^{2}(X, 𝔾_{m}) → H^{2}(X̄, 𝔾_{m})) ⊆ H^{2}(X, 𝔾_{m}).  (4.1)

Here the successive quotients of the filtration are isomorphic to: the image of Br k; H^{1}(k, Pic X̄); and some subgroup of H^{0}(k, H^{2}(X̄, 𝔾_{m})). These three pieces of H^{2}(X, 𝔾_{m}) might be described as follows.

• The image of Br k is the constant Brauer group of X, written Br_{0} X. Constant elements can produce no Brauer–Manin obstruction. The study of Brauer groups of fields is an essential part of class field theory.

• The group consisting of those elements which split over some finite extension of the base field k is written Br_{1} X. The second piece of the filtration is Br_{1} X/ Br_{0} X. This is the arithmetic part, in which we are interested. The isomorphism between this part of the Brauer group and H^{1}(k, Pic X̄) will be our means of calculating this part of the group.

• The final piece of H^{2}(X, 𝔾_{m}) consists of elements of H^{2}(X̄, 𝔾_{m}). These have been studied in a geometric context, being related to those homology classes on X̄ which do not contain an algebraic cycle. This might be described as the geometric part of the Brauer group of X. These elements can under some circumstances give rise to non-trivial Brauer–Manin obstructions: see Harari’s example [20].

We now prove some general and well-known results about the cohomology of various sheaves.

Let η = Spec k(X) denote the generic point of X, and j the inclusion map η → X. The direct image sheaf j_{∗}𝔾_{m} is simply the sheaf of rational functions on X; there is a canonical inclusion, of sheaves on X, of 𝔾_{m} into j_{∗}𝔾_{m}. The quotient sheaf is by definition the sheaf of Cartier divisors on X, denoted by D_{X}. In the case we are considering, this is isomorphic to the sheaf of Weil divisors on X:

D_{X} ≅ ⨁_{x∈X^{(1)}} i_{x∗}ℤ

where X^{(1)} is the set of points of X of codimension 1 (that is, generic points of closed subschemes of codimension 1) and i_{x} is the inclusion x → X. The short exact sequence

0 → 𝔾_{m} → j_{∗}𝔾_{m} → D_{X} → 0

gives rise to a long exact sequence in cohomology, part of which is

H^{1}(X, D_{X}) → H^{2}(X, 𝔾_{m}) → H^{2}(X, j_{∗}𝔾_{m}) → H^{2}(X, D_{X})  (4.2)

The second term of this sequence is the one which interests us. We will now describe the other terms.

Lemma 4.3. H^{1}(X, D_{X}) = 0.

Proof. Cohomology commutes with direct sums of sheaves, so it is enough to show that H^{1}(X, i_{x∗}ℤ) = 0 for all points x ∈ X^{(1)}. The Leray spectral sequence [28, Theorem 12.8] shows that H^{1}(X, i_{x∗}ℤ) injects into H^{1}(x, ℤ); and this last group is a Galois cohomology group which is equal to

Hom(Gal(\overline{k(x)}/k(x)), ℤ)

and therefore is zero, as the Galois group is by definition profinite and ℤ is free.

Corollary 4.4. The map H^{2}(X, 𝔾_{m}) → H^{2}(X, j_{∗}𝔾_{m}) in (4.2) is injective.

Lemma 4.5. H^{i}(X, j_{∗}𝔾_{m}) = H^{i}(k(X), 𝔾_{m}) for all i ≥ 0.

Proof. By the Leray spectral sequence, it is enough to show that R^{i}j_{∗}𝔾_{m} = 0 for all i ≥ 1. If x̄ is any geometric point of X, then the stalk of R^{i}j_{∗}𝔾_{m} at x̄ is given by H^{i}(Spec O_{X,x̄}×_{X} η, 𝔾_{m}). Now the fibre product in that expression is simply the spectrum of the field of fractions of O_{X,x̄}; and by a theorem of Lang [28, 13.5] such fields are quasi-algebraically closed, hence the result.

Corollary 4.6. The exact sequence (4.2) becomes

0 → H^{2}(X, 𝔾_{m}) → Br k(X) → H^{2}(X, D_{X}).

We now suppose that X is such that Br X = H^{2}(X, 𝔾_{m}). As stated above, we are primarily interested in the arithmetic part Br_{1} X of the Brauer group of X. It turns out that to study this part of the Brauer group we only have to use Galois cohomology:

Corollary 4.7. There is an exact sequence

0 → Br_{1} X → Br(k(X̄)/k(X)) → H^{2}(k, Div X̄)

that is, a central simple algebra over k(X), which is split by some finite extension of the base field k, is Azumaya if and only if it maps to zero in H^{2}(k, Div X̄).

Proof. By Corollary 4.6 applied to X and X̄, we have a commutative diagram with exact rows

0 ⟶ Br X ⟶ Br k(X) ⟶ H^{2}(X, D_{X})
↓ ↓ ↓
0 ⟶ Br X̄ ⟶ Br k(X̄) ⟶ H^{2}(X̄, DX̄)

where the kernels of the vertical maps are, from left to right: Br_{1} X (by definition); Br(k(X̄)/k(X)) (by the inflation-restriction exact sequence); and H^{2}(k, Div X̄) (by Lemma 4.3 and the Hochschild–Serre spectral sequence). Hence the exact sequence of kernels is the desired result.

Since we are interested in studying the Brauer–Manin obstruction, we only wish to consider elements of Br X up to constant algebras. There is an exact sequence of Galois cohomology

H^{2}(k, k̄^{×}) → H^{2}(k, k(X̄)^{×}) → H^{2}(k, k(X̄)^{×}/k̄^{×}) → H^{3}(k, k̄^{×})

and Tate has proved [46, p. 199] that the last group is always trivial when k is a number field; thus the middle map is surjective. Moreover, the map k(X̄)^{×} → Div X̄ is zero on k̄^{×} and so factors through k(X̄)^{×}/k̄^{×}. We may therefore define the following group, whose calculation will concern us in this chapter:

H_{Az}^{2}(X) := ker (H^{2}(k, k(X̄)^{×}/k̄^{×}) → H^{2}(k, Div X̄)).  (4.3)

4.1.2 Change of fields

It will sometimes be necessary to consider how the various groups defined above vary as the base field of X changes. We state here some simple results. If X is defined over a field k, and K/k is a field extension, we write X_{K} for X ×_{k} K.

Lemma 4.8. Let L/K be a Galois extension of fields. Then the following two conditions are equivalent:

1. Pic X_{K} = H^{0}(L/K, Pic X_{L});

2. H^{1}(L/K, k(X_{L})^{×}/L^{×}) = 0.

Proof. We consider the long exact sequence in cohomology

0 → H^{0}(L/K, k(X_{L})^{×}/L^{×}) → H^{0}(L/K, Div X_{L}) → H^{0}(L/K, Pic X_{L}) → H^{1}(L/K, k(X_{L})^{×}/L^{×}) → 0.

Firstly, H^{0}(L/K, k(X_{L})^{×}/L^{×}) is identified with the group of principal divisors on X_{K}: this follows from the fact that H^{1}(L/K, L^{×}) = 0. The next term is equal to Div X_{K}. The sequence therefore becomes

0 → Pic X_{K} → H^{0}(L/K, Pic X_{L}) → H^{1}(L/K, k(X_{L})^{×}/L^{×}) → 0

which implies the result.

Note that, in particular, these equivalent conditions hold when X has points in every completion of k (see Proposition 2.21).

We now note a consequence of these conditions.

Lemma 4.9. Let L/K/k be Galois extensions, and let X be a scheme over k as above. If the equivalent conditions of Lemma 4.8 are satisfied, then the following two results hold:

1. The map H^{1}(K/k, Pic X_{K}) → H^{1}(L/k, Pic X_{L}) is injective;

2. The map H^{2}(K/k, k(X_{K})^{×}/K^{×}) → H^{2}(L/k, k(X_{L})^{×}/L^{×}) is injective.

Proof. Each result is an immediate consequence of the conditions together with the appropriate inflation-restriction exact sequence.

4.2 Explicit descriptions

We now return to the surface V, as defined in (1.1). The coefficients a_{i} are fixed, and we write α_{ij} for a fourth root of a_{i}/a_{j}. We define

k = ℚ
K = ℚ(α_{10}, α_{20}, α_{30}, ϵ)/ℚ
G = Gal(K/ℚ)

The previous section shows that there are isomorphisms

Br_{1} V/ Br k
↕
ker(H^{2}(k, k(V̄)^{×}) → H^{2}(k, Div V̄))/ Br k
↕
H_{Az}^{2}(V)
↕
H^{1}(k, Pic V̄)

In this section, we will make these isomorphisms explicit; this will allow the calculation of elements of Br_{1} V. The procedure is not entirely satisfactory, because lifting elements from k(V̄)^{×}/k^{×} (the group of principal divisors) to k(V̄^{×}) is a rather tedious process; in the case discussed in Section 4.4, where a fibration exists from V to ℙ^{1}, we will see that this step may be avoided.

4.2.1 Calculation of H^{1}(k, Pic V̄)

Firstly we consider the group H^{1}(k, Pic V̄). As in Section 3.6.2, we let Λ denote the subgroup of Div V̄ generated by the 48 lines on V̄, and Λ^{0} the group consisting of those divisors linearly equivalent to 0; then Pic V̄ is isomorphic to the quotient Λ/Λ^{0}. Lemma 3.1 shows that the least field of definition of Λ is K. Then the inflation-restriction exact sequence gives:

Lemma 4.10. The map

H^{1}(G, Λ/Λ^{0}) ⟶^{inf} H^{1}(k, Pic V̄)

is an isomorphism.

This group may therefore be calculated using the algorithms described in Chapter 3.

4.2.2 Calculation of H_{Az}^{2}(V)

We now describe the isomorphism between H^{1}(k, Pic V̄), which we can compute, and H_{Az}^{2}(V). This comes from the exact sequence

0 → k(V̄)^{×}/k̄^{×} → Div V̄ → Pic V̄ → 0

where we think of k(V̄)^{×}/k̄^{×} as being the group of principal divisors on V̄. Part of the long exact sequence in cohomology associated to this sequence is

H^{1}(k, Div V̄) → H^{1}(k, Pic V̄) ⟶^{∂} H^{2}(k, k(V̄)^{×}/k̄^{×}) → H^{2}(k, Div V̄)

and Lemma 4.3 shows that H^{1}(k, Div V̄) = 0; thus ∂ gives an isomorphism from H^{1}(k, Pic V̄) to H_{Az}^{2}(V). The definition of this coboundary map is well known, and may be explicitly described as follows:

1. Given a 1-cocycle with values in Pic V̄, lift it to a 1-cochain with values in Div V̄.

2. Take the coboundary of this cochain to get a 2-coboundary with values in Div V̄.

3. This 2-coboundary maps to 0 in Pic V̄; thus it may be pulled back to obtain a 2-cochain with values in k(V̄)^{×}/k̄^{×}.

Computationally, how should this be accomplished? We are representing the group H^{1}(k, Pic V̄) by the isomorphic, but easily computable, group H^{1}(G, Λ^{0}/Λ). Is there a more manageable group with which we might replace H_{Az}^{2}(V)? Lemma 4.11. H_{Az}^{2}(V) is isomorphic to the kernel of

H^{2}(G, Λ^{0}) → H^{2}(G, Λ).

In other words, every class in H_{Az}^{2}(V) contains a cocycle whose support is contained in Λ^{0}, and which is obtained by inflation from G; and two such cocycles lie in the same class if and only if they differ by a coboundary satisfying the same criteria.

Proof. There is a commutative diagram with exact rows

0 ⟶ H^{1}(G, Λ/Λ^{0}) ⟶^{∂} H^{2}(G, Λ^{0}) ⟶ H^{2}(G, Λ)
α↓  ^{β}↓  ^{γ}↓
0 ⟶ H^{1}(k, Pic V̄) ⟶^{∂} H^{2}(k, k(V̄^{×})/k^{×}) ⟶ H^{2}(k, Div V̄)

where the vertical maps all come from inflation and inclusion of modules. The map α is, as we have seen above, an isomorphism. The map γ is the composite

H^{2}(G, Λ) ⟶^{i_{∗}} H^{2}(G, Div V_{K}) ⟶^{inf} H^{2}(k, Div V̄)

where V_{K} means V ×_{Spec k}Spec K and i is the inclusion of Λ into Div V_{K}. Now Λ is a direct summand of Div V_{K}, so i_{∗} is injective; and the inflation map is also injective, by the inflation-restriction exact sequence. Thus γ is injective, and hence so is β. Now, by the commutativity of the left-hand square, β identifies the images of the two coboundary maps ∂; hence the result.

It is easy to see that if we follow through the explicit definition of ∂ given above, starting from an element of H^{1}(G, Λ/Λ^{0}), then we will obtain an element of H^{2}(G, Λ^{0}).

The inverse mapping ∂^{−1} is slightly harder to compute, but still presents no real problem. The explicit description is as follows:

1. Any 2-cocycle lying in H_{Az}^{2}(V) by definition maps to a coboundary in H^{2}(k, Div V̄); so it may be written as dα say, for some 1-cochain α with values in Div V̄.

2. Now consider α as a 1-cochain with values in Pic V̄; since dα takes values in the principal divisors, α is a 1-cocycle for Pic V̄.

There is what looks like a problem in step 1: given an element of H^{2}(k, Div V̄) which we know to be a coboundary, we must find a suitable element α. This may however be reduced to a smaller problem. Write Ḡ for the absolute Galois group of k. Now Div V̄ is the direct sum of the Galois modules ℤ[Ḡ]D, where D runs through a set of non-conjugate prime divisors on V̄. These modules are all of finite rank over ℤ. Any element of H^{2}(k, Div V̄) factors through a finite quotient H of Ḡ, hence has support containing only finitely many prime divisors, and so takes values in some finite direct sum ∑_{i} ℤ[H]D_{i}. Cohomology commutes with direct sums, so we are looking for an element α ∈ ∑_{i} C^{1}(H, ℤ[H]D_{i}) with the desired coboundary. This gives a finite system of equations to be solved.

More concretely, this means that we can take the divisors of V one at a time when calculating ∂^{−1}. Each divisor D on V splits over V̄ into divisors D_{j}. We start with a 2-cocycle c in H_{Az}^{2}(V), which by definition maps to 0 in H^{2}(k, Div V̄) and hence maps to 0 in H^{2}(k, ∑ ℤD_{j}), which is the smallest submodule of Div V̄, fixed by the Galois action, which contains the D_{j}. So we can write the image of c (a 2-cocycle with values in ∑ ℤD_{j}) as dα, where α is a 1-cochain with values in ∑ ℤD_{j}; searching for a suitable α is a small and simple problem. The image of α in Pic V̄ will not necessarily be a cocycle: but when we add together all the α from all the divisors D on V, the sum will indeed be a cocycle.

We state here a variation on the usual description by cochains of the first cohomology group, which is useful in light of the above discussion.

Lemma 4.12. Let M be a torsion-free Abelian group acted on by a group G. As usual, we write C^{i}(G, M), Z^{i}(G, M), B^{i}(G, M) for the groups of icochains, -cocycles and -coboundaries respectively with values in M. Then

H^{1}(G, M) ≅ \frac{Z^{1}(G, M) + C^{1}(G, M^{G})}{B^{1}(G, M) + C^{1}(G, M^{G})}.  (4.4)

Proof. This is the usual isomorphism theorem: if K is a normal subgroup of a group H, and H′ is any subgroup of H, then H′K/K ≅ H′/(H′ ∩ K). We take H′ to be Z^{1}(G, M) and K to be the sum of B^{1}(G, M) and C^{1}(G, M^{G}). Then

H′ ∩ K = B^{1}(G, M) + (Z^{1}(G, M) ∩ C^{1}(G, M^{G})) = B^{1}(G, M) + Z^{1}(G, M^{G})

but M^{G} is a torsion-free G-module with trivial G-action, so Z^{1}(G, M^{G}) = Hom(G, M^{G}) = 0.

Corollary 4.13. When calculating the map from H_{Az}^{2}(V) to H^{1}(k, Pic V̄), we may use this description of H^{1}(k, Pic V̄) and then forget the contributions at any divisors on V̄ which are defined over k.

Proof. Such divisors give rise to cochains in Pic V̄ which take values in H^{0}(k, Pic V̄); thus they lie in the bottom half of the quotient (4.4).

Recall that, in Section 3.6.1, we computed H^{1}(G, M) for a torsion-free G-module M by the isomorphism

H^{1}(G, M) ≅ \frac{H^{0}(G, M ⊗ ℚ/ℤ)}{im H^{0}(G, M ⊗ ℚ)}.

The right hand side here is a quotient of a subgroup of M ⊗ℚ; so we will often be storing elements of H^{1}(k, Pic V̄), after lifting, as elements of Pic V̄ ⊗ ℚ. Now k(V̄)^{×}/k̄^{×} is also a torsion-free group: this is easily seen to be true, as any element of k̄ already has as many roots as possible. Hence there is an isomorphism

H^{1}(k, (k(V̄)^{×}/k̄^{×}) ⊗ ℚ/ℤ) ≅ H^{2}(k, k(V̄)^{×}/k̄^{×}).

By the functorial properties of cohomology, the diagram

Pic V̄ ⊗ ℚ ⟶^{∂} H^{1}(k, (k(V̄)^{×}/k̄^{×}) ⊗ ℚ/ℤ)
↓  ↓
H^{1}(k, Pic V̄) ⟶^{∂}  H^{2}(k, k(V̄)^{×}/k̄^{×})

commutes: we may compute the coboundary either way, and the comments above still apply.

4.2.3 From H^{2} to algebras

As mentioned in the introduction to this section, it is not always straightforward to lift elements from H_{Az}^{2}(V) to H^{2}(k, k(V̄)^{×}). In specific cases, this can be accomplished by hand: given a principal divisor, we need to find the function (unique up to a constant multiple) whose divisor it is. Now the only divisors whose functions we can instantly write down are those which are hyperplane sections or, more generally, intersections of V with some hypersurface in ℙ^{3}. So we must express the given principal divisor as a difference of sums of hypersurface sections, which is always possible, and then it is possible to write down the required function. The function will in general be a messy ratio of high-degree polynomials.

The correspondence between H^{2}(k, k(V̄)^{×}) and Br k(V̄), on the other hand, is well-known and is reproduced here only for reference. This material is taken from Chapter V of [15], which has the advantage that the calculations are very explicit.

Briefly, let A be a central simple algebra over a field K; then it is known that A contains a maximal subfield L, of degree n, say, over K; and A is then a vector space of dimension n over L. Now let σ be an element of Gal(L/K); then it can be shown that the action of σ on L is the same as conjugating by an element u_{σ} of A:

σa = u_{σ}au_{σ}^{−1} for all a ∈ L.  (4.5)

If we now define ϕ : Gal(L/K)^{2} → L by

ϕ(σ, τ) = u_{σ}u_{τ}u_{στ}^{−1}  (4.6)

then it is easily checked that the values do indeed lie in L, and that ϕ defines a 2-cocycle. Different choices of u_{σ} change ϕ by a coboundary, so we have defined a map from Br(L/K) to H^{2}(L/K, L^{×}).

The inverse to this correspondence is simple to construct. Given a 2cocycle ϕ ∈ H^{2}(L/K, L^{×}), we take A to be the vector space over L generated by the symbols u_{σ}, and define multiplication in A to satisfy the equations (4.5) and (4.6). It turns out that A is indeed a central simple algebra over K, as required.

4.3 Cyclic algebras

We now investigate in detail the situation for cyclic algebras on V. Apart from being generally instructive, this case will be important when we consider algebras arising from fibrations in Section 4.4. The details of the Galois cohomology used in this section may be found in several places, for example the on-line notes [27] of Milne.

Temporarily, we let L/K be any finite cyclic extension of fields; let G be the Galois group of L/K, which has order n and is generated by some element σ. Let u denote a symbol such that u^{n} = b lies in K^{×}; then we may define an algebra A as follows:

• As a vector space over L, A has basis {u^{i} | i = 0, …, n − 1};

• Multiplication is defined by

(σx)u = ux for all x ∈ L.

This algebra A is by definition a cyclic algebra over K, which is its centre. This algebra will be denoted by (L/K, b); we will now see exactly how it depends on both L/K and b.

4.3.1 Cohomology of cyclic groups

If G is any finite group, then H^{i}(G, ℚ) = 0 for all i ≥ 1; this gives us, in particular, an isomorphism H^{1}(G, ℚ/ℤ) → H^{2}(G, ℤ). Now H^{1}(G, ℚ/ℤ) is simply the character group of G; if G is cyclic of order n, then so is its character group. We obtain the following result:

Lemma 4.14. Let G be cyclic of order n. Then H^{2}(G, ℤ) is cyclic of order n, and a generator is given by

w(σ^{i}, σ^{j}) = \begin{cases}0 & if i + j < n; \\ 1 & if i + j ≥ n. \end{cases}

Proof. A generator for Hom(G, ℚ/ℤ) is given by the homomorphism χ mapping σ to 1/n. To calculate the image of χ in H^{2}(G, ℤ), we first imagine that χ is now mapping into ℚ, and then take its coboundary:

\begin{aligned}dχ(σ^{i}, σ^{j}) & = σ^{i}χ(σ^{j}) + χ(σ^{i}) − χ(σ^{i+j}) \\ & = (i + j − (i + j mod n))/n \end{aligned}

which gives the result stated.

The next result, which is the well-known and fundamental result on the cohomology of cyclic groups, is best stated in terms of the Tate cohomology groups. Let M be any G-module, and let N denote the norm element ∑_{i=0}^{n−1} σ^{i} in ℤ[G]. The Tate cohomology groups are defined as follows:

Ĥ^{i}(G, M) = \begin{cases}H^{i}(G, M) & i ≥ 1; \\ H^{0}(G, M)/NM & i = 0; \\ _{N}M/(σ − 1)M & i = −1; \\ H_{−i−1}(G, M) & i ≤ −2. \end{cases}

Here _{N}M denotes the kernel of N : M → M. The norm map takes the place of the coboundary in dimension 0. The result we need is this:

Lemma 4.15. Let G be cyclic of order n, and let M be any G-module. Then, for each i, there is an isomorphism

Ĥ^{i}(G, M) ⟶^{∼} Ĥ^{i+2}(G, M).

This isomorphism is given by the cup-product with the generator of H^{2}(G, ℤ) given in Lemma 4.14.

The proof is omitted here; it is Proposition 2.11 in [27]. We will need the explicit construction of the isomorphism, given there, later.

This result immediately shows that, for L/K as at the beginning of this section, H^{2}(G, L^{×}) is isomorphic to Ĥ^{0}(G, L^{×}). In fact, we have Lemma 4.16. Let L/K be a cyclic extension with Galois group G. Then

H^{2}(G, L^{×}) ≅ K^{×}/NL^{×}

where the isomorphism identifies an element b of K^{×} with the 2-cocycle b∪w. The algebra associated to this cocycle is the cyclic algebra (L/K, b).

Proof. The first statement follows immediately from the preceding lemma. The second statement is a consequence of the definition of (L/K, b) and the correspondence between algebras and 2-cocycles described in Section 4.2.3.

More generally, let χ be any homomorphism from Gal(K̄/K) to ℚ/ℤ, and b be any element of K^{×}; then we can form an element of H^{2}(K, K̄^{×}) by letting (χ, b) := b ∪ dχ. The element (L/K, b) which we have been examining is the special case of this when χ is chosen to the cyclic character associated to the extension L/K.

4.3.2 Cyclic algebras on V

We now apply the general study of cyclic algebras in the previous section to the surface V, and in particular describe the elements of H^{1}(G, Pic V̄) corresponding to cyclic Azumaya algebras. In this section, k is once again the field of definition of V (normally ℚ, but the theory is quite general). We let L be some cyclic extension of k, with Galois group G, and we will study those Azumaya algebras on V which split when the base field is extended to L.

The inflation-restriction sequence says that an algebra on V splits over L if and only if the corresponding 2-cocycle lies in H^{2}(L/k, k(V_{L})^{×}). Now, we are considering algebras modulo elements of the image of Br k; but it is not in general true that an algebra which splits over L, and is equivalent to a constant algebra over k̄, is therefore equivalent to a constant algebra over L. The diagram looks like this:

Br(L/k) ⟶ Br(k(V_{L})/k(V)) ⟶ H^{2}(L/k, k(V_{L})^{×}/L^{×}) ⟶ H^{3}(L/k, L^{×})
↓ ↓ ↓
Br(k) ⟶ Br(k(V̄)/k(V)) ⟶ H^{2}(k, k(V̄)^{×}/k̄^{×}) ⟶  0

where the two leftmost vertical arrows are injective. The right-hand vertical arrow has no reason to be injective, unless the conditions of Lemma 4.8 are satisfied: for example, when V has points in every completion of k. We do know, however, that H^{3}(L/k, L^{×}) = 0, for by Lemma 4.15 this is isomorphic to H^{1}(L/k, L^{×}), which is trivial.

There are no problems with deciding whether an algebra is Azumaya. For the map from H^{2}(L/k, Div V_{L}) to H^{2}(k, Div V̄) is injective by the inflationrestriction sequence; therefore an element of H^{2}(L/k, k(V_{L})^{×}/L^{×}) is Azumaya if and only if it maps to 0 in H^{2}(L/k, Div V_{L}). We define

H_{Az}^{2}(L/k, V) := ker(H^{2}(L/k, k(V_{L})^{×}/L^{×}) → H^{2}(L/k, Div V_{L})).

In light of the discussion in the preceding paragraph, we keep in mind that the map from H_{Az}^{2}(L/k, V) to H_{Az}^{2}(V) is not necessarily injective.

A cyclic algebra which is split by L/k can be written as (L/k, f) for some rational function f in k(V). We will now answer two questions about such algebras: when are they Azumaya, and what do such algebras map to in H^{1}(k, Pic V̄)?

The isomorphism described in Lemma 4.15 commutes with the coboundary morphism, so we have a commutative diagram

Ĥ^{−1}(L/k, Pic V_{L}) ⟶^{∂} Ĥ^{0}(L/k, k(V_{L})^{×}/L^{×}) ⟶ Ĥ^{0}(L/k, Div V_{L})
↓  ↓  ↓
H^{1}(L/k, Pic V_{L}) ⟶^{∂} H^{2}(L/k, k(V_{L})^{×}/L^{×}) ⟶ H^{2}(L/k, Div V_{L})

(4.7)

where the vertical arrows are all isomorphisms, given by cup-product with the generator w of H^{2}(L/k, ℤ). By Lemma 4.16, the cyclic algebra (L/k, f) corresponds to the element f of Ĥ^{0}(L/k, k(V_{L})^{×}/L^{×}). The question of whether (L/k, f) is Azumaya can be answered immediately, and we get the following generalisation of Lemma 11 of [44]:

Proposition 4.17. Let A = (L/k, f) be a cyclic algebra on V, where L 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 L. 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.

Proof. For the first statement, we need only recall the definition of the Tate cohomology group:

Ĥ^{0}(L/k, Div V_{L}) = Div V/N Div V_{L}.

The result then comes straight from the diagram (4.7). The second statement is similar, but we must also apply Lemma 4.9 to show that in this case the map from H_{Az}^{2}(L/k, V) to H_{Az}^{2}(V) is injective.

The first step towards calculating the image of (L/k, f) in H^{1}(k, Pic V̄) is to work out the corresponding element of Ĥ^{−1}(L/k, Pic V_{L}). Recall the definition:

Ĥ^{−1}(L/k, Pic V_{L}) =_{N} Pic V_{L}/(σ − 1) Pic V_{L}

where σ is a generator for Gal(L/k).

Lemma 4.18. The element of Ĥ^{−1}(L/k, Pic V_{L}) corresponding to the algebra (L/k, f) is simply the class of D from Proposition 4.17.

Proof. First of all, we note that ND = (f), which is principal, so D does indeed lie in _{N} Pic V_{L}. We then apply the definition of the coboundary morphism: lift to Div V_{L} to obtain the divisor D; take coboundary, which in this case is the norm, to obtain (f); and pull back to Ĥ^{0}(L/k, k(V_{L})^{×}/L^{×}) to regain the function f. By Lemma 4.16, this corresponds to the algebra (L/k, f) in H_{Az}^{2}(L/k, V).

The next step is to describe the isomorphism between H^{1}(L/k, Pic V_{L}) and Ĥ^{−1}(L/k, Pic V_{L}). One definition of this isomorphism can be found in [27], in the proof of Proposition 2.11. Let G be a cyclic group, generated by an element σ, and let M be a G-module. We consider the exact sequence

0 → M ⟶^{a↦N⊗a} ℤ[G] ⊗_{ℤ} M ⟶^{σ−1} ℤ[G] ⊗_{ℤ} M ⟶^{ϵ} M → 0  (4.8)

where N is the norm element of ℤ[G], the middle arrow is multiplication by (σ − 1), and ϵ is the augmentation map which takes σ^{i} ⊗ a to a for all i. The group action on ℤ[G] ⊗_{ℤ} M is the diagonal action:

σ(g ⊗ a) = σg ⊗ σa.

The sequence (4.8) defines a “double coboundary” map from Ĥ^{i}(G, M) to Ĥ^{i+2}(G, M) for all i. Simple arguments (which we will not need) show that ℤ[G] ⊗_{ℤ} M has trivial cohomology, so this map is an isomorphism. The map may be explicitly described.

1. Given an i-cocycle with values in M, lift it to an i-cochain with values in ℤ[G] ⊗ M.

2. Take the coboundary of this i-cochain.

3. This (i + 1)-coboundary maps to 0 under ϵ, so may be written as the product of (σ−1) with another (i+1)-cochain with values in ℤ[G]⊗M.

4. Take the coboundary of this (i + 1)-cochain.

5. The resulting (i+2)-coboundary maps to 0 when multiplied by (σ −1), so may be written as the norm tensored with an (i + 2)-cocycle with values in M.

We now unpick this definition to get the following result.

Lemma 4.19. Let a be an element of Ĥ^{−1}(G, M). Then the image of a in H^{1}(G, M) is represented by the cocycle α, defined by

α(1) = 0
α(σ^{r}) = ∑_{i=0}^{r−1} σ^{i}a  for r ≥ 1.

Proof. We simply follow the definition of the “double coboundary” map given above.

1. Lift a to 1 ⊗ a in ℤ[G] ⊗ M.

2. The coboundary in this case is the norm: we get

N(1 ⊗ a) = ∑_{i=0}^{n−1}(σ^{i} ⊗ σ^{i}a).

3. As Na = 0, we may subtract 1 ⊗ Na to get

\begin{aligned}N(1 ⊗ a) & = ∑_{i=1}^{n−1}(σ^{i} − 1) ⊗ σ^{i}a \\ & = (σ − 1) ∑_{i=1}^{n−1} ∑_{j=0}^{i−1}(σ^{j} ⊗ σ^{i}a) \end{aligned}

4. Dividing by (σ −1) and taking the coboundary, we obtain the 1-cocycle

σ^{r} ↦ ∑_{i=1}^{n−1} ∑_{j=0}^{i−1}(σ^{j+r} ⊗ σ^{i+r}a − σ^{j} ⊗ σ^{i}a).  (4.9)

5. Now we must do some tedious manipulation of (4.9) to show that it can be written in the form N ⊗ α. If r = 0, this is trivial; so assume CHAPTER 4. THE BRAUER GROUP 96

that r ≥ 1. We split the sum up into four separate sums:

σ^{r} ↦ ∑_{i=1}^{n−r−1} ∑_{j=0}^{i−1}(σ^{j+r} ⊗ σ^{i+r}a) + ∑_{i=n−r}^{n−1} ∑_{j=0}^{n−r−1}(σ^{j+r} ⊗ σ^{i+r}a)

+ ∑_{i=n−r+1}^{n−1} ∑_{j=n−r}^{n−1}(σ^{j+r} ⊗ σ^{i+r}a) − ∑_{i=1}^{n−1} ∑_{j=0}^{i−1}(σ^{j} ⊗ σ^{i}a)

= ∑_{i=r}^{n−1} ∑_{j=r}^{i−1}(σ^{j} ⊗ σ^{i}a) + ∑_{i=0}^{r−1} ∑_{j=r}^{n−1}(σ^{j} ⊗ σ^{i}a)

+ ∑_{i=1}^{r−1} ∑_{j=0}^{i−1}(σ^{j} ⊗ σ^{i}a) − ∑_{i=1}^{n−1} ∑_{j=0}^{i−1}(σ^{j} ⊗ σ^{i}a)

= ∑_{i=0}^{r−1} ∑_{j=r}^{n−1}(σ^{j} ⊗ σ^{i}a) − ∑_{i=r}^{n−1} ∑_{j=0}^{r−1}(σ^{j} ⊗ σ^{i}a).

Now once again Na = 0, so we may add

0 = ∑_{j=0}^{r−1}(σ^{j} ⊗ Na) = ∑_{j=0}^{r−1} ∑_{i=0}^{n−1}(σ^{j} ⊗ σ^{i}a)

to the expression above to obtain

\begin{aligned}σ^{r} & ↦ ∑_{i=0}^{r−1} ∑_{j=0}^{n−1}(σ^{j} ⊗ σ^{i}a) \\ & = N ⊗ ∑_{i=0}^{r−1} σ^{i}a \end{aligned}

as desired.

Corollary 4.20. Let (L/k, f) be a cyclic Azumaya algebra on V. Then the element of H^{1}(L/k, Pic V_{L}) corresponding to the algebra (L/k, f) is represented by the cocycle

1 ↦ 0
σ^{r} ↦ ∑_{i=0}^{r−1} σ^{i}D  for r ≥ 1.

where D is the divisor from Proposition 4.17.

4.4 The vertical Brauer group

We now consider the case when V is equipped with a fibration π : V → ℙ^{1}. The geometry of this situation was discussed in Section 2.2, and in this section we will investigate how such a fibration might be used to give other ways of computing the Brauer group of V. Specifically, we will show how the methods used in [44] for computing Brauer groups fit in with the rest of this chapter.

Let η denote the generic point of ℙ^{1}, so that V_{η} is the generic fibre of the fibration, a curve of genus 1 over k(ℙ^{1}). We recall from Section 2.2 that the vertical Picard group, denoted Pic_{vert}, associated to this fibration is defined by

Pic_{vert} = ker(Pic V̄ → Pic V̄_{η}).

Any Azumaya algebra on ℙ^{1} can be lifted to an Azumaya algebra on V. This is not very useful, however, as the Brauer group of ℙ^{1} consists only of the classes of the constant algebras Br k. Instead, we consider those elements of Br k(ℙ^{1}) which, although not Azumaya over ℙ^{1}, give Azumaya algebras when lifted to V. The group of these algebras modulo constant algebras will be written as B:

B = {A ∈ Br k(ℙ^{1}) | π^{∗}A ∈ Br V }/ Br k.

Any such algebra must necessarily split over k̄, since Br k(ℙ1 ; k̄) is trivial by Tsen’s theorem [28, 13.5]. We identify k(ℙ^{1}) with k(t). Since once again H^{3}(k, k̄^{×}) = 0, we have that Br k(t)/ Br k is isomorphic to H^{2}(k, k̄(t)^{×}/k̄^{×}).

Therefore, by Corollary 4.7,

B ≅ ker (H^{2}(k, k̄(t)^{×}/k̄^{×}) ⟶^{π^{∗}} H^{2}(k, k(V̄)^{×}/k̄^{×}) → H^{2}(k, Div V̄)). (4.10)

As with H_{Az}^{2}(V), this isomorphism is not straightforward to calculate in general, although in some specific cases it can be done without too much trouble. However, we will see below that in the vertical case it is possible to write down the elements of B without coming through this isomorphism.

We can calculate the structure of B by cohomological methods, just as for H_{Az}^{2}(V).

Proposition 4.21. 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) ⟶^{∼} H_{Az}^{2}(V)

commutes.

Proof. Consider the commutative diagram with exact rows and columns

0  0  0
↓  ↓  ↓
0 ⟶ k̄(t)^{×}/k̄^{×} ⟶ Div_{vert} ⟶ Pic_{vert} ⟶ 0
↓ ↓ ↓
0 ⟶ k(V̄)^{×}/k̄^{×} ⟶ Div V̄ ⟶ Pic V̄ ⟶ 0
α↓ ^{β}↓ ^{γ}↓
0 ⟶ k(V̄_{η})^{×}/k̄(t) ⟶ Div V̄_{η} ⟶ Pic V̄_{η} ⟶ 0

where the top row consists simply of the kernels of the maps α, β, γ; and the exactness of the top row comes from the Snake Lemma and the surjectivity of α. Taking cohomology of the top two rows gives us the diagram

H^{1}(k, Div_{vert}) ⟶ H^{1}(k, Pic_{vert}) ⟶ H^{2}(k, k̄(t)^{×}/k̄^{×}) ⟶ H^{2}(k, Div_{vert})
↓ ↓ ↓ ↓
0  ⟶ H^{1}(k, Pic V̄) ⟶ H^{2}(k, k(V̄)^{×}/k̄^{×}) ⟶ H^{2}(k, Div V̄)

with exact rows. Now Div_{vert} is a direct summand of Div V̄, so therefore H^{1}(k, Div_{vert}) = 0 and H^{2}(k, Div_{vert}) injects into H^{2}(k, Div V̄). The result then follows from the form (4.10) of B.

As was the case with H_{Az}^{2}(V), we may calculate H^{1}(k, Pic_{vert}) using the 48 lines on V as generators. The difficulty with this approach is that it is not always straightforward to calculate the classes of the irreducible components of the reducible fibres in terms of the 48 lines.

4.4.1 Identifying B

We now turn to trying to write down the group B, defined in (4.4). The material in this section is well known, and indeed is quoted by Swinnerton-Dyer in [44] to write down Azumaya algebras on V. Following Section 1 of [13], we will write elements of Br k(t) in a unique way; for this we need some notation.

Let P be any closed point of ℙ_{k}^{1}; then, over the normal closure of the residue field k(P), P splits into geometric points P_{1}, …, P_{n}, where n is the degree of the extension k(P)/k. We fix P_{1} and choose t_{P} ∈ k(P) such that P_{1} is the point given by t = t_{P}; then the original point P is defined over k by the equation N_{k(P)/k}(t − t_{P}) = 0. Here we abuse notation slightly by writing N_{k(P)/k} instead of N_{k(P)(t)/k(t)}. If now c_{P} is an element of H^{1}(k(P), ℚ/ℤ) and f an element of k(P)(t)^{×}, denote the element f ∪ dc_{P} of Br k(P)(t) by (c_{P} , f). In this manner we obtain an element of Br k(t) by the formula cores_{k(P)/k}(c_{P} , (t − t_{P})). The result we now quote is this:

Lemma 4.22. Let A be a central simple algebra over k(t). Then A is equivalent to an algebra of the form

∑_{P} n_{P} cores_{k(P)/k}(c_{P} , (t − t_{P})) + a  (4.11)

where the sum is over finitely many distinct closed points P of ℙ_{k}^{1}, the n_{P} are integers, c_{P} is a non-trivial element of H^{1}(k(P), ℚ/ℤ), t_{P} is as above and a is a constant algebra, that is, an element of Br k.

Proof. See [13].

The problem in identifying B is to compute the image of an algebra (4.11) in H^{2}(k, Div V̄). The constant algebra a does not interest us. Now the cup product commutes with the map k(P)(t) → Div V̄, so we must find the value of

cores_{k(P)/k}((π^{∗}P_{1} − π^{∗}∞) ∪ dc_{P}) ∈ H^{2}(k, Div V̄)  (4.12)

where ∞ is the point at infinity on ℙ_{k(P)}^{1}.

Let Div_{P} and Div_{∞} denote the subgroups of Div V̄ generated by the irreducible components of the geometric fibres above P and ∞, respectively. We assume that coordinates have been chosen such that the fibre at infinity is geometrically irreducible. Both Div_{P} and Div_{∞} are direct summands of Div V̄, so we may treat the two terms of (4.12), in π^{∗}P_{1} and in π^{∗}∞, separately.

First we consider the fibre at infinity. We can treat this fibre easily, for the point ∞ is defined over k. So Div_{∞} is simply a copy of ℤ, and π^{∗}∞ is the element 1 in this copy of ℤ. The restriction map

res : H^{0}(k, Div_{∞}) → H^{0}(k(P), Div_{∞})

is the identity, and we therefore have

cores_{k(P)/k}(π^{∗}∞ ∪ dc_{P}) = cores_{k(P)/k}(res(1) ∪ dc_{P}) = 1 ∪ cores_{k(P)/k} dc_{P}.

This gives the following lemma:

Lemma 4.23. The algebra (4.11) maps to 0 in H^{2}(k, Div_{∞}) if and only if

∑_{P} n_{P} cores_{k(P)/k} c_{P} = 0.

Now we consider the term cores_{k(P)/k}(π^{∗}P_{1} ∪ dc_{P}). It turns out that the corestriction here generates no problems: for, if Div_{P_{1}} denotes the subgroup of Div V̄ generated by the irreducible components of the fibre over P_{1}, then Div_{P} as a module is induced from Div_{P_{1}}. Formally, let G be the absolute Galois group of k, and H the subgroup of G which is the absolute Galois group of k(P). Then we define the induced module

Ind_{G}^{H} Div_{P_{1}} = Hom_{H}(ℤ[G], Div_{P_{1}})

with the action of G given by gϕ(x) = ϕ(xg). Then we have the following lemma:

Lemma 4.24. Let g_{1}, …, g_{n} be a set of left coset representatives of H in G. Then the map ϕ ↦ ∑ g_{i}ϕ(g_{i}^{−1}) is an isomorphism of G-modules from Ind_{G}^{H} Div_{P_{1}} to Div_{P}.

Proof. The map is injective: for the image of ϕ lies in Div_{P_{1}}, and so the n elements g_{i}ϕ(g_{i}^{−1}) each lie above a different one of the P_{i}; if their sum is zero, it therefore follows that each ϕ(g_{i}^{−1}) is zero; and as ϕ is determined by its values on a single element of each coset, so ϕ is zero. The map is surjective: given an irreducible divisor D in D_{P}, choose j such that D lies above g_{j}P_{1}; then define ϕ such that g_{j}ϕ(g_{j}^{−1}) = D, and ϕ is zero on the other cosets of H in G; then ϕ clearly maps to D. Finally, the map commutes with the action of G: first note that, as ϕ commutes with elements of H, our map is independent of the choice of coset representatives g_{i}; then the image of gϕ is

∑ g_{i}(gϕ)(g_{i}^{−1}) = g ∑(g^{−1}g_{i})ϕ(g_{i}^{−1}g)

but the g^{−1}g_{i} are another set of coset representatives; the result follows.

Corollary 4.25. The corestriction map

cores_{k(P)/k} : H^{2}(k(P), Div_{P_{1}}) → H^{2}(k, Div_{P})

is an isomorphism.

Proof. This is essentially Shapiro’s Lemma (see [27], Chapter II, Proposition 1.11).

Corollary 4.26. Suppose that the algebra (4.11) is Azumaya when lifted to V. Then the geometric components P_{i} of each P all lie below geometrically reducible fibres of π.

Proof. Suppose without loss of generality that the fibre above P_{1} is geometrically irreducible (and therefore so are those above the other P_{i}). Then Div_{P_{1}} is simply a copy of ℤ, acted on trivially by Gal(k̄/k(P)); and π^{∗}P_{1} is 1 in this copy of ℤ; so the cup-product map

c_{P} ↦ π^{∗}P_{1} ∪ dc_{P}

is an isomorphism. If the algebra maps to 0 in H^{2}(k, Div V̄), then it must map to 0 in each H^{2}(k, Div_{P}), hence in H^{2}(k(P), Div_{P_{1}}) by Corollary 4.4.1. So by the isomorphism above, c_{P} = 0.

Suppose now that the fibre over P_{1} is geometrically reducible. It may already split, over k(P), into several components which we will call D, D′, D′′ and so on. The geometrically irreducible components of D generate a subgroup Div_{D} of Div_{P_{1}}, as similarly do those of D′, …; an element of H^{2}(k(P), Div_{P_{1}}) is zero if and only if it gives zero in each of Div_{D}, Div_{D′}, … since Div_{P_{1}} is the direct sum of these components. We now consider Div_{D} in more detail.

Lemma 4.27. Let D be a divisor on V, defined over a field l; write Div_{D} for the subgroup of Div V̄ generated by the geometrically irreducible components of D. Let D̄ be one of those geometrically irreducible components, and let l′ be the least extension of l over which D splits completely into its irreducible components; that is, l′ is the least field of definition of D̄. Let c be an element of H^{1}(l, ℚ/ℤ). Then D∪dc is zero in H^{2}(l, Div_{D}) if and only if the restriction of c to H^{1}(l′, ℚ/ℤ) is trivial.

Proof. This is the same game as the one we played to show that we need only consider one of the P_{i} above. The Galois theory here is described in [19]: briefly, the group Gal(l′/l) acts faithfully and transitively on the geometrically irreducible components of D. This means that we can apply the same proof as in Lemma 4.24 to show that Div_{D} is isomorphic to the Gal(l̄/l)-module induced from the Gal(l̄/l′)-module ℤD̄. The corestriction map

cores_{l′/l} : H^{2}(l′, ℤD̄) → H^{2}(l, Div_{D})

is therefore an isomorphism (Shapiro’s Lemma).

Now we can write D as N_{l′/l}D̄, and then by standard properties of the cup-product

D ∪ dc = cores_{l′/l}(D̄ ∪ d(res_{l′/l} c))

which, since the corestriction map is an isomorphism on H^{2}(l′, ℤD̄), is zero if and only if res_{l′/l} c is trivial.

We can now put all the results in this section together to get a useful description of the group B:

Proposition 4.28. Let A be an element of Br k(t), written in the form shown in (4.11). For each point P, pick a closed point P_{1}, defined over k(P), lying above P. For each P, the fibre π^{∗}P_{1} splits over k(P) into components; let D̄ run through one geometrically irreducible component of each of these components. Then the lift of A to V is Azumaya if and only if:

1. ∑_{P} n_{P} cores_{k(P)/k} c_{P} = 0;

2. the restriction of c to the least field of definition of each D̄ is trivial.

Proof. Combine Lemmas 4.23 and 4.27.

4.5 An example

The results of Sections 4.2 and 4.3 can be adapted to give corresponding results about the vertical Brauer group, the group B and its relation to H^{1}(k, Pic_{vert}). Rather than repeating the details of those results here, we instead illustrate their applications by working through the calculation of B given in Section 6 of [44]; we then compute the elements of H^{1}(k, Pic_{vert}) corresponding to the algebras exhibited there, and show that those algebras all lift to equivalent Azumaya algebras on V.

4.5.1 Definitions

We work on the diagonal quartic surface V, defined by the equation (1.1), where the product a_{0}a_{1}a_{2}a_{3} of the four coefficients is a square. We also assume that no other relation holds between the coefficients; in other words, the surface falls into case A1 of the classification in Appendix A.

The fibration π : V → ℙ^{1} is given explicitly in [44]; the only piece of information we will need is the description of the reducible fibres of π. It turns out that the coordinates of the points of ℙ^{1} lying below the six geometrically reducible fibres fall into conjugate pairs. We will call these coordinates t_{1}, t′_{1}, t_{2}, t′_{2}, t_{3}, t′_{3}. Writing u_{1} = \sqrt{−a_{2}a_{3}}, u_{2} = \sqrt{−a_{1}a_{3}}, u_{3} = \sqrt{−a_{1}a_{2}}, we have each conjugate pair t_{i}, t′_{i} lying in ℚ(u_{i}). The fibres above each point all consist of straight lines, as follows:

Point Fibre
t_{1}  L_{11}^{123} + L_{15}^{123} + L_{51}^{123} + L_{55}^{123}
t′_{1}  L_{33}^{123} + L_{37}^{123} + L_{73}^{123} + L_{77}^{123}
t_{2}  L_{11}^{231} + L_{15}^{231} + L_{51}^{231} + L_{55}^{231}
t′_{2}  L_{33}^{231} + L_{37}^{231} + L_{73}^{231} + L_{77}^{231}
t_{3}  L_{11}^{312} + L_{15}^{312} + L_{51}^{312} + L_{55}^{312}
t′_{3}  L_{33}^{312} + L_{37}^{312} + L_{73}^{312} + L_{77}^{312}

It is easily checked that each of these fibres is indeed defined over the appropriate ℚ(t_{i}) = ℚ(u_{i}), and that the automorphism of ℚ(u_{i})/ℚ interchanges the fibres above t_{i} and t′_{i}.

We also need to know the field of definition of one, hence all, of the geometrically irreducible components of each fibre. As an illustration, L_{11}^{123} is defined by the equations

X_{0} = \sqrt[4]{−a_{1}/a_{0}}X_{1}, X_{2} = \sqrt[4]{−a_{3}/a_{2}}X_{3}.

Now, as a_{0}a_{1}a_{2}a_{3} is a rational square, we have

ℚ(u_{1}) = ℚ(\sqrt{−a_{2}a_{3}}) = ℚ(\sqrt{−a_{1}/a_{0}}) = ℚ(\sqrt{−a_{3}/a_{2}})

and so the field of definition of L_{11}^{123} is given by

K_{1} = ℚ(u_{1}, \sqrt[4]{−a_{1}/a_{0}}, \sqrt[4]{−a_{3}/a_{2}}) = ℚ(u_{1}, \sqrt{−a_{1}u_{1}θ}, \sqrt{a_{2}u_{1}})

where θ is the (rational) square root of a_{0}a_{1}a_{2}a_{3}. Thus K_{1} is a biquadratic extension of ℚ(u_{1}). Being deliberately vague about the choice of roots here does not matter: it merely gives another component of the same fibre. Replacing u_{1} with −u_{1} takes us to the conjugate fibre, that over t′_{1}. Permuting the indices cyclically gives the definitions of K_{2} and K_{3}, the least fields of definition for the geometrically irreducible components of the fibres above t_{2} and t_{3} respectively.

4.5.2 Computing B

The computation of the subgroup B of Br ℚ(t) is carried out in detail in [44]; we reproduce it here to illustrate the use of Proposition 4.28. Knowing that, in the case under consideration, H^{1}(ℚ, Pic V̄) is of order 2, we look first for quaternion algebras; it will turn out that these are sufficient to list all elements of B.

Suppose that an algebra of the form (4.11) lies in B. We suppose that each c_{P} is a quadratic character on k(P) = ℚ(u_{i}); let the corresponding elements of ℚ(u_{i})^{×}/(ℚ(u_{i})^{×})^{2} be called α_{i}. The first condition of Proposition 4.28 then says that the product

∏ N_{ℚ(u_{i})/ℚ}α_{i}^{n_{i}}  (4.13)

must be a square in ℚ; and the second condition says that each α_{i} must be a square in K_{i}. Now (K_{i}^{×})^{2}/(ℚ(u_{i})^{×})^{2} is a 4-group, whose non-trivial elements are a_{j}u_{i}, −a_{i}u_{i}θ and their product a_{i}a_{k}θ, where i, j, k is a cyclic permutation of 1, 2, 3. Choosing α_{i} from these possibilities, we find that the only solutions making (4.13) a square are

N_{ℚ(u_{i})/ℚ}a_{i}a_{k}θ = (a_{i}a_{k}θ)^{2} ∈ (ℚ^{×})^{2} (i = 1, 2, 3)

and

∏_{i=1}^{3} N_{ℚ(u_{i})/ℚ}(−a_{i}u_{i}θ) = ∏(a_{i}^{2}θ^{2}a_{j}a_{k}) ∈ (ℚ^{×})^{2}

and therefore B is of order 16, generated by the four elements

cores_{ℚ(u_{i})/ℚ}(a_{i}a_{k}θ, t − t_{i}) (i = 1, 2, 3)  (4.14)

and

∑_{i=1}^{3} cores_{ℚ(u_{i})/ℚ}(−a_{i}u_{i}θ, t − t_{i}).  (4.15)

4.5.3 The map to H^{1}(ℚ, Pic_{vert})

The subgroup Pic_{vert} of Pic V̄ is that generated by the irreducible components of the fibres of π. Now each reducible fibre has four irreducible components, each of which is a straight line; each fibre is linearly equivalent to every other fibre, and by general theory about fibrations there are no non-trivial linear equivalences. Thus Pic_{vert} is of rank 19, generated by the class of a fibre together with the classes of three irreducible components of each reducible fibre. The Galois action is that induced from Pic V̄, and so we can use the methods of Section 3.6.1 to compute

\frac{H^{0}(G, Pic_{vert} ⊗ℚ/ℤ)}{im H^{0}(G, Pic_{vert} ⊗ℚ)} ≅ H^{1}(ℚ, Pic_{vert})  (4.16)

where again G is the Galois group over ℚ of the least field of definition of the 48 lines. The calculation shows that the group on the left is indeed generated by four elements of order 2, which we may lift to obtain

\frac{1}{2}(L_{11}^{123} + L_{33}^{123} + L_{55}^{123} + L_{77}^{123}),

\frac{1}{2}(L_{11}^{231} + L_{33}^{231} + L_{55}^{231} + L_{77}^{231}),  (4.17)

\frac{1}{2}(L_{11}^{312} + L_{33}^{312} + L_{55}^{312} + L_{77}^{312})

and

\frac{1}{2}(L_{11}^{123} + L_{15}^{123} + L_{33}^{123} + L_{37}^{123}

+ L_{11}^{231} + L_{15}^{231} + L_{33}^{231} + L_{37}^{231}
+ L_{11}^{312} + L_{15}^{312} + L_{33}^{312} + L_{37}^{312}). (4.18)

Now we show how these generators could instead be obtained by the map in Proposition 4.21. Note that the map in (4.16) is defined in precisely the same way as usual if we use the form (4.4) of H^{1}(ℚ, Pic_{vert}); so we may forget about the components at infinity of the algebras (4.14) and (4.15), as described in Corollary 4.13.

First consider the algebra (4.14) with i = 1. This algebra is easy because a_{1}a_{3}θ is already rational; so

cores_{ℚ(u_{1})/ℚ}(a_{1}a_{3}θ, t − t_{1}) = (a_{1}a_{3}θ, N_{ℚ(u_{1})/ℚ}(t − t_{1})).

By the appropriate variant of Corollary 4.20, we must write the fibres above t_{1} and t′_{1} as norms from ℚ(\sqrt{a_{1}a_{3}θ}). A short calculation reveals that

π^{∗}t_{1} + π^{∗}t′_{1} = N_{ℚ(\sqrt{a_{1}a_{3}θ})/ℚ}(L_{11}^{123} + L_{33}^{123} + L_{55}^{123} + L_{77}^{123})

which by Corollary 4.20 and the isomorphism (4.16) corresponds to the first of the generators (4.17). Similarly the i = 2 and i = 3 cases of (4.14) give the other two generators of (4.17).

For the fourth generator (4.15) of B, we must write each divisor π^{∗}t_{i} as a norm from ℚ(u_{i}, \sqrt{−a_{i}u_{i}θ}) to ℚ(u_{i}). Again, calculation shows that the appropriate form is

π^{∗}t_{1} = N_{ℚ(u_{1},\sqrt{−a_{1}u_{1}θ})/ℚ(u_{1})}(L_{11}^{123} + L_{15}^{123})

and similarly for the other two components. The map between B and H^{1}(ℚ, Pic_{vert}) commutes with corestriction, as does the isomorphism (4.16), so the element we obtain from the algebra (4.15) is

N_{ℚ(u_{1})/ℚ}(L_{11}^{123} + L_{15}^{123}) + N_{ℚ(u_{2})/ℚ}(L_{11}^{231} + L_{15}^{231}) + N_{ℚ(u_{3})/ℚ}(L_{11}^{312} + L_{15}^{312})

which does indeed give the fourth generator (4.18) as desired.

4.5.4 Lifting to Br V

The calculations in the previous section agree with the results obtained in [44]: that B is generated by four elements of order 2. However, we now show that those four generators all map to the same element of Br V. That is, these four distinct elements of Br ℚ(t) give equivalent Azumaya algebras when lifted to V.

In light of Proposition 4.21, all we need to show is that the four generators of H^{1}(ℚ, Pic_{vert}) given in (4.17) and (4.18) all map to the same non-trivial element of H^{1}(ℚ, Pic V̄). This map is the same as the map

\frac{H^{0}(G, Pic_{vert} ⊗ℚ/ℤ)}{im H^{0}(G, Pic_{vert} ⊗ℚ)} → \frac{H^{0}(G, Pic V̄ ⊗ ℚ/ℤ)}{im H^{0}(G, Pic V̄ ⊗ ℚ)}

whose kernel is the image, in the object on the left, of H^{0}(G, Pic V̄ ⊗ ℚ). An easy computation shows that the differences of the four generators of H^{1}(ℚ, Pic_{vert}) do indeed lie in this kernel.

4.5.5 Other examples

The computations carried out for this example are easy to repeat with other cases and other fibrations; for example, the fibrations of Section 2.3.3 all have Pic_{vert} groups which are straightforward to compute. It is not always the case that the map B → Br_{1} V/ Br_{0} V is surjective, as in the example above. For instance, in the case where each quotient a_{i}/a_{j} is a square (case A3 in the tables of Appendix A), the group Br_{1} V/ Br_{0} V is isomorphic to (ℤ/2ℤ)^{5}. The fibration used in the previous example is defined, and the associated group B this time is isomorphic to (ℤ/2ℤ)^{6}; but its image in Br_{1} V/ Br_{0} V is only isomorphic to (ℤ/2ℤ)^{3}.

Numerical methods and examples

The material in the previous chapters has been somewhat removed from the original aim of studying the Brauer–Manin obstruction on diagonal quartic surfaces. This chapter contains some concrete examples, which show how the methods developed can be used to look at the Brauer–Manin obstruction. First we explain the algorithms used for two important peripheral procedures: testing local solubility of the equations, and searching for small solutions. Together, these give us the ability to look for surfaces which might have an interesting obstruction.

The rest of this chapter contains examples of calculating the Brauer– Manin obstruction for some diagonal quartic surfaces.

Note that most of the cases listed in Appendix A contain both locally soluble and locally insoluble surfaces. Some cases are always insoluble at some place: for example, case A3, where all the coefficients are squares, is insoluble in ℝ; and certain other cases are always insoluble in ℚ_{2}.

5.1 Local solubility

Testing whether an equation has solutions everywhere locally is the natural first step to determining whether there are any integer solutions. We will now show how this is accomplished. The main tool is Hensel’s Lemma, which we state in the following form. Lemma 5.1 (Hensel). Let f(X_{0}, …, X_{n}) be a polynomial in n+1 variables with integer coefficients. Fix a prime p, and suppose that there exist integers (x_{i}) and 0 ≤ j ≤ n such that 0 ≤ 2k < m, where

k = v_{p} (\frac{∂f}{∂X_{j}} (x_{0}, …, x_{n}))
m = v_{p}(f(x_{0}, …, x_{n})).

Then there exists α ∈ ℚ_{p} such that, replacing x_{j} by α, we have

f(x_{0}, …, α, …, x_{n}) = 0

and α is congruent to x_{j} modulo p^{m−k}. Proof. See Serre [36, Chapter 2, 2.2, Theorem 1].

In particular, Hensel’s Lemma shows this: let f̄ be the polynomial over 𝔽_{p} obtained by reducing f modulo p; then each smooth point of the variety over 𝔽_{p} defined by f̄ lifts to a point of the variety over ℚ_{p} defined by f.

To begin, we show that there are only finitely many primes at which local solubility of the diagonal quartic is not automatic. Lemma 5.2. Let p be a prime, not equal to 2 or 5 and not dividing a_{0}a_{1}a_{2}a_{3}. Then the equation (1.1) has solutions in ℚ_{p}. Proof. It is enough to check solubility in 𝔽_{p}: for the conditions on p imply that any solution in 𝔽_{p} must satisfy the conditions for Hensel’s Lemma to be applied, and hence lifts to a solution in ℚ_{p}.

It follows from the Weil conjectures (see [28, Chapter 26]) that the number N_{p} of points on our surface over 𝔽_{p} satisfies

|N_{p} − (p^{2} + 1)| ≤ 22p

where the 22 here is the second Betti number of the surface over ℂ. Therefore, for p ≥ 23, the surface always has points.

Next, suppose that p ≡ 3 (mod 4). Then squares in 𝔽_{p} are all fourth powers: for if a is a square, with square roots ±x, then the fact that −1 is not a square means that either x or −x must be a square, and hence a is a fourth power. Hence solving (1.1) is equivalent to solving the equation

a_{0}X_{0}^{2} + a_{1}X_{1}^{2} + a_{2}X_{2}^{2} + a_{3}X_{3}^{2} = 0

where the coefficients are all non-zero in 𝔽_{p}; but this has solutions, by the Chevalley–Warning theorem (see [36, Chapter I, 2.2, Corollary 2]).

This leaves the primes 13 and 17. For each of these, a short computer search shows that all diagonal quartics with non-zero coefficients have a solution.

Next, we show how to test local solubility at one given prime p, which may divide a_{0}a_{1}a_{2}a_{3}. Continuity shows that near every smooth point of V (ℚ_{p}) is an integer point which satisfies the hypothesis of Hensel’s Lemma. A naïve approach to testing local solubility might work thus: search for all solutions to (1.1) in 𝔽_{p}. If a solution is found which fulfils the conditions for applying Hensel’s Lemma, we are done; otherwise search through the possible lifts of each solution to ℤ/p^{2}ℤ, and so on. This is essentially the approach we will use, though in a slightly different form. Our method is the same as that used by Pinch for his calculations [31].

We may assume that the coefficients a_{i} are coprime integers, and that no a_{i} is divisible by a fourth power. If we multiply each a_{i} by p and then remove any factors of p^{4} which might have appeared, we obtain a new equation which clearly has rational solutions if and only if the original one has; but the new equation has a different reduction modulo p. Repeating this process gives four polynomials over 𝔽_{p} which are all “valid reductions modulo p” of the original variety. We call these polynomials f̄_{j}, where j takes values from 0 to 3. They might be described thus:

f̄_{j}(X_{0}, X_{1}, X_{2}, X_{3}) = b_{0j}X_{0}^{4} + b_{1j}X_{1}^{4} + b_{2j}X_{2}^{4} + b_{3j}X_{3}^{4}

where

b_{ij} ≡ \begin{cases}p^{−v_{p}(a_{i})}a_{i} & if v_{p}(a_{i}) ≡ j (mod 4) \\ 0 & otherwise. \end{cases}

Proposition 5.3. Let p be an odd prime. The equation (1.1) has a solution in ℚ_{p} if and only if there exists a smooth point on the variety over 𝔽_{p} defined by one of the polynomials f̄_{j}.

Proof. If there is such a smooth point, we can apply Hensel’s Lemma to lift it to a point in ℚ_{p}.

Conversely, suppose that there is a solution P in ℙ^{3}(ℚ_{p}). By continuity, we can find integers (x_{0}, x_{1}, x_{2}, x_{3}) such that

f(x_{0}, x_{1}, x_{2}, x_{3}) = a_{0}x_{0}^{4} + a_{1}x_{1}^{4} + a_{2}x_{2}^{4} + a_{3}x_{3}^{4}

has arbitrarily high p-adic valuation. Now choose i_{0} such that v_{p}(4a_{i_{0}}x_{i_{0}}^{3}) is minimal; we may in addition ensure that v_{p}(x_{i_{0}}) = 0, as no a_{i} has valuation higher than 3 by our earlier assumptions. Write j for v_{p}(a_{i_{0}}). We now define new coefficients a′_{i} by

a′_{i} = \begin{cases}p^{−j}a_{i} & if v_{p}(a_{i}) ≥ j \\ p^{4−j}a_{i} & if v_{p}(a_{i}) < j \end{cases}

and new values x′_{i} by

x′_{i} = \begin{cases}x_{i} & if v_{p}(a_{i}) ≥ j \\ x_{i}/p & if v_{p}(a_{i}) < j \end{cases}

where the expression x_{i}/p is an integer, because otherwise we would have chosen i_{0} differently. The polynomial

f′(X_{0}, X_{1}, X_{2}, X_{3}) = a′_{0}X_{0}^{4} + a′_{1}X_{1}^{4} + a′_{2}X_{2}^{4} + a′_{3}X_{3}^{4}

reduces modulo p to f̄_{j}. The new point (x′_{0}, x′_{1}, x′_{2}, x′_{3}) is a solution to f̄_{j} provided that we chose our original integer solution close enough to the padic solution; and moreover 4a′_{i_{0}}x′_{i_{0}} is a unit in 𝔽_{p}, so the point is smooth.

For p = 2, small changes are needed. To apply Hensel’s Lemma, we need a solution in the ring ℤ/32ℤ with not all the a_{i}x_{i} divisible by 2. The same argument works, though, and we obtain

Proposition 5.4. The equation (1.1) has a solution in ℚ_{2} if and only if one of the polynomials f̄_{j}, considered over the ring ℤ/32ℤ, has a solution (x_{0}, x_{1}, x_{2}, x_{3}) where some a_{i}x_{i} is not divisible by 2.

5.2 Finding small solutions

Searching for small solutions of the equation (1.1), with given coefficients a_{i}, is not an entirely straightforward procedure and merits some discussion.

We will consider only those cases where two coefficients are positive and the other two negative; this does not appear to be a great restriction, as a wide variety of cases are still available for experimentation. Anyway, the modifications needed to deal with other cases are not substantial. The equation can then be written as

ax^{4} + by^{4} = cu^{4} + dv^{4}  (5.1)

where a, b, c and d are coprime positive integers, free of fourth powers; and we seek solutions with x, y, u and v coprime and positive.

The “small” here means that we seek points of height less than a given bound; the height used is the standard height on ℙ^{3}(ℚ), so that

H(x : y : u : v) = max{x, y, u, v}

where x, y, u, v are coprime integers.

There is a reasonably straightforward way to find solutions to (5.1) of height less than or equal to H, which works as follows.

1. For each pair (x, y) with 0 ≤ x, y ≤ H, put the triple (x, y, ax^{4} + by^{4}) in a list.

2. For each pair (u, v) with 0 ≤ u, v ≤ H, put the triple (u, v, cu^{4} + dv^{4}) in a second list.

3. Sort both lists on the third elements of the triples.

4. Merge the two lists: for each triple (x, y, A) from the first list and (u, v, B) from the second list such that A = B, print the solution (x, y, u, v).

This algorithm is moderately successful for small H, but rapidly runs out of space to store the lists. To work for H = 10^{4}, say, each of the two lists must contain 10^{8} elements, each of size 10 bytes or so; that is already a gigabyte of storage. This may be written to a file, but the sorting either requires that the entire file fit in memory, or else is extremely slow. The sorting can be avoided by using some data structure which sorts the data as it is inserted; but storage space is still a serious constraint. A more sophisticated algorithm is needed.

The answer is found in a paper by Bernstein [4]. Here an algorithm is given for enumerating the values of p(x) + q(y) in increasing order, where p and q are increasing functions of x and y respectively. The algorithm makes use of a heap, which is a data structure particularly suited to the operation needed, namely insertion of a new element whilst removing the smallest element. Briefly, the algorithm works roughly as follows. The heap contains triples (x, y, p(x) + q(y)), and there is always precisely one element in the heap for each value of y from 0 to H. On each iteration, the triple with the smallest value of p(x) + q(y) is removed, and the new triple inserted into the heap is (x + 1, y, p(x + 1) + q(y)). When x reaches H, no new triple is inserted.

The storage required for this algorithm is only of the order of H, compared with H^{2} for the previous naïve method. Applying it to diagonal quartic surfaces with small integral coefficients allows us to search for solutions of height up to about 10^{4}. A barrier is reached, though, when the quantities ax^{4} + by^{4} can no longer be stored in 64 bits: for that is the largest size of integer which most desktop computers can use in arithmetic operations.

There are several ways round this problem. One is to perform all operations modulo some prime which is slightly less than 2^{64}, and then find out which of these solutions come from integer solutions. Another is to work modulo two or more primes, whose product is greater than the largest number which will arise. Finally, one can use a multiple-precision integer arithmetic library: this is the solution used to produce the results in this chapter. The sortedsums package written by Bernstein was modified to use the PARI library. Using this algorithm on a Pentium III running at 800 MHz, it takes less than four seconds to list all solutions of height less than 1000 to a given diagonal quartic equation of the form (5.1). Solutions of height less than 10^{4} can be listed in about ten minutes.

It would be possible to speed up this algorithm by a constant factor by using various local constraints. In particular, fourth powers can only be congruent to 0 or 1 modulo 16, so taking advantage of this would be an easy way to gain some speed. Similarly one can use the fact that fourth powers are 0 or 1 modulo 5.

Appendix B contains a list of all the diagonal quartic surfaces of the form (5.1), with integral coefficients between 1 and 15, which are everywhere locally soluble but have no points of height less than 10^{4}. This list was produced using the algorithms described above, and will be useful to provide examples for more detailed study.

5.3 Example with a fibration

In this example, we look at the Brauer–Manin obstruction on the particular surface V defined by

X_{0}^{4} + X_{1}^{4} = 6X_{2}^{4} + 12X_{3}^{4}.  (5.2)

This surface appears in the list of Appendix B, so has no rational points of height less than 10^{4}. We will describe an element of the unique non-trivial class of the Brauer group of V, and show that the resulting Brauer–Manin obstruction is non-trivial. This proves that the equation (5.2) has no rational solutions.

The surface (5.2) falls into case A121 of the classification listed in Appendix A. Using the methods of Chapter 3, we determine that the Picard group over ℚ is of rank 2. There is an elliptic fibration on this surface defined over the rational numbers: two singular fibres of the fibration are

F = L_{11}^{123} + L_{15}^{123} + L_{31}^{123} + L_{35}^{123}
F′ = L_{53}^{123} + L_{57}^{123} + L_{73}^{123} + L_{77}^{123}.

We can easily write down defining equations for F:

F : X_{0}^{2} − \sqrt{−2}X_{0}X_{1} − X_{1}^{2} = 0, X_{2}^{2} − \sqrt{−2}X_{3}^{2} = 0.

The equations for F′ are the conjugates over ℚ of those for F. Note that, although the fibration is defined over ℚ, these two singular fibres are not: they are defined over K = ℚ(\sqrt{−2}). In fact, we do not at this stage know the equation for any rational fibre; neither do we have any explicit description of a map from V to ℙ^{1} defined over ℚ. What we can do is write down a function with a zero along F and a pole along F′:

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}} .

Here the numerator of the rational function is zero along F and on the remaining L_{1n}^{123} and L_{3n}^{123}. The denominator is zero on F′ and also along those same straight lines; thus the divisor of f is as desired. The morphism π = (1 : f) defines a map from V to ℙ^{1}, albeit only over K.

Proposition 4.28 states that we can write down elements of the Brauer group of the surface by using these singular fibres. The group Br_{1} V/ Br_{0} V has order 2, so any Azumaya algebra we can find which is not equivalent to a constant algebra lies in the non-trivial class. To use Proposition 4.28, though, the fibre at infinity must be non-singular and defined over ℚ; our map π has fibre at infinity equal to F′, which is neither. We must find a fibre of π which is defined over ℚ, which after a change of coordinates on ℙ^{1} will be our new fibre at infinity.

A fibre is defined over ℚ if and only if it is fixed by the Galois action on V. To find such a fibre, we write the equation for a general fibre

f(X_{0}, X_{1}, X_{2}, X_{3}) = c

and seek a value of c such that {f^{σ} = σc} defines the same variety as {f = c}, where σ is the non-trivial element of Gal(K/ℚ). Now note that f^{σ} has divisor F′ − F, and therefore N_{K/ℚ}f = f^{σ}f is a constant. In fact we have

f^{σ}f = \frac{X_{0}^{4} + X_{1}^{4}}{X_{2}^{4} + 2X_{3}^{4}} = 6.

Now, suppose that the two varieties {f = c} and {f^{σ} = σc} are the same; then, multiplying the two equations, we have

N_{K/ℚ}c = N_{K/ℚ}f = 6

and, conversely, if N_{K/ℚ}c = N_{K/ℚ}f then f = c if and only if f^{σ} = σc, that is, the fibre over c is fixed by the Galois action and is therefore defined over ℚ.

A solution is

c = 2 − \sqrt{−2}.

The function f/(f − c) therefore has a zero at the singular fibre F, and a pole at some fibre of π which is defined over the rationals. Therefore this function can play the rôle of the function π^{∗}(t − t_{P}) of Proposition 4.28.

Were we interested in having an equation for a map from V to ℙ^{1} defined over ℚ, we would note that by Hilbert’s theorem 90 the general solution to N_{K/ℚ}(c) = 6 is

c = \frac{(2 − \sqrt{−2})(u − \sqrt{−2}v)}{u + \sqrt{−2}v}.

Setting f equal to this expression, we can derive the desired equation. In this situation, though, all we need are defining equations for one rational fibre.

According to Proposition 4.28, the other piece of information needed to write down an Azumaya algebra on V is a character on Gal(ℚ̄/K) which becomes trivial when restricted to the least field of definition of one of the geometrically irreducible components of F. That field is

L = K(\sqrt{−1}, \sqrt[4]{−2})

which is a biquadratic extension of K. The characters which satisfy the above requirement are the quadratic characters associated to −1 and ±\sqrt{−2}. To satisfy the first condition of Proposition 4.28 as well, we choose the character associated to −1. We now have a non-trivial Azumaya algebra of V: it is the algebra A = (−1, G), where

G = N_{K/ℚ} (\frac{f}{f − c})

= \frac{Nf}{(f − c)(f − c)^{σ}}

= \frac{Nf}{Nf + Nc − fc^{σ} − f^{σ}c}

= \frac{6}{12 − Tr(fc^{σ})}.

Using the identity Tr(a/b) = Tr(ab^{σ})/Nb, we get

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

= \frac{4X_{0}^{2}X_{2}^{2} + 4X_{0}^{2}X_{3}^{2} − 8X_{0}X_{1}X_{3}^{2} + 4X_{0}X_{1}X_{2}^{2} − 4X_{1}^{2}X_{2}^{2} − 4X_{1}^{2}X_{3}^{2}}{X_{2}^{4} + 2X_{3}^{4}}

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

and, substituting into the above expression for G,

G = \frac{6(X_{2}^{4} + 2X_{3}^{4})}{12(X_{2}^{4} + 2X_{3}^{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}}

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

We now calculate the Brauer–Manin obstruction associated to A. For each place v of ℚ, including the real place, Br(ℚ_{v})[2] is cyclic of order 2; the invariant inv_{v} is directly related [35, Chapter XIV] to the Hilbert symbol ( , )_{v}. For each adèlic point (x_{v}) of V, we must evaluate

(−1)^{2 ∑_{v} inv_{v} A(x_{v})} = ∏_{v}(−1, G(x_{v}))_{v}.

At each place, the function (−1, G(x_{v}))_{v} is continuous, hence locally constant, on V (ℚ_{v}). If we were to find a place where this function were not constant, there could be no Brauer–Manin obstruction: for by changing our adèlic point at that place, we could always make the sum of the invariants equal zero.

We note (see [10, Lemma 3.1.2]) that the complement of any union of divisors on V is dense in V, not only in the Zariski topology but also in the real or p-adic topology on each V (ℚ_{v}). Therefore we need not check the Brauer–Manin condition on the support of (G).

For primes p not equal to 2 or 3, both the surface V and the algebra A have good reduction. It follows (see [25, Chapter VI, §44]) that inv_{p} A(x_{p}) is zero on all of V (ℚ_{p}).

At p = 3, we note that −2 is a square in ℚ_{3}. The algebra A splits over K = ℚ(\sqrt{−2}), hence over ℚ_{3} which contains a copy of K. Thus the local invariant is constantly zero on V (ℚ_{3}).

At p = 2, looking at the equation (5.2) modulo 16 shows that any primitive solution must have all four X_{i} odd. To evaluate the obstruction, it turns out that we must look at the X_{i} modulo 32. Now, if X_{i} is known modulo 2^{5}, then X_{i}^{4} is known modulo 2^{7}. This means that the value of

P = X_{0}^{4} + X_{1}^{4} − 6X_{2}^{4} − 12X_{3}^{4}

is known modulo 2^{7}. Suppose that we know a solution to

P (X_{0}, X_{1}, X_{2}, X_{3}) ≡ 0 (mod 2^{7}).  (5.3)

We apply Hensel’s Lemma, considering P as a polynomial in the one variable X_{0}. The derivative is P′ = 4X_{0}^{3}, which has 2-adic valuation 2 since X_{0} is odd. The conclusion of Hensel’s Lemma states that our solution lifts to one in ℚ_{2} which is unique modulo

2^{v_{2}(P (X_{0},X_{1},X_{2},X_{3}))−v_{2}(P′(X_{0},X_{1},X_{2},X_{3}))} = 2^{5}

and therefore the primitive solutions to (5.3) are, modulo 2^{5}, equivalent to all primitive solutions to P in ℚ_{2}.

Using a computer, we list all the solutions modulo 32 to (5.3). Now, up to squares, G is equivalent 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}) (5.4)

and knowing the X_{i} modulo 32 gives the value of (5.4) modulo 2^{8}. We evaluate the expression (5.4) for each solution found, and discover that the value is always congruent to either 96 or 176 modulo 2^{8}. In either case, by the description of the Hilbert symbol in [36, Chapter III, 1.2, Theorem 1], we can evaluate (−1, G(x_{2}))_{2} = −1 for all points x_{v} of V (ℚ_{2}).

Over ℝ, we note that the function G has no zeros in V (ℝ): if so, we would have

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

and hence all the X_{i} equal to zero. The one pole of G lies on the curve of genus 1 which is the fibre defined by f = c, and is a double pole; so in a neighbourhood of any simple point of this curve, G takes either only positive or only negative values. We deduce that G is either everywhere positive or everywhere negative. Evaluation at the point (2 : 2 : 1 : \sqrt[4]{13/6}) reveals that G is everywhere positive. Hence (−1, G)_{ℝ} is everywhere equal to +1.

Combining these results, we obtain

∑_{v} inv_{v} A(x_{v}) = \frac{1}{2}

for all adèlic points (x_{v}) of V (𝔸_{ℚ}). Thus the equation (5.2) has no rational solutions.

5.4 Example with no fibration

Our second example is a surface which falls into the most general class, that is, case A222 in the table of Appendix A. From the list of Appendix B we choose the surface (2, 6, 7, 15), which we rearrange as

7X_{0}^{4} + 15X_{1}^{4} = 2X_{2}^{4} + 6X_{3}^{4}.  (5.5)

This surface has Picard group of rank 1, generated by the class Π of a plane section. The group Br_{1} V/ Br_{0} V is again of order 2, so any non-trivial Azumaya algebra will give the same obstruction.

Although the surface clearly has no elliptic fibration, we can still use the fibrations studied earlier to construct an Azumaya algebra. We will use Proposition 4.17. For this we must find a cyclic extension L/ℚ, a nonprincipal divisor D defined over L, and a function f ∈ k(V) such that (f) = N_{L/ℚ}D.

The case studied in detail in [44] shows that if we take L = ℚ(\sqrt{a_{0}a_{1}a_{2}a_{3}}), then the surface V defined by (5.5) has a pencil of curves of genus 1 defined over L. Explicit descriptions are given in [44] of equations defining this fibration. Let F be one fibre, defined over L; then we will construct a function f defined over ℚ such that

(f) = F + σF − 2Π = N_{L/ℚ}(F − Π)  (5.6)

to satisfy Proposition 4.17.

The surface (5.5) has a_{0}a_{1}a_{2}a_{3} = 6^{2} × 35, and so we take L = ℚ(\sqrt{35}). To write down equations for the fibration, the recipe given in [44] requires that we solve the equation

15r_{1}^{2} − 2r_{2}^{2} − 6r_{3}^{2} = 0  (5.7)

over L. If instead we were to use the approach of the previous example, we would take a singular fibre consisting of straight lines but not defined over L; we would use that to produce an equation for the fibration, not defined over L; and we would then try to turn that equation into one defined over L. In the previous example, that last step involved solving a simple norm equation. If we try that approach in this example, we must again solve a norm equation; that turns out to be equivalent to solving (5.7). We therefore stick to the recipe given in [44].

Solving the conic equation (5.7) is not entirely straightforward; a good discussion of how to approach such an equation is given in Simon’s thesis [40]. We first apply Legendre’s technique of descent to reduce the equation to

x^{2} − ωy^{2} = 3z^{2}

where ω = 6 + \sqrt{35} is a fundamental unit in L. Then a computer search reveals a solution, which gives

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

Putting these into Swinnerton–Dyer’s recipe and removing some spare factors gives polynomials

A = 3(\sqrt{35}X_{0}^{2} − (20 + 5\sqrt{35})X_{1}^{2} + (20 + 2\sqrt{35})X_{3}^{2})
B = (35 + 4\sqrt{35})X_{0}^{2} + 15X_{1}^{2} + (20 + 2\sqrt{35})X_{2}^{2}
C = (−35 − 4\sqrt{35})X_{0}^{2} − 15X_{1}^{2} + (20 + 2\sqrt{35})X_{2}^{2}
D = \sqrt{35}X_{0}^{2} − (20 + 5\sqrt{35})X_{1}^{2} − (20 + 2\sqrt{35})X_{3}^{2}

such that AD = BC gives the equation (5.5) of the original surface. The two fibrations π_{i} : V → ℙ^{1} are defined by

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

and the classes in Pic V_{L} of these two fibrations are conjugates over ℚ. If we write (A) for the divisor of zeros of the polynomial A, and so on, then

(A) = F_{1} + F_{2} (B) = F′_{1} + F_{2}
(C) = F_{1} + F′_{2} (D) = F′_{1} + F′_{2}

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

The goal now is to find a function f satisfying (5.6), where we take F = F_{1}. This requires that we know some more about σF_{1}. It is a fibre, defined over L, of π_{2}, and therefore defined by an equation of the form B/D = c for some constant c ∈ L. To find c, it suffices to evaluate B/D at any point of σF_{1}. 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

\frac{B}{D}(σF_{1}) = −6

We can now use this to write down a suitable function f:

\frac{C}{X_{0}^{2}} (\frac{B}{D} + 6) = \frac{A + 6C}{X_{0}^{2}}

which after removing a constant factor gives

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

To evaluate the obstruction, we may ignore the X_{0}^{2} in the denominator of f. At each place v of ℚ, we look at (35, g)_{v} where g = 7X_{0}^{2}+5X_{1}^{2}−4X_{2}^{2}−2X_{3}^{2}.

At p = 2, looking at the equation (5.5) modulo 16 shows that X_{2} must be even and the other X_{i} all odd. Evaluating g at each possible solution modulo 16 always gives a result congruent to 6 or 14 modulo 16, and hence (35, g)_{2} = −1 everywhere.

At p = 3, we list the solutions modulo 27 and find that the value of g at each point is either 6, 15 or 24 modulo 27; hence (35, g)_{3} = −1.

At p = 5, the situation is quite different. Using the formula in [36, Chapter III, 1.2, Theorem 1], we have

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

where b = 5^{β}v with v a 5-adic unit. Listing the solutions modulo 25 shows that the value of g is always a unit, but is sometimes a quadratic residue modulo 5 and sometimes not. For example, the point (2 : 1 : 1 : 5) satisfies Hensel’s Lemma, so lifts to a point of V (ℚ_{5}), and we have

g(2, 1, 1, 5) ≡ 4 (mod 25).

Now any element of ℚ_{5} congruent to 4 modulo 25 is a square, and so at that point (35, g)_{5} = 1. However, the point (2 : 2 : 7 : 5) also lifts to a point of V (ℚ_{5}); but there, we have

g(2, 2, 7, 5) ≡ 2 (mod 25)

which cannot be a square in ℚ_{5}, so (35, g)_{5} = −1 there.

At p = 7, a similar phenomenon occurs: (35, g)_{7} takes both values ±1 on V (ℚ_{7}). Therefore there is no arithmetic Brauer–Manin obstruction to the existence of rational points on the surface (5.5).

Given the number of surfaces we have found in this category with no small rational points, it seems reasonable to conjecture that the Brauer–Manin obstruction is not the only obstruction to the Hasse principle for diagonal quartic surfaces, at least when no elliptic fibration is defined.

5.5 Trivial Brauer group

The calculations leading to the tables of Appendix A reveal certain families of diagonal quartic surfaces with Brauer group as small as possible: that is, Br_{1} V/ Br_{0} V is trivial. For these surfaces, no Brauer–Manin obstruction can possibly come from the arithmetic part of the Brauer group; and so studying such a family of surfaces is an apparently attractive way of investigating whether there can be surfaces with trivial Brauer–Manin obstruction but no rational points.

One such class is case A127 of the table, where the equation of the surface looks like

X_{0}^{4} + aX_{1}^{4} = 4(aX_{2}^{4} + 2b^{2}X_{3}^{4})  (5.8)

for suitably general constants a and b. It is straightforward to produce a list of surfaces like this, to test local solubility and to search for solutions. Appendix C contains a list of such surfaces with no point of height less than 10^{4}.

Unfortunately, it is not possible to draw many conclusions from these results. For the coefficients of the surfaces quickly become moderately large (or at least not small). It seems reasonable to believe that a diagonal quartic surface with large coefficients might also have large solutions; therefore the non-existence of a point of height less that 10^{4} on a surface with even slightly large coefficients should not be taken as an indication that the surface is unlikely to have any points at all.

This rather vague idea is borne out by some numerical experimentation: surfaces with coefficients of the order of 15 appear far less likely to have a point of height less that 10^{4} than surfaces with coefficients all less that 4, say. This effect occurs partly because surfaces with small coefficients are more likely to fall into a narrow case of our classification and so have rational curves. However, even if we exclude points on the 48 lines, which account for much of that phenomenon, then surfaces with small coefficients still appear in general to have more small points than surfaces with large coefficients. We conclude that, if meaningful results were to be obtained when searching for points on surfaces such as those in Appendix C, then it would be necessary CHAPTER 5. NUMERICAL METHODS AND EXAMPLES 127 to search to a higher height than we have been able to.

Appendix A

Tables of cases

This appendix contains tables listing the 546 different cases into which the Galois action of Gal(k̄/k) on Pic V̄ may fall. The cases are split first by the base field k: each of the five tables corresponds to one of the possible fields k ∩ ℚ(ϵ). Within each table, the cases are split according to whether the product a_{0}a_{1}a_{2}a_{3} is a square, minus a square, twice a square, minus twice a square, or none of these. Where k contains square roots of −1 or 2, some of these cases merge.

The first column in each table contains the index number of the case: this has no meaning and comes simply from the order in which the cases were originally listed. The second column shows a generic example of the coefficients (a_{0}, a_{1}, a_{2}, a_{3}) for this case, as described in Section 3.5.2. The third column contains the rank of the group H^{0}(k, Pic V̄). The fourth column shows the structure, as a product of cyclic groups, of the group Br_{1} V/ Br_{0} V in each case. This is isomorphic to H^{1}(k, Pic V̄). The final column list all those immediate subcases of each case which appear in the same table.

A.1 k ∩ ℚ(ϵ) = ℚ

Number Example  ρ Br_{1}  Subcases
a_{0}a_{1}a_{2}a_{3} square

A1 [1, c_{1}, c_{2}, c_{3}^{2}c_{2}^{3}c_{1}^{3}] 2 [2] A2, A4, A6, A8,

A36, A37, A56, A84

A2 [1, c_{1}^{2}, c_{2}, c_{3}^{2}c_{2}^{3}] 2 [2, 2, 2] A3, A5, A7, A9, A11, A13, A15, A25, A38, A50, A57, A87

A3  [1, c_{1}^{2}, c_{2}^{2}, c_{3}^{2}]  2 [2, 2, 2, 2, 2] A16, A26, A60, A96
A4  [1, −4c_{1}^{2}, c_{2}, −4c_{3}^{2}c_{2}^{3}]  2 [2]  A5, A10, A12, A14,

A21, A31, A43, A53, A65, A99

A5  [1, c_{1}^{2}, −4c_{2}^{2}, −4c_{3}^{2}]  2 [2, 2, 2]  {A17, A22, A27, A32, \atop A68, A105}
A6  [1, 2c_{1}^{2}, c_{2}, 2c_{3}^{2}c_{2}^{3}]  2 [2]  {A7, A10, A48, A76, \atop A85}
A7  [1, c_{1}^{2}, 2c_{2}^{2}, 2c_{3}^{2}]  2 [2, 2, 2]  A18, A28, A77, A90
A8  [1, −8c_{1}^{2}, c_{2}, −8c_{3}^{2}c_{2}^{3}]  2 [2]  {A9, A10, A49, A79, \atop A86}
A9  [1, c_{1}^{2}, −8c_{2}^{2}, −8c_{3}^{2}]  2 [2, 2, 2]  A19, A29, A80, A93
A10  [1, −4c_{1}^{2}, 2c_{2}^{2}, −8c_{3}^{2}]  2 [2]  {A23, A33, A82, \atop A102}
A11  [1, c_{1}, c_{2}^{2}c_{1}^{3}, c_{1}^{2}]  2 [2, 2, 2]  A16, A17, A28, A29,

A41, A52, A58, A89

A12  [1, c_{1}, −4c_{2}^{2}c_{1}^{3}, −4c_{1}^{2}]  2 [2]  {A22, A33, A46, A55, \atop A66, A101}
A13  [1, c_{1}, c_{2}^{2}c_{1}^{3}, 4c_{1}^{2}]  2 [2, 2, 2]  A18, A19, A26, A27,

A42, A52, A59, A88

A14  [1, c_{1}, −4c_{2}^{2}c_{1}^{3}, −c_{1}^{2}]  2 [2, 2]  {A23, A32, A47, A55, \atop A67, A100}

Number Example  ρ Br_{1}  Subcases
A15  [1, c_{1}^{2}, c_{2}, c_{2}]  2 [2, 2, 2]  A16, A17, A18, A19,

A39, A51, A58, A88

A16  [1, c_{1}^{2}, c_{2}^{2}, 1]  2 [2, 2, 2, 2, 2] A20, A30, A61, A97
A17  [1, −4c_{1}^{2}, −4c_{2}^{2}, 1]  2 [2, 2, 2, 2] {A24, A34, A69, \atop A106}
A18  [1, 2c_{1}^{2}, 2c_{2}^{2}, 1]  2 [2, 2, 2]  A78, A91
A19  [1, −8c_{1}^{2}, −8c_{2}^{2}, 1]  2 [2, 2, 2]  A81, A94
A20  [1, c_{1}^{2}, 1, 1]  2 [2, 2, 2, 2, 2] A62, A98
A21  [1, −4c_{1}^{2}, c_{2}, −4c_{2}]  2 [2]  {A22, A23, A44, A54, \atop A66, A100}
A22  [1, c_{1}^{2}, −4c_{2}^{2}, −4]  2 [2, 2, 2]  {A24, A35, A70, \atop A107}
A23  [1, 2c_{1}^{2}, −8c_{2}^{2}, −4]  2 [2, 2]  A83, A103
A24  [1, c_{1}^{2}, −4, −4]  2 [4, 2, 2, 2] A71, A108
A25  [1, c_{1}^{2}, c_{2}, 4c_{2}]  3 [2, 2]  A26, A27, A28, A29,

A40, A51, A59, A89

A26  [1, c_{1}^{2}, c_{2}^{2}, 4]  3 [2, 2, 2, 2] A30, A63, A97
A27  [1, −4c_{1}^{2}, −4c_{2}^{2}, 4]  3 [2, 2]  A35, A72, A106
A28  [1, 2c_{1}^{2}, 2c_{2}^{2}, 4]  3 [2, 2]  A78, A92
A29  [1, −8c_{1}^{2}, −8c_{2}^{2}, 4]  3 [2, 2]  A81, A95
A30  [1, c_{1}^{2}, 1, 4]  4 [2, 2, 2]  A64, A98
A31  [1, −4c_{1}^{2}, c_{2}, −c_{2}]  3 [2]  {A32, A33, A45, A54, \atop A67, A101}
A32  [1, c_{1}^{2}, −4c_{2}^{2}, −1]  3 [2, 2, 2]  {A34, A35, A73, \atop A107}
A33  [1, 2c_{1}^{2}, −8c_{2}^{2}, −1]  3 [2]  A83, A104
A34  [1, −4c_{1}^{2}, 1, −1]  4 [2, 2, 2]  A74, A108
A35  [1, c_{1}^{2}, −4, −1]  4 [2, 2]  A75, A108
A36  [1, c_{1}, c_{2}, c_{2}^{3}c_{1}]  3 []  A38, A43, A48, A49,

A57, A65, A85, A86

A37 [1, c_{1}, c_{2}, 4c_{2}^{3}c_{1}] 2 [2] A48, A49, A50, A53, A76, A79, A87, A99

Number Example  ρ Br_{1}  Subcases
A38  [1, c_{1}, c_{2}^{2}c_{1}^{3}, c_{2}^{2}c_{1}^{2}]  4 [2]  A39, A40, A41, A42,

A60, A68, A90, A93

A39  [1, 1, c_{2}, c_{2}]  4 [2]  A61, A69, A91, A94
A40  [1, 4, c_{2}, 4c_{2}]  6 []  A63, A72, A92, A95
A41  [1, c_{1}^{2}, c_{2}^{2}c_{1}^{3}, c_{2}^{2}c_{1}]  4 [2]  A61, A69, A92, A95
A42  [1, c_{1}^{2}, 2c_{2}^{2}c_{1}^{3}, 2c_{2}^{2}c_{1}]  4 [2, 2]  A63, A72, A91, A94
A43  [1, c_{1}, −4c_{2}^{2}c_{1}^{3}, −4c_{2}^{2}c_{1}^{2}] 4 []  {A44, A45, A46, A47, \atop A68, A102}
A44  [1, −4, c_{2}, −4c_{2}]  4 []  A70, A103
A45  [1, −1, c_{2}, −c_{2}]  6 []  A73, A104
A46  [1, −4c_{1}^{2}, c_{2}^{2}c_{1}^{3}, −4c_{2}^{2}c_{1}] 4 []  A70, A104
A47  [1, −4c_{1}^{2}, 2c_{2}^{2}c_{1}^{3}, −8c_{2}^{2}c_{1}] 4 [2]  A73, A103
A48  [1, c_{1}, 2c_{2}^{2}c_{1}^{3}, 8c_{2}^{2}c_{1}^{2}]  3 []  {A77, A82, A90, \atop A102}
A49  [1, c_{1}, −8c_{2}^{2}c_{1}^{3}, −2c_{2}^{2}c_{1}^{2}] 3 []  {A80, A82, A93, \atop A102}
A50  [1, c_{1}, c_{2}^{2}c_{1}^{3}, 4c_{2}^{2}c_{1}^{2}]  2 [2, 2, 2]  {A51, A52, A77, A80, \atop A96, A105}
A51  [1, 1, c_{2}, 4c_{2}]  3 [2, 2]  {A78, A81, A97, \atop A106}
A52  [1, c_{1}^{2}, c_{2}^{2}c_{1}^{3}, 4c_{2}^{2}c_{1}]  2 [2, 2, 2]  {A78, A81, A97, \atop A106}
A53  [1, c_{1}, −4c_{2}^{2}c_{1}^{3}, −c_{2}^{2}c_{1}^{2}]  2 [2]  {A54, A55, A82, \atop A105}
A54  [1, −4, c_{2}, −c_{2}]  3 [2]  A83, A107
A55  [1, −4c_{1}^{2}, c_{2}^{2}c_{1}^{3}, −c_{2}^{2}c_{1}]  2 [2, 2]  A83, A107
A56  [1, c_{1}, c_{2}, c_{2}^{3}c_{1}^{3}]  2 [4]  A57, A65, A76, A79
A57  [1, c_{1}^{2}, c_{2}, c_{2}^{3}c_{1}^{2}]  3 [2, 2]  {A58, A59, A60, A68, \atop A77, A80}
A58  [1, 1, c_{2}, c_{2}^{3}]  3 [2, 2]  A61, A69, A78, A81
A59  [1, 4, c_{2}, 4c_{2}^{3}]  4 [2]  A63, A72, A78, A81
A60  [1, c_{1}^{2}, c_{2}^{2}, c_{2}^{2}c_{1}^{2}]  5 [2, 2]  A61, A63

Number Example  ρ Br_{1}  Subcases
A61  [1, 1, c_{2}^{2}, c_{2}^{2}]  5 [2, 2, 2]  A62, A64
A62  [1, 1, 1, 1]  5 [2, 2, 2, 2]
A63  [1, 4, c_{2}^{2}, 4c_{2}^{2}]  7 [2]  A64
A64  [1, 1, 4, 4]  9 []
A65  [1, −4c_{1}^{2}, c_{2}, −4c_{2}^{3}c_{1}^{2}]  3 [2]  A66, A67, A68, A82
A66  [1, −4, c_{2}, −4c_{2}^{3}]  3 [2]  A70, A83
A67  [1, −1, c_{2}, −c_{2}^{3}]  4 [2]  A73, A83
A68  [1, c_{1}^{2}, −4c_{2}^{2}, −4c_{2}^{2}c_{1}^{2}]  5 [2]  A69, A70, A72, A73
A69  [1, 1, −4c_{2}^{2}, −4c_{2}^{2}]  5 [2, 2]  A71, A74
A70  [1, −4, c_{2}^{2}, −4c_{2}^{2}]  5 [2]  A71, A75
A71  [1, 1, −4, −4]  5 [4, 2]
A72  [1, 4, −4c_{2}^{2}, −c_{2}^{2}]  7 []  A75
A73  [1, −1, c_{2}^{2}, −c_{2}^{2}]  7 [2]  A74, A75
A74  [1, 1, −1, −1]  9 [2]
A75  [1, −4, 4, −1]  9 []
A76  [1, 2c_{1}^{2}, c_{2}, 8c_{2}^{3}c_{1}^{2}]  2 [8]  A77, A82
A77  [1, c_{1}^{2}, 2c_{2}^{2}, 8c_{2}^{2}c_{1}^{2}]  3 [4, 2]  A78
A78  [1, 1, 2c_{2}^{2}, 8c_{2}^{2}]  4 [4]
A79  [1, −8c_{1}^{2}, c_{2}, −2c_{2}^{3}c_{1}^{2}]  2 [8]  A80, A82
A80  [1, c_{1}^{2}, −8c_{2}^{2}, −2c_{2}^{2}c_{1}^{2}]  3 [4, 2]  A81
A81  [1, 1, −8c_{2}^{2}, −2c_{2}^{2}]  4 [4]
A82  [1, −4c_{1}^{2}, 2c_{2}^{2}, −2c_{2}^{2}c_{1}^{2}]  3 [4]  A83
A83  [1, −4, 2c_{2}^{2}, −2c_{2}^{2}]  4 [4]
A84  [1, c_{1}, c_{2}, 4c_{2}^{3}c_{1}^{3}]  2 [2]  A85, A86, A87, A99
A85  [1, 2c_{1}^{2}, c_{2}, 2c_{2}^{3}c_{1}^{2}]  3 []  A90, A102
A86  [1, −8c_{1}^{2}, c_{2}, −8c_{2}^{3}c_{1}^{2}]  3 []  A93, A102
A87  [1, c_{1}^{2}, c_{2}, 4c_{2}^{3}c_{1}^{2}]  2 [2, 2, 2]  {A88, A89, A90, A93, \atop A96, A105}
A88  [1, 1, c_{2}, 4c_{2}^{3}]  2 [2, 2, 2]  {A91, A94, A97, \atop A106}
A89  [1, 4, c_{2}, c_{2}^{3}]  3 [2, 2]  {A92, A95, A97, \atop A106}

Number Example  ρ Br_{1}  Subcases
A90  [1, c_{1}^{2}, 2c_{2}^{2}, 2c_{2}^{2}c_{1}^{2}]  4 [2]  A91, A92
A91  [1, 1, 2c_{2}^{2}, 2c_{2}^{2}]  4 [2, 2]
A92  [1, 4, 2c_{2}^{2}, 8c_{2}^{2}]  6 []
A93  [1, c_{1}^{2}, −8c_{2}^{2}, −8c_{2}^{2}c_{1}^{2}]  4 [2]  A94, A95
A94  [1, 1, −8c_{2}^{2}, −8c_{2}^{2}]  4 [2, 2]
A95  [1, 4, −8c_{2}^{2}, −2c_{2}^{2}]  6 []
A96  [1, c_{1}^{2}, c_{2}^{2}, 4c_{2}^{2}c_{1}^{2}]  2 [2, 2, 2, 2, 2] A97
A97  [1, 1, c_{2}^{2}, 4c_{2}^{2}]  3 [2, 2, 2, 2] A98
A98  [1, 1, 1, 4]  5 [2, 2]
A99  [1, −4c_{1}^{2}, c_{2}, −c_{2}^{3}c_{1}^{2}]  2 [2]  {A100, A101, A102, \atop A105}
A100  [1, −4, c_{2}, −c_{2}^{3}]  2 [2, 2]  A103, A107
A101  [1, −1, c_{2}, −4c_{2}^{3}]  3 [2]  A104, A107
A102  [1, −4c_{1}^{2}, 2c_{2}^{2}, −8c_{2}^{2}c_{1}^{2}]  4 []  A103, A104
A103  [1, −4, 2c_{2}^{2}, −8c_{2}^{2}]  4 [2]
A104  [1, −1, 2c_{2}^{2}, −2c_{2}^{2}]  6 []
A105  [1, c_{1}^{2}, −4c_{2}^{2}, −c_{2}^{2}c_{1}^{2}]  2 [2, 2, 2]  A106, A107
A106  [1, 1, −4c_{2}^{2}, −c_{2}^{2}]  3 [4, 2, 2]  A108
A107  [1, −4, c_{2}^{2}, −c_{2}^{2}]  3 [2, 2, 2]  A108
A108  [1, 1, −4, −1]  5 [4, 2]
2a_{0}a_{1}a_{2}a_{3} square
A109  [1, c_{1}, c_{2}, 2c_{3}^{2}c_{2}^{3}c_{1}^{3}]  1 [2]  A110, A112
A110  [1, c_{1}^{2}, c_{2}, 2c_{3}^{2}c_{2}^{3}]  1 [2, 2]  {A111, A113, A114, \atop A116, A121}
A111  [1, c_{1}^{2}, c_{2}^{2}, 2c_{3}^{2}]  1 [2, 2, 2]  A117, A122
A112  [1, −4c_{1}^{2}, c_{2}, −8c_{3}^{2}c_{2}^{3}]  1 [2]  {A113, A115, A119, \atop A127}
A113  [1, c_{1}^{2}, −4c_{2}^{2}, −8c_{3}^{2}]  1 [2, 2]  {A118, A120, A123, \atop A128}
A114  [1, 2c_{1}^{2}, c_{2}, 4c_{2}]  1 [4, 2]  A117, A118
A115  [1, −8c_{1}^{2}, c_{2}, −c_{2}]  1 [4]  A120

Number Example  ρ Br_{1}  Subcases
A116  [1, c_{1}, 2c_{2}^{2}c_{1}^{3}, c_{1}^{2}]  1 [2, 2]  {A117, A118, A122, \atop A123}
A117  [1, c_{1}^{2}, 2c_{2}^{2}, 4]  1 [4, 2, 2]  A124
A118  [1, −4c_{1}^{2}, −8c_{2}^{2}, 4]  1 [4, 2, 2]  A129
A119  [1, c_{1}, −8c_{2}^{2}c_{1}^{3}, −4c_{1}^{2}]  1 [4]  A120, A128
A120  [1, c_{1}^{2}, −8c_{2}^{2}, −1]  1 [4, 2]  A125, A129
A121  [1, 2c_{1}^{2}, c_{2}, c_{2}]  2 [2]  A122, A123
A122  [1, c_{1}^{2}, 2c_{2}^{2}, 1]  2 [2, 2]  A124, A126
A123  [1, −4c_{1}^{2}, −8c_{2}^{2}, 1]  2 [2, 2]  A125, A130
A124  [1, 2c_{1}^{2}, 1, 4]  2 [4, 4]
A125  [1, −8c_{1}^{2}, 1, −1]  2 [4, 4]
A126  [1, 2c_{1}^{2}, 1, 1]  4 []
A127  [1, −8c_{1}^{2}, c_{2}, −4c_{2}]  2 []  A128
A128  [1, c_{1}^{2}, −8c_{2}^{2}, −4]  2 [2]  A129, A130
A129  [1, 2c_{1}^{2}, −4, −1]  2 [4, 2]
A130  [1, 2c_{1}^{2}, −4, −4]  4 []
−a_{0}a_{1}a_{2}a_{3} square
A131  [1, c_{1}, c_{2}, −4c_{3}^{2}c_{2}^{3}c_{1}^{3}]  1 [2]  {A132, A134, A157, \atop A158, A170, A185}
A132  [1, c_{1}^{2}, c_{2}, −4c_{3}^{2}c_{2}^{3}]  1 [2, 2]  A133, A135, A136,

A137, A138, A139, A140, A144, A148, A152, A159, A165, A171, A187

A133  [1, c_{1}^{2}, c_{2}^{2}, −4c_{3}^{2}]  1 [2, 2, 2]  {A141, A145, A149, \atop A153, A176, A195}
A134  [1, 2c_{1}^{2}, c_{2}, −8c_{3}^{2}c_{2}^{3}]  1 [2]  {A135, A164, A181, \atop A186}
A135  [1, c_{1}^{2}, 2c_{2}^{2}, −8c_{3}^{2}]  1 [2, 2]  {A142, A146, A150, \atop A154, A182, A192}

Number Example  ρ Br_{1}  Subcases
A136  [1, c_{1}, c_{2}^{2}c_{1}^{3}, −4c_{1}^{2}]  1 [2, 2]  {A145, A154, A162, \atop A169, A172, A190}
A137  [1, c_{1}, −4c_{2}^{2}c_{1}^{3}, c_{1}^{2}]  1 [2, 2]  {A141, A150, A162, \atop A168, A173, A191}
A138  [1, c_{1}, c_{2}^{2}c_{1}^{3}, −c_{1}^{2}]  1 [4, 2]  {A146, A153, A163, \atop A168, A174, A188}
A139  [1, c_{1}, −4c_{2}^{2}c_{1}^{3}, 4c_{1}^{2}]  1 [2, 2, 2]  {A142, A149, A163, \atop A169, A175, A189}
A140  [1, −4c_{1}^{2}, c_{2}, c_{2}]  1 [2, 2]  {A141, A142, A160, \atop A166, A172, A188}
A141  [1, c_{1}^{2}, −4c_{2}^{2}, 1]  1 [2, 2, 2, 2] {A143, A147, A151, \atop A155, A177, A196}
A142  [1, 2c_{1}^{2}, −8c_{2}^{2}, 1]  1 [2, 2, 2]  A183, A193
A143  [1, −4c_{1}^{2}, 1, 1]  1 [2, 2, 2, 2, 2] A178, A197
A144  [1, c_{1}^{2}, c_{2}, −4c_{2}]  1 [2, 2]  {A145, A146, A160, \atop A167, A173, A189}
A145  [1, c_{1}^{2}, c_{2}^{2}, −4]  1 [4, 2, 2]  {A147, A156, A177, \atop A198}
A146  [1, 2c_{1}^{2}, 2c_{2}^{2}, −4]  1 [4, 2]  A184, A193
A147  [1, c_{1}^{2}, 1, −4]  1 [4, 2, 2, 2] A178, A199
A148  [1, −4c_{1}^{2}, c_{2}, 4c_{2}]  2 [2, 2]  {A149, A150, A161, \atop A167, A174, A190}
A149  [1, c_{1}^{2}, −4c_{2}^{2}, 4]  2 [2, 2, 2]  {A151, A156, A179, \atop A198}
A150  [1, 2c_{1}^{2}, −8c_{2}^{2}, 4]  2 [2, 2]  A184, A194
A151  [1, −4c_{1}^{2}, 1, 4]  3 [2, 2, 2]  A180, A199
A152  [1, c_{1}^{2}, c_{2}, −c_{2}]  2 [2]  {A153, A154, A161, \atop A166, A175, A191}
A153  [1, c_{1}^{2}, c_{2}^{2}, −1]  2 [2, 2]  {A155, A156, A179, \atop A196}
A154  [1, 2c_{1}^{2}, 2c_{2}^{2}, −1]  2 [2]  A183, A194
A155  [1, c_{1}^{2}, 1, −1]  3 [2, 2, 2]  A180, A197

Number Example  ρ Br_{1}  Subcases
A156  [1, c_{1}^{2}, −4, 4]  3 [2, 2]  A180, A199
A157  [1, c_{1}, c_{2}, −4c_{2}^{3}c_{1}]  2 []  {A159, A164, A171, \atop A186}
A158  [1, c_{1}, c_{2}, −c_{2}^{3}c_{1}]  1 [2]  {A164, A165, A181, \atop A187}
A159  [1, c_{1}, c_{2}^{2}c_{1}^{3}, −4c_{2}^{2}c_{1}^{2}]  3 []  {A160, A161, A162, \atop A163, A176, A192}
A160  [1, 1, c_{2}, −4c_{2}]  3 []  A177, A193
A161  [1, 4, c_{2}, −c_{2}]  5 []  A179, A194
A162  [1, c_{1}^{2}, −4c_{2}^{2}c_{1}^{3}, c_{2}^{2}c_{1}]  3 []  A177, A194
A163  [1, c_{1}^{2}, −8c_{2}^{2}c_{1}^{3}, 2c_{2}^{2}c_{1}]  3 [2]  A179, A193
A164  [1, c_{1}, 2c_{2}^{2}c_{1}^{3}, −2c_{2}^{2}c_{1}^{2}]  2 []  A182, A192
A165  [1, c_{1}, c_{2}^{2}c_{1}^{3}, −c_{2}^{2}c_{1}^{2}]  1 [2, 2]  {A166, A167, A168, \atop A169, A182, A195}
A166  [1, 1, c_{2}, −c_{2}]  2 [2]  A183, A196
A167  [1, −4, c_{2}, 4c_{2}]  2 [2, 2]  A184, A198
A168  [1, c_{1}^{2}, −4c_{2}^{2}c_{1}^{3}, 4c_{2}^{2}c_{1}]  1 [4, 2]  A184, A196
A169  [1, −4c_{1}^{2}, −4c_{2}^{2}c_{1}^{3}, −c_{2}^{2}c_{1}] 1 [2, 2, 2]  A183, A198
A170  [1, c_{1}, c_{2}, −4c_{2}^{3}c_{1}^{3}]  1 [4]  A171, A181
A171  [1, c_{1}^{2}, c_{2}, −4c_{2}^{3}c_{1}^{2}]  2 [2]  {A172, A173, A174, \atop A175, A176, A182}
A172  [1, 1, c_{2}, −4c_{2}^{3}]  2 [4]  A177, A183
A173  [1, −4, c_{2}, c_{2}^{3}]  2 [4]  A177, A184
A174  [1, 4, c_{2}, −c_{2}^{3}]  3 [2]  A179, A184
A175  [1, −1, c_{2}, 4c_{2}^{3}]  3 [2]  A179, A183
A176  [1, c_{1}^{2}, c_{2}^{2}, −4c_{2}^{2}c_{1}^{2}]  4 []  A177, A179
A177  [1, 1, c_{2}^{2}, −4c_{2}^{2}]  4 [2]  A178, A180
A178  [1, 1, 1, −4]  4 [2, 2]
A179  [1, 4, c_{2}^{2}, −c_{2}^{2}]  6 []  A180
A180  [1, 1, 4, −1]  8 []
A181  [1, 2c_{1}^{2}, c_{2}, −2c_{2}^{3}c_{1}^{2}]  1 [8]  A182
A182  [1, c_{1}^{2}, 2c_{2}^{2}, −2c_{2}^{2}c_{1}^{2}]  2 [4]  A183, A184

Number Example  ρ Br_{1}  Subcases
A183  [1, 1, 2c_{2}^{2}, −2c_{2}^{2}]  3 [4]
A184  [1, −4, 2c_{2}^{2}, 8c_{2}^{2}]  3 [2, 2]
A185  [1, c_{1}, c_{2}, −c_{2}^{3}c_{1}^{3}]  1 [2]  A186, A187
A186  [1, 2c_{1}^{2}, c_{2}, −8c_{2}^{3}c_{1}^{2}]  2 []  A192
A187  [1, c_{1}^{2}, c_{2}, −c_{2}^{3}c_{1}^{2}]  1 [2, 2]  {A188, A189, A190, \atop A191, A192, A195}
A188  [1, 1, c_{2}, −c_{2}^{3}]  1 [4, 2]  A193, A196
A189  [1, −4, c_{2}, 4c_{2}^{3}]  1 [2, 2, 2]  A193, A198
A190  [1, 4, c_{2}, −4c_{2}^{3}]  2 [2, 2]  A194, A198
A191  [1, −1, c_{2}, c_{2}^{3}]  2 [2]  A194, A196
A192  [1, c_{1}^{2}, 2c_{2}^{2}, −8c_{2}^{2}c_{1}^{2}]  3 []  A193, A194
A193  [1, 1, 2c_{2}^{2}, −8c_{2}^{2}]  3 [2]
A194  [1, 4, 2c_{2}^{2}, −2c_{2}^{2}]  5 []
A195  [1, c_{1}^{2}, c_{2}^{2}, −c_{2}^{2}c_{1}^{2}]  1 [4, 2, 2]  A196, A198
A196  [1, 1, c_{2}^{2}, −c_{2}^{2}]  2 [2, 2, 2]  A197, A199
A197  [1, 1, 1, −1]  4 [2, 2, 2, 2]
A198  [1, −4, c_{2}^{2}, 4c_{2}^{2}]  2 [4, 2, 2]  A199
A199  [1, 1, −4, 4]  4 [2, 2]
−2a_{0}a_{1}a_{2}a_{3} square
A200  [1, c_{1}, c_{2}, −8c_{3}^{2}c_{2}^{3}c_{1}^{3}]  1 [2]  A201, A203
A201  [1, c_{1}^{2}, c_{2}, −8c_{3}^{2}c_{2}^{3}]  1 [2, 2]  {A202, A204, A206, \atop A209, A212}
A202  [1, c_{1}^{2}, c_{2}^{2}, −8c_{3}^{2}]  1 [2, 2, 2]  A210, A213
A203  [1, −4c_{1}^{2}, c_{2}, 2c_{3}^{2}c_{2}^{3}]  1 [2]  {A204, A205, A207, \atop A218}
A204  [1, c_{1}^{2}, −4c_{2}^{2}, 2c_{3}^{2}]  1 [2, 2]  {A208, A211, A214, \atop A219}
A205  [1, 2c_{1}^{2}, c_{2}, −c_{2}]  1 [4]  A208
A206  [1, −8c_{1}^{2}, c_{2}, 4c_{2}]  1 [4, 2]  A210, A211
A207  [1, c_{1}, 2c_{2}^{2}c_{1}^{3}, −4c_{1}^{2}]  1 [4]  A208, A219
A208  [1, c_{1}^{2}, 2c_{2}^{2}, −1]  1 [4, 2]  A216, A220

Number Example  ρ Br_{1}  Subcases
A209  [1, c_{1}, −8c_{2}^{2}c_{1}^{3}, c_{1}^{2}]  1 [2, 2]  {A210, A211, A213, \atop A214}
A210  [1, c_{1}^{2}, −8c_{2}^{2}, 4]  1 [4, 2, 2]  A215
A211  [1, −4c_{1}^{2}, 2c_{2}^{2}, 4]  1 [4, 2, 2]  A220
A212  [1, −8c_{1}^{2}, c_{2}, c_{2}]  2 [2]  A213, A214
A213  [1, c_{1}^{2}, −8c_{2}^{2}, 1]  2 [2, 2]  A215, A217
A214  [1, −4c_{1}^{2}, 2c_{2}^{2}, 1]  2 [2, 2]  A216, A221
A215  [1, −8c_{1}^{2}, 1, 4]  2 [4, 2, 2]
A216  [1, 2c_{1}^{2}, 1, −1]  2 [4, 4]
A217  [1, −8c_{1}^{2}, 1, 1]  4 [2]
A218  [1, 2c_{1}^{2}, c_{2}, −4c_{2}]  2 []  A219
A219  [1, c_{1}^{2}, 2c_{2}^{2}, −4]  2 [2]  A220, A221
A220  [1, 2c_{1}^{2}, −4, 4]  2 [4, 2]
A221  [1, 2c_{1}^{2}, 1, −4]  4 []

Other cases

A222 [1, c_{1}, c_{2}, c_{3}] 1 [2] A1, A109, A131, A200, A223, A224, A225, A226

A223 [1, c_{1}, c_{2}, c_{3}^{2}] 1 [2, 2] A2, A110, A132, A201, A227, A229, A231, A232, A233, A234, A236, A244, A257, A268

A224 [1, c_{1}, c_{2}, −4c_{3}^{2}] 1 [2] A4, A112, A132, A203, A228, A230, A232, A235, A241, A249, A263, A276

A225  [1, c_{1}, c_{2}, 2c_{3}^{2}]  1 [2]  {A6, A110, A134, \atop A203, A233, A235}
A226  [1, c_{1}, c_{2}, −8c_{3}^{2}]  1 [2]  {A8, A112, A134, \atop A201, A234, A235}

Number Example  ρ Br_{1}  Subcases
A227  [1, c_{1}, c_{2}, 1]  1 [2, 2]  A15, A121, A140,

A212, A237, A238, A247, A248

A228  [1, c_{1}, c_{2}, −4]  1 [2]  {A21, A127, A144, \atop A218, A242, A251}
A229  [1, c_{1}, c_{2}, 4]  1 [4, 2]  A25, A114, A148,

A206, A239, A240, A245, A246

A230  [1, c_{1}, c_{2}, −1]  1 [4]  {A31, A115, A152, \atop A205, A243, A250}
A231  [1, c_{1}, c_{2}^{2}, c_{3}^{2}]  1 [2, 2, 2]  A3, A111, A133,

A202, A237, A245, A258, A269

A232 [1, c_{1}, c_{2}^{2}, −4c_{3}^{2}] 1 [2, 2] A5, A113, A133, A204, A238, A242, A246, A250, A259, A264, A270, A277

A233 [1, c_{1}, c_{2}^{2}, 2c_{3}^{2}] 1 [2, 2] A7, A111, A135, A204, A239, A247, A260, A271

A234 [1, c_{1}, c_{2}^{2}, −8c_{3}^{2}] 1 [2, 2] A9, A113, A135, A202, A240, A248, A261, A272

A235 [1, c_{1}, −4c_{2}^{2}, 2c_{3}^{2}] 1 [2] A10, A113, A135, A204, A243, A251, A265, A278

A236 [1, c_{1}, c_{2}, c_{2}^{2}] 1 [2, 2] A11, A116, A137, A209, A237, A238, A239, A240, A258, A259, A271, A272

Number Example  ρ Br_{1}  Subcases
A237  [1, c_{1}, c_{2}^{2}, 1]  1 [2, 2, 2]  A16, A122, A141,

A213, A252, A254, A262, A275

A238 [1, c_{1}, −4c_{2}^{2}, 1] 1 [2, 2, 2] A17, A123, A141, A214, A253, A255, A266, A281

A239  [1, c_{1}, 2c_{2}^{2}, 4]  1 [4, 2]  {A28, A117, A150, \atop A211}
A240  [1, c_{1}, −8c_{2}^{2}, 4]  1 [4, 2]  {A29, A118, A150, \atop A210}
A241  [1, c_{1}, c_{2}, −4c_{2}^{2}]  1 [2]  A12, A119, A136,

A207, A242, A243, A264, A278

A242 [1, c_{1}, c_{2}^{2}, −4] 1 [2, 2] A22, A128, A145, A219, A253, A256, A267, A282

A243  [1, c_{1}, 2c_{2}^{2}, −1]  1 [4]  {A33, A120, A154, \atop A208}
A244  [1, c_{1}, c_{2}, 4c_{2}^{2}]  1 [2, 2]  A13, A116, A139,

A209, A245, A246, A247, A248, A260, A261, A269, A270

A245  [1, c_{1}, c_{2}^{2}, 4]  1 [4, 2, 2]  {A26, A117, A149, \atop A210, A254, A273}
A246  [1, c_{1}, −4c_{2}^{2}, 4]  1 [4, 2]  {A27, A118, A149, \atop A211, A256, A279}
A247  [1, c_{1}, 2c_{2}^{2}, 1]  1 [2, 2]  {A18, A122, A142, \atop A214}
A248  [1, c_{1}, −8c_{2}^{2}, 1]  1 [2, 2]  {A19, A123, A142, \atop A213}

Number Example  ρ Br_{1}  Subcases
A249  [1, c_{1}, c_{2}, −c_{2}^{2}]  1 [4]  A14, A119, A138,

A207, A250, A251, A265, A277

A250 [1, c_{1}, c_{2}^{2}, −1] 1 [4, 2] A32, A120, A153, A208, A255, A256, A274, A280

A251  [1, c_{1}, 2c_{2}^{2}, −4]  1 [4]  {A23, A128, A146, \atop A219}
A252  [1, c_{1}, 1, 1]  1 [2, 2, 2]  {A20, A126, A143, \atop A217}
A253  [1, c_{1}, 1, −4]  1 [2, 2, 2]  {A24, A130, A147, \atop A221}
A254  [1, c_{1}, 1, 4]  1 [4, 4, 2]  {A30, A124, A151, \atop A215}
A255  [1, c_{1}, 1, −1]  1 [4, 2, 2]  {A34, A125, A155, \atop A216}
A256  [1, c_{1}, −4, 4]  1 [4, 4]  {A35, A129, A156, \atop A220}
A257  [1, c_{1}, c_{2}, c_{2}^{2}c_{1}^{2}]  1 [2, 2]  A15, A114, A140,

A206, A258, A259, A260, A261

A258  [1, c_{1}^{2}, c_{2}, c_{2}^{2}]  1 [2, 2, 2]  {A16, A117, A141, \atop A210, A262, A273}
A259  [1, −4c_{1}^{2}, c_{2}, c_{2}^{2}]  1 [2, 2, 2]  {A17, A118, A141, \atop A211, A267, A280}
A260  [1, 2c_{1}^{2}, c_{2}, 4c_{2}^{2}]  1 [2, 2]  {A18, A117, A142, \atop A211}
A261  [1, −8c_{1}^{2}, c_{2}, 4c_{2}^{2}]  1 [2, 2]  {A19, A118, A142, \atop A210}
A262  [1, c_{1}, 1, c_{1}^{2}]  1 [2, 2, 2]  {A20, A124, A143, \atop A215}

Number Example  ρ Br_{1}  Subcases
A263  [1, c_{1}, c_{2}, −4c_{2}^{2}c_{1}^{2}]  1 [2]  {A21, A115, A144, \atop A205, A264, A265}
A264  [1, c_{1}^{2}, c_{2}, −4c_{2}^{2}]  1 [2, 2]  A22, A120, A145,

A208, A266, A267, A274, A279

A265  [1, 2c_{1}^{2}, c_{2}, −c_{2}^{2}]  1 [4]  {A23, A120, A146, \atop A208}
A266  [1, c_{1}, 1, −4c_{1}^{2}]  1 [2, 2, 2]  {A24, A125, A147, \atop A216}
A267  [1, c_{1}, −4, c_{1}^{2}]  1 [2, 2, 2]  {A24, A129, A147, \atop A220}
A268  [1, c_{1}, c_{2}, 4c_{2}^{2}c_{1}^{2}]  2 [2]  A25, A121, A148,

A212, A269, A270, A271, A272

A269  [1, c_{1}^{2}, c_{2}, 4c_{2}^{2}]  2 [2, 2]  {A26, A122, A149, \atop A213, A273, A275}
A270  [1, −4c_{1}^{2}, c_{2}, 4c_{2}^{2}]  2 [2]  {A27, A123, A149, \atop A214, A274, A282}
A271  [1, 2c_{1}^{2}, c_{2}, c_{2}^{2}]  2 [2]  {A28, A122, A150, \atop A214}
A272  [1, −8c_{1}^{2}, c_{2}, c_{2}^{2}]  2 [2]  {A29, A123, A150, \atop A213}
A273  [1, c_{1}, 4, c_{1}^{2}]  2 [4, 2]  {A30, A124, A151, \atop A215}
A274  [1, c_{1}, −1, −4c_{1}^{2}]  2 [4]  {A35, A125, A156, \atop A216}
A275  [1, c_{1}, 1, 4c_{1}^{2}]  3 [2]  {A30, A126, A151, \atop A217}
A276  [1, c_{1}, c_{2}, −c_{2}^{2}c_{1}^{2}]  2 []  {A31, A127, A152, \atop A218, A277, A278}

Number Example  ρ Br_{1}  Subcases
A277  [1, c_{1}^{2}, c_{2}, −c_{2}^{2}]  2 [2]  A32, A128, A153,

A219, A279, A280, A281, A282

A278  [1, 2c_{1}^{2}, c_{2}, −4c_{2}^{2}]  2 []  {A33, A128, A154, \atop A219}
A279  [1, c_{1}, 4, −4c_{1}^{2}]  2 [4]  {A35, A129, A156, \atop A220}
A280  [1, c_{1}, −1, c_{1}^{2}]  2 [2, 2]  {A34, A129, A155, \atop A220}
A281  [1, c_{1}, 1, −c_{1}^{2}]  3 [2]  {A34, A130, A155, \atop A221}
A282  [1, c_{1}, −4, 4c_{1}^{2}]  3 []  {A35, A130, A156, \atop A221}

A.2 k ∩ ℚ(ϵ) = ℚ(i)

Number Example  ρ Br_{1}  Subcases
a_{0}a_{1}a_{2}a_{3} square
B1  [1, c_{1}, c_{2}, c_{3}^{2}c_{2}^{3}c_{1}^{3}] 2 [2]  {B2, B4, B16, B17, \atop B27, B39}
B2  [1, c_{1}^{2}, c_{2}, c_{3}^{2}c_{2}^{3}]  2 [2, 2, 2]  B3, B5, B6, B7, B8,

B12, B18, B24, B28, B41

B3  [1, c_{1}^{2}, c_{2}^{2}, c_{3}^{2}]  2 [2, 2, 2, 2, 2]  B9, B13, B31, B47
B4  [1, 2c_{1}^{2}, c_{2}, 2c_{3}^{2}c_{2}^{3}] 2 [2]  B5, B23, B36, B40
B5  [1, c_{1}^{2}, 2c_{2}^{2}, 2c_{3}^{2}]  2 [2, 2, 2]  B10, B14, B37, B44
B6  [1, c_{1}, c_{2}^{2}c_{1}^{3}, c_{1}^{2}]  2 [2, 2, 2]  {B9, B14, B21, B26, \atop B29, B43}
B7  [1, c_{1}, c_{2}^{2}c_{1}^{3}, 4c_{1}^{2}]  2 [2, 2, 2, 2]  {B10, B13, B22, B26, \atop B30, B42}

Number Example  ρ Br_{1}  Subcases
B8  [1, c_{1}^{2}, c_{2}, c_{2}]  2 [2, 2, 2]  {B9, B10, B19, B25, \atop B29, B42}
B9  [1, c_{1}^{2}, c_{2}^{2}, 1]  2 [4, 2, 2, 2, 2, 2] B11, B15, B32, B48
B10  [1, 2c_{1}^{2}, 2c_{2}^{2}, 1]  2 [2, 2, 2, 2]  B38, B45
B11  [1, c_{1}^{2}, 1, 1]  2 [4, 4, 2, 2, 2, 2, 2] B33, B49
B12  [1, c_{1}^{2}, c_{2}, 4c_{2}]  4 [2, 2]  {B13, B14, B20, B25, \atop B30, B43}
B13  [1, c_{1}^{2}, c_{2}^{2}, 4]  4 [2, 2, 2, 2]  B15, B34, B48
B14  [1, 2c_{1}^{2}, 2c_{2}^{2}, 4]  4 [2, 2]  B38, B46
B15  [1, c_{1}^{2}, 1, 4]  6 [2, 2, 2, 2]  B35, B49
B16  [1, c_{1}, c_{2}, c_{2}^{3}c_{1}]  4 []  B18, B23, B28, B40
B17  [1, c_{1}, c_{2}, 4c_{2}^{3}c_{1}]  2 [2]  B23, B24, B36, B41
B18  [1, c_{1}, c_{2}^{2}c_{1}^{3}, c_{2}^{2}c_{1}^{2}] 6 []  {B19, B20, B21, B22, \atop B31, B44}
B19  [1, 1, c_{2}, c_{2}]  6 []  B32, B45
B20  [1, 4, c_{2}, 4c_{2}]  10 []  B34, B46
B21  [1, c_{1}^{2}, c_{2}^{2}c_{1}^{3}, c_{2}^{2}c_{1}] 6 []  B32, B46
B22  [1, c_{1}^{2}, 2c_{2}^{2}c_{1}^{3}, 2c_{2}^{2}c_{1}] 6 [2, 2]  B34, B45
B23  [1, c_{1}, 2c_{2}^{2}c_{1}^{3}, 8c_{2}^{2}c_{1}^{2}] 4 []  B37, B44
B24  [1, c_{1}, c_{2}^{2}c_{1}^{3}, 4c_{2}^{2}c_{1}^{2}] 2 [2, 2, 2]  B25, B26, B37, B47
B25  [1, 1, c_{2}, 4c_{2}]  4 [2, 2]  B38, B48
B26  [1, c_{1}^{2}, c_{2}^{2}c_{1}^{3}, 4c_{2}^{2}c_{1}] 2 [2, 2, 2, 2]  B38, B48
B27  [1, c_{1}, c_{2}, c_{2}^{3}c_{1}^{3}]  2 [4, 2]  B28, B36
B28  [1, c_{1}^{2}, c_{2}, c_{2}^{3}c_{1}^{2}]  4 [2, 2]  B29, B30, B31, B37
B29  [1, 1, c_{2}, c_{2}^{3}]  4 [4, 2]  B32, B38
B30  [1, 4, c_{2}, 4c_{2}^{3}]  6 [2]  B34, B38
B31  [1, c_{1}^{2}, c_{2}^{2}, c_{2}^{2}c_{1}^{2}]  8 []  B32, B34
B32  [1, 1, c_{2}^{2}, c_{2}^{2}]  8 [2, 2]  B33, B35
B33  [1, 1, 1, 1]  8 [2, 2, 2, 2]
B34  [1, 4, c_{2}^{2}, 4c_{2}^{2}]  12 []  B35
B35  [1, 1, 4, 4]  16 []
B36  [1, 2c_{1}^{2}, c_{2}, 8c_{2}^{3}c_{1}^{2}] 2 [8, 4]  B37

Number Example  ρ Br_{1}  Subcases
B37  [1, c_{1}^{2}, 2c_{2}^{2}, 8c_{2}^{2}c_{1}^{2}] 4 [4, 4]  B38
B38  [1, 1, 2c_{2}^{2}, 8c_{2}^{2}]  6 [4, 2]
B39  [1, c_{1}, c_{2}, 4c_{2}^{3}c_{1}^{3}]  2 [2]  B40, B41
B40  [1, 2c_{1}^{2}, c_{2}, 2c_{2}^{3}c_{1}^{2}] 4 []  B44
B41  [1, c_{1}^{2}, c_{2}, 4c_{2}^{3}c_{1}^{2}]  2 [2, 2, 2]  B42, B43, B44, B47
B42  [1, 1, c_{2}, 4c_{2}^{3}]  2 [2, 2, 2, 2]  B45, B48
B43  [1, 4, c_{2}, c_{2}^{3}]  4 [2, 2]  B46, B48
B44  [1, c_{1}^{2}, 2c_{2}^{2}, 2c_{2}^{2}c_{1}^{2}] 6 []  B45, B46
B45  [1, 1, 2c_{2}^{2}, 2c_{2}^{2}]  6 [2, 2]
B46  [1, 4, 2c_{2}^{2}, 8c_{2}^{2}]  10 []
B47  [1, c_{1}^{2}, c_{2}^{2}, 4c_{2}^{2}c_{1}^{2}]  2 [2, 2, 2, 2, 2, 2] B48
B48  [1, 1, c_{2}^{2}, 4c_{2}^{2}]  4 [4, 2, 2, 2, 2]  B49
B49  [1, 1, 1, 4]  8 [2, 2, 2, 2]

2a_{0}a_{1}a_{2}a_{3} square

B50  [1, c_{1}, c_{2}, 2c_{3}^{2}c_{2}^{3}c_{1}^{3}] 1 [2]  B51
B51  [1, c_{1}^{2}, c_{2}, 2c_{3}^{2}c_{2}^{3}]  1 [2, 2]  B52, B53, B54, B56
B52  [1, c_{1}^{2}, c_{2}^{2}, 2c_{3}^{2}]  1 [4, 2, 2]  B55, B57
B53  [1, 2c_{1}^{2}, c_{2}, 4c_{2}]  1 [4, 4]  B55
B54  [1, c_{1}, 2c_{2}^{2}c_{1}^{3}, c_{1}^{2}]  1 [4, 2]  B55, B57
B55  [1, c_{1}^{2}, 2c_{2}^{2}, 4]  1 [4, 4, 2, 2]  B58
B56  [1, 2c_{1}^{2}, c_{2}, c_{2}]  3 []  B57
B57  [1, c_{1}^{2}, 2c_{2}^{2}, 1]  3 [2, 2]  B58, B59
B58  [1, 2c_{1}^{2}, 1, 4]  3 [4, 4, 2, 2]
B59  [1, 2c_{1}^{2}, 1, 1]  7 []

Other cases

B60  [1, c_{1}, c_{2}, c_{3}]  1 [2]  B1, B50, B61, B62
B61  [1, c_{1}, c_{2}, c_{3}^{2}]  1 [2, 2]  B2, B51, B63, B64,

B65, B66, B67, B70, B75, B79

B62  [1, c_{1}, c_{2}, 2c_{3}^{2}]  1 [2]  B4, B51, B66

Number Example  ρ Br_{1}  Subcases
B63  [1, c_{1}, c_{2}, 1]  1 [2, 2]  B8, B56, B68, B72
B64  [1, c_{1}, c_{2}, 4]  1 [4, 4]  B12, B53, B69, B71
B65  [1, c_{1}, c_{2}^{2}, c_{3}^{2}]  1 [2, 2, 2]  {B3, B52, B68, B71, \atop B76, B80}
B66  [1, c_{1}, c_{2}^{2}, 2c_{3}^{2}]  1 [2, 2]  {B5, B52, B69, B72, \atop B77, B81}
B67  [1, c_{1}, c_{2}, c_{2}^{2}]  1 [2, 2]  {B6, B54, B68, B69, \atop B76, B81}
B68  [1, c_{1}, c_{2}^{2}, 1]  1 [2, 2, 2, 2]  {B9, B57, B73, B74, \atop B78, B83}
B69  [1, c_{1}, 2c_{2}^{2}, 4]  1 [4, 4]  B14, B55
B70  [1, c_{1}, c_{2}, 4c_{2}^{2}]  1 [4, 2]  {B7, B54, B71, B72, \atop B77, B80}
B71  [1, c_{1}, c_{2}^{2}, 4]  1 [4, 4, 2]  B13, B55, B74, B82
B72  [1, c_{1}, 2c_{2}^{2}, 1]  1 [4, 2]  B10, B57
B73  [1, c_{1}, 1, 1]  1 [2, 2, 2, 2, 2]  B11, B59
B74  [1, c_{1}, 1, 4]  1 [4, 4, 4, 4]  B15, B58
B75  [1, c_{1}, c_{2}, c_{2}^{2}c_{1}^{2}]  1 [2, 2]  B8, B53, B76, B77
B76  [1, c_{1}^{2}, c_{2}, c_{2}^{2}]  1 [2, 2, 2, 2]  B9, B55, B78, B82
B77  [1, 2c_{1}^{2}, c_{2}, 4c_{2}^{2}]  1 [4, 2]  B10, B55
B78  [1, c_{1}, 1, c_{1}^{2}]  1 [2, 2, 2, 2, 2]  B11, B58
B79  [1, c_{1}, c_{2}, 4c_{2}^{2}c_{1}^{2}]  3 []  B12, B56, B80, B81
B80  [1, c_{1}^{2}, c_{2}, 4c_{2}^{2}]  3 [2]  B13, B57, B82, B83
B81  [1, 2c_{1}^{2}, c_{2}, c_{2}^{2}]  3 []  B14, B57
B82  [1, c_{1}, 4, c_{1}^{2}]  3 [4, 2]  B15, B58
B83  [1, c_{1}, 1, 4c_{1}^{2}]  5 []  B15, B59

A.3 k ∩ ℚ(ϵ) = ℚ(\sqrt{2})

Number Example  ρ Br_{1}  Subcases
a_{0}a_{1}a_{2}a_{3} square

Number Example  ρ Br_{1}  Subcases
C1  [1, c_{1}, c_{2}, c_{3}^{2}c_{2}^{3}c_{1}^{3}]  2 [2]  C2, C4, C15, C22
C2  [1, c_{1}^{2}, c_{2}, c_{3}^{2}c_{2}^{3}]  2 [2, 2, 2]  {C3, C5, C6, C8, \atop C16, C23}
C3  [1, c_{1}^{2}, c_{2}^{2}, c_{3}^{2}]  2 [2, 2, 2, 2, 2] C9, C25
C4  [1, −4c_{1}^{2}, c_{2}, −4c_{3}^{2}c_{2}^{3}] 2 [2]  {C5, C7, C12, C19, \atop C28}
C5  [1, c_{1}^{2}, −4c_{2}^{2}, −4c_{3}^{2}]  2 [2, 2, 2]  C10, C13, C30
C6  [1, c_{1}, c_{2}^{2}c_{1}^{3}, c_{1}^{2}]  2 [2, 2, 2]  C9, C10, C18, C24
C7  [1, c_{1}, −4c_{2}^{2}c_{1}^{3}, −4c_{1}^{2}] 2 [2, 2]  C13, C21, C29
C8  [1, c_{1}^{2}, c_{2}, c_{2}]  3 [2, 2]  C9, C10, C17, C24
C9  [1, c_{1}^{2}, c_{2}^{2}, 1]  3 [2, 2, 2, 2] C11, C26
C10  [1, −4c_{1}^{2}, −4c_{2}^{2}, 1]  3 [2, 2, 2, 2] C14, C31
C11  [1, c_{1}^{2}, 1, 1]  5 [2, 2, 2]  C27
C12  [1, −4c_{1}^{2}, c_{2}, −4c_{2}]  3 [2]  C13, C20, C29
C13  [1, c_{1}^{2}, −4c_{2}^{2}, −4]  3 [2, 2, 2]  C14, C32
C14  [1, c_{1}^{2}, −4, −4]  5 [2, 2, 2]  C33
C15  [1, c_{1}, c_{2}, c_{2}^{3}c_{1}]  3 []  C16, C19, C23, C28
C16  [1, c_{1}, c_{2}^{2}c_{1}^{3}, c_{2}^{2}c_{1}^{2}]  4 [2]  C17, C18, C25, C30
C17  [1, 1, c_{2}, c_{2}]  6 []  C26, C31
C18  [1, c_{1}^{2}, c_{2}^{2}c_{1}^{3}, c_{2}^{2}c_{1}]  4 [2, 2]  C26, C31
C19  [1, c_{1}, −4c_{2}^{2}c_{1}^{3}, −4c_{2}^{2}c_{1}^{2}] 4 []  C20, C21, C30
C20  [1, −4, c_{2}, −4c_{2}]  6 []  C32
C21  [1, −4c_{1}^{2}, c_{2}^{2}c_{1}^{3}, −4c_{2}^{2}c_{1}] 4 [2]  C32
C22  [1, c_{1}, c_{2}, c_{2}^{3}c_{1}^{3}]  2 [8]  C23, C28
C23  [1, c_{1}^{2}, c_{2}, c_{2}^{3}c_{1}^{2}]  3 [4, 2]  C24, C25, C30
C24  [1, 1, c_{2}, c_{2}^{3}]  4 [4]  C26, C31
C25  [1, c_{1}^{2}, c_{2}^{2}, c_{2}^{2}c_{1}^{2}]  5 [2, 2, 2]  C26
C26  [1, 1, c_{2}^{2}, c_{2}^{2}]  7 [2, 2]  C27
C27  [1, 1, 1, 1]  11 []
C28  [1, −4c_{1}^{2}, c_{2}, −4c_{2}^{3}c_{1}^{2}] 3 [4]  C29, C30
C29  [1, −4, c_{2}, −4c_{2}^{3}]  4 [4]  C32
C30  [1, c_{1}^{2}, −4c_{2}^{2}, −4c_{2}^{2}c_{1}^{2}] 5 [4]  C31, C32

Number Example  ρ Br_{1}  Subcases
C31  [1, 1, −4c_{2}^{2}, −4c_{2}^{2}]  7 [2, 2]  C33
C32  [1, −4, c_{2}^{2}, −4c_{2}^{2}]  7 [4]  C33
C33  [1, 1, −4, −4]  11 [2]

−a_{0}a_{1}a_{2}a_{3} square

C34  [1, c_{1}, c_{2}, −4c_{3}^{2}c_{2}^{3}c_{1}^{3}]  1 [2]  C35, C45, C49
C35  [1, c_{1}^{2}, c_{2}, −4c_{3}^{2}c_{2}^{3}]  1 [2, 2]  {C36, C37, C38, C39, \atop C42, C46, C50}
C36  [1, c_{1}^{2}, c_{2}^{2}, −4c_{3}^{2}]  1 [2, 2, 2, 2] C40, C43, C53
C37  [1, c_{1}, c_{2}^{2}c_{1}^{3}, −4c_{1}^{2}]  1 [4, 2]  C43, C48, C51
C38  [1, c_{1}, −4c_{2}^{2}c_{1}^{3}, c_{1}^{2}]  1 [2, 2, 2]  C40, C48, C52
C39  [1, −4c_{1}^{2}, c_{2}, c_{2}]  2 [2, 2]  C40, C47, C51
C40  [1, c_{1}^{2}, −4c_{2}^{2}, 1]  2 [2, 2, 2, 2] C41, C44, C54
C41  [1, −4c_{1}^{2}, 1, 1]  4 [2, 2, 2, 2] C55
C42  [1, c_{1}^{2}, c_{2}, −4c_{2}]  2 [2]  C43, C47, C52
C43  [1, c_{1}^{2}, c_{2}^{2}, −4]  2 [2, 2, 2]  C44, C54
C44  [1, c_{1}^{2}, 1, −4]  4 [2, 2, 2]  C55
C45  [1, c_{1}, c_{2}, −4c_{2}^{3}c_{1}]  2 []  C46, C50
C46  [1, c_{1}, c_{2}^{2}c_{1}^{3}, −4c_{2}^{2}c_{1}^{2}]  3 []  C47, C48, C53
C47  [1, 1, c_{2}, −4c_{2}]  5 []  C54
C48  [1, c_{1}^{2}, −4c_{2}^{2}c_{1}^{3}, c_{2}^{2}c_{1}]  3 [2]  C54
C49  [1, c_{1}, c_{2}, −4c_{2}^{3}c_{1}^{3}]  1 [8]  C50
C50  [1, c_{1}^{2}, c_{2}, −4c_{2}^{3}c_{1}^{2}]  2 [4]  C51, C52, C53
C51  [1, 1, c_{2}, −4c_{2}^{3}]  3 [2, 2]  C54
C52  [1, −4, c_{2}, c_{2}^{3}]  3 [4]  C54
C53  [1, c_{1}^{2}, c_{2}^{2}, −4c_{2}^{2}c_{1}^{2}]  4 [2]  C54
C54  [1, 1, c_{2}^{2}, −4c_{2}^{2}]  6 [2]  C55
C55  [1, 1, 1, −4]  10 []

Other cases

C56  [1, c_{1}, c_{2}, c_{3}]  1 [2]  C1, C34, C57, C58

Number Example  ρ Br_{1}  Subcases
C57  [1, c_{1}, c_{2}, c_{3}^{2}]  1 [2, 2]  {C2, C35, C59, C61, \atop C62, C63, C70}
C58  [1, c_{1}, c_{2}, −4c_{3}^{2}]  1 [2]  {C4, C35, C60, C62, \atop C66, C74}
C59  [1, c_{1}, c_{2}, 1]  1 [4, 2]  C8, C39, C64, C65
C60  [1, c_{1}, c_{2}, −4]  1 [4]  C12, C42, C67
C61  [1, c_{1}, c_{2}^{2}, c_{3}^{2}]  1 [2, 2, 2]  C3, C36, C64, C71
C62  [1, c_{1}, c_{2}^{2}, −4c_{3}^{2}]  1 [2, 2]  {C5, C36, C65, C67, \atop C72, C75}
C63  [1, c_{1}, c_{2}, c_{2}^{2}]  1 [2, 2]  {C6, C38, C64, C65, \atop C71, C72}
C64  [1, c_{1}, c_{2}^{2}, 1]  1 [4, 2, 2]  C9, C40, C68, C73
C65  [1, c_{1}, −4c_{2}^{2}, 1]  1 [4, 2, 2]  C10, C40, C69, C76
C66  [1, c_{1}, c_{2}, −4c_{2}^{2}]  1 [4]  C7, C37, C67, C75
C67  [1, c_{1}, c_{2}^{2}, −4]  1 [4, 2]  C13, C43, C69, C77
C68  [1, c_{1}, 1, 1]  1 [4, 4, 2, 2] C11, C41
C69  [1, c_{1}, 1, −4]  1 [4, 4, 4]  C14, C44
C70  [1, c_{1}, c_{2}, c_{2}^{2}c_{1}^{2}]  2 [2]  C8, C39, C71, C72
C71  [1, c_{1}^{2}, c_{2}, c_{2}^{2}]  2 [2, 2]  C9, C40, C73
C72  [1, −4c_{1}^{2}, c_{2}, c_{2}^{2}]  2 [2, 2]  C10, C40, C77
C73  [1, c_{1}, 1, c_{1}^{2}]  3 [2, 2]  C11, C41
C74  [1, c_{1}, c_{2}, −4c_{2}^{2}c_{1}^{2}]  2 []  C12, C42, C75
C75  [1, c_{1}^{2}, c_{2}, −4c_{2}^{2}]  2 [2]  C13, C43, C76, C77
C76  [1, c_{1}, 1, −4c_{1}^{2}]  3 [4]  C14, C44
C77  [1, c_{1}, −4, c_{1}^{2}]  3 [2]  C14, C44

A.4 k ∩ ℚ(ϵ) = ℚ(\sqrt{−2})

Number Example  ρ Br_{1}  Subcases
a_{0}a_{1}a_{2}a_{3} square
D1  [1, c_{1}, c_{2}, c_{3}^{2}c_{2}^{3}c_{1}^{3}]  2 [2]  D2, D4, D15, D22

Number Example  ρ Br_{1}  Subcases
D2  [1, c_{1}^{2}, c_{2}, c_{3}^{2}c_{2}^{3}]  2 [2, 2, 2]  {D3, D5, D6, D8, \atop D16, D23}
D3  [1, c_{1}^{2}, c_{2}^{2}, c_{3}^{2}]  2 [2, 2, 2, 2, 2] D9, D25
D4  [1, −4c_{1}^{2}, c_{2}, −4c_{3}^{2}c_{2}^{3}] 2 [2]  {D5, D7, D12, D19, \atop D28}
D5  [1, c_{1}^{2}, −4c_{2}^{2}, −4c_{3}^{2}]  2 [2, 2, 2]  D10, D13, D30
D6  [1, c_{1}, c_{2}^{2}c_{1}^{3}, c_{1}^{2}]  2 [2, 2, 2]  D9, D10, D18, D24
D7  [1, c_{1}, −4c_{2}^{2}c_{1}^{3}, −4c_{1}^{2}] 2 [2, 2]  D13, D21, D29
D8  [1, c_{1}^{2}, c_{2}, c_{2}]  3 [2, 2]  D9, D10, D17, D24
D9  [1, c_{1}^{2}, c_{2}^{2}, 1]  3 [2, 2, 2, 2] D11, D26
D10  [1, −4c_{1}^{2}, −4c_{2}^{2}, 1]  3 [2, 2, 2, 2] D14, D31
D11  [1, c_{1}^{2}, 1, 1]  5 [2, 2, 2]  D27
D12  [1, −4c_{1}^{2}, c_{2}, −4c_{2}]  3 [2]  D13, D20, D29
D13  [1, c_{1}^{2}, −4c_{2}^{2}, −4]  3 [2, 2, 2]  D14, D32
D14  [1, c_{1}^{2}, −4, −4]  5 [2, 2, 2]  D33
D15  [1, c_{1}, c_{2}, c_{2}^{3}c_{1}]  3 []  D16, D19, D23, D28
D16  [1, c_{1}, c_{2}^{2}c_{1}^{3}, c_{2}^{2}c_{1}^{2}]  4 [2]  D17, D18, D25, D30
D17  [1, 1, c_{2}, c_{2}]  6 []  D26, D31
D18  [1, c_{1}^{2}, c_{2}^{2}c_{1}^{3}, c_{2}^{2}c_{1}]  4 [2, 2]  D26, D31
D19  [1, c_{1}, −4c_{2}^{2}c_{1}^{3}, −4c_{2}^{2}c_{1}^{2}] 4 []  D20, D21, D30
D20  [1, −4, c_{2}, −4c_{2}]  6 []  D32
D21  [1, −4c_{1}^{2}, c_{2}^{2}c_{1}^{3}, −4c_{2}^{2}c_{1}] 4 [2]  D32
D22  [1, c_{1}, c_{2}, c_{2}^{3}c_{1}^{3}]  2 [8]  D23, D28
D23  [1, c_{1}^{2}, c_{2}, c_{2}^{3}c_{1}^{2}]  3 [4, 2]  D24, D25, D30
D24  [1, 1, c_{2}, c_{2}^{3}]  4 [4]  D26, D31
D25  [1, c_{1}^{2}, c_{2}^{2}, c_{2}^{2}c_{1}^{2}]  5 [2, 2, 2]  D26
D26  [1, 1, c_{2}^{2}, c_{2}^{2}]  7 [2, 2]  D27
D27  [1, 1, 1, 1]  11 []
D28  [1, −4c_{1}^{2}, c_{2}, −4c_{2}^{3}c_{1}^{2}] 3 [4]  D29, D30
D29  [1, −4, c_{2}, −4c_{2}^{3}]  4 [4]  D32
D30  [1, c_{1}^{2}, −4c_{2}^{2}, −4c_{2}^{2}c_{1}^{2}] 5 [4]  D31, D32
D31  [1, 1, −4c_{2}^{2}, −4c_{2}^{2}]  7 [2, 2]  D33

Number Example  ρ Br_{1}  Subcases
D32  [1, −4, c_{2}^{2}, −4c_{2}^{2}]  7 [4]  D33
D33  [1, 1, −4, −4]  11 [2]

−a_{0}a_{1}a_{2}a_{3} square

D34  [1, c_{1}, c_{2}, −4c_{3}^{2}c_{2}^{3}c_{1}^{3}]  1 [2]  D35, D45, D49
D35  [1, c_{1}^{2}, c_{2}, −4c_{3}^{2}c_{2}^{3}]  1 [2, 2]  {D36, D37, D38, D39, \atop D42, D46, D50}
D36  [1, c_{1}^{2}, c_{2}^{2}, −4c_{3}^{2}]  1 [2, 2, 2, 2] D40, D43, D53
D37  [1, c_{1}, c_{2}^{2}c_{1}^{3}, −4c_{1}^{2}]  1 [4, 2]  D43, D48, D51
D38  [1, c_{1}, −4c_{2}^{2}c_{1}^{3}, c_{1}^{2}]  1 [2, 2, 2]  D40, D48, D52
D39  [1, −4c_{1}^{2}, c_{2}, c_{2}]  2 [2, 2]  D40, D47, D51
D40  [1, c_{1}^{2}, −4c_{2}^{2}, 1]  2 [2, 2, 2, 2] D41, D44, D54
D41  [1, −4c_{1}^{2}, 1, 1]  4 [2, 2, 2]  D55
D42  [1, c_{1}^{2}, c_{2}, −4c_{2}]  2 [2]  D43, D47, D52
D43  [1, c_{1}^{2}, c_{2}^{2}, −4]  2 [2, 2, 2]  D44, D54
D44  [1, c_{1}^{2}, 1, −4]  4 [2, 2, 2]  D55
D45  [1, c_{1}, c_{2}, −4c_{2}^{3}c_{1}]  2 []  D46, D50
D46  [1, c_{1}, c_{2}^{2}c_{1}^{3}, −4c_{2}^{2}c_{1}^{2}]  3 []  D47, D48, D53
D47  [1, 1, c_{2}, −4c_{2}]  5 []  D54
D48  [1, c_{1}^{2}, −4c_{2}^{2}c_{1}^{3}, c_{2}^{2}c_{1}]  3 [2]  D54
D49  [1, c_{1}, c_{2}, −4c_{2}^{3}c_{1}^{3}]  1 [8]  D50
D50  [1, c_{1}^{2}, c_{2}, −4c_{2}^{3}c_{1}^{2}]  2 [4]  D51, D52, D53
D51  [1, 1, c_{2}, −4c_{2}^{3}]  3 [2, 2]  D54
D52  [1, −4, c_{2}, c_{2}^{3}]  3 [4]  D54
D53  [1, c_{1}^{2}, c_{2}^{2}, −4c_{2}^{2}c_{1}^{2}]  4 [2]  D54
D54  [1, 1, c_{2}^{2}, −4c_{2}^{2}]  6 [2]  D55
D55  [1, 1, 1, −4]  10 []

Other cases

D56  [1, c_{1}, c_{2}, c_{3}]  1 [2]  D1, D34, D57, D58
D57  [1, c_{1}, c_{2}, −4c_{3}^{2}]  1 [2]  {D4, D35, D60, D62, \atop D66, D74}

Number Example  ρ Br_{1}  Subcases
D58  [1, c_{1}, c_{2}, c_{3}^{2}]  1 [2, 2]  {D2, D35, D59, D61, \atop D62, D63, D70}
D59  [1, c_{1}, c_{2}, 1]  1 [4, 2]  D8, D39, D64, D65
D60  [1, c_{1}, c_{2}, −4]  1 [4]  D12, D42, D67
D61  [1, c_{1}, c_{2}^{2}, c_{3}^{2}]  1 [2, 2, 2]  D3, D36, D64, D71
D62  [1, c_{1}, c_{2}^{2}, −4c_{3}^{2}]  1 [2, 2]  {D5, D36, D65, D67, \atop D72, D75}
D63  [1, c_{1}, c_{2}, c_{2}^{2}]  1 [2, 2]  {D6, D38, D64, D65, \atop D71, D72}
D64  [1, c_{1}, c_{2}^{2}, 1]  1 [4, 2, 2]  D9, D40, D68, D73
D65  [1, c_{1}, −4c_{2}^{2}, 1]  1 [4, 2, 2]  D10, D40, D69, D76
D66  [1, c_{1}, c_{2}, −4c_{2}^{2}]  1 [4]  D7, D37, D67, D75
D67  [1, c_{1}, c_{2}^{2}, −4]  1 [4, 2]  D13, D43, D69, D77
D68  [1, c_{1}, 1, 1]  1 [4, 4, 4]  D11, D41
D69  [1, c_{1}, 1, −4]  1 [4, 4, 4]  D14, D44
D70  [1, c_{1}, c_{2}, c_{2}^{2}c_{1}^{2}]  2 [2]  D8, D39, D71, D72
D71  [1, c_{1}^{2}, c_{2}, c_{2}^{2}]  2 [2, 2]  D9, D40, D73
D72  [1, −4c_{1}^{2}, c_{2}, c_{2}^{2}]  2 [2, 2]  D10, D40, D77
D73  [1, c_{1}, 1, c_{1}^{2}]  3 [4]  D11, D41
D74  [1, c_{1}, c_{2}, −4c_{2}^{2}c_{1}^{2}]  2 []  D12, D42, D75
D75  [1, c_{1}^{2}, c_{2}, −4c_{2}^{2}]  2 [2]  D13, D43, D76, D77
D76  [1, c_{1}, 1, −4c_{1}^{2}]  3 [4]  D14, D44
D77  [1, c_{1}, −4, c_{1}^{2}]  3 [2]  D14, D44

A.5 k ∩ ℚ(ϵ) = ℚ(ϵ)

Number Example  ρ Br_{1}  Subcases
a_{0}a_{1}a_{2}a_{3} square
E1  [1, c_{1}, c_{2}, c_{3}^{2}c_{2}^{3}c_{1}^{3}] 2 [2]  E2, E8, E12
E2  [1, c_{1}^{2}, c_{2}, c_{3}^{2}c_{2}^{3}] 2 [2, 2, 2]  E3, E4, E5, E9, E13
E3  [1, c_{1}^{2}, c_{2}^{2}, c_{3}^{2}]  2 [2, 2, 2, 2, 2, 2] E6, E15

Number Example  ρ Br_{1}  Subcases
E4  [1, c_{1}, c_{2}^{2}c_{1}^{3}, c_{1}^{2}] 2 [2, 2, 2, 2]  E6, E11, E14
E5  [1, c_{1}^{2}, c_{2}, c_{2}]  4 [2, 2]  E6, E10, E14
E6  [1, c_{1}^{2}, c_{2}^{2}, 1]  4 [2, 2, 2, 2, 2, 2] E7, E16
E7  [1, c_{1}^{2}, 1, 1]  8 [2, 2, 2, 2, 2] E17
E8  [1, c_{1}, c_{2}, c_{2}^{3}c_{1}] 4 []  E9, E13
E9  [1, c_{1}, c_{2}^{2}c_{1}^{3}, c_{2}^{2}c_{1}^{2}] 6 []  E10, E11, E15
E10  [1, 1, c_{2}, c_{2}]  10 []  E16
E11  [1, c_{1}^{2}, c_{2}^{2}c_{1}^{3}, c_{2}^{2}c_{1}] 6 [2, 2]  E16
E12  [1, c_{1}, c_{2}, c_{2}^{3}c_{1}^{3}] 2 [8, 4]  E13
E13  [1, c_{1}^{2}, c_{2}, c_{2}^{3}c_{1}^{2}] 4 [4, 4]  E14, E15
E14  [1, 1, c_{2}, c_{2}^{3}]  6 [4, 2]  E16
E15  [1, c_{1}^{2}, c_{2}^{2}, c_{2}^{2}c_{1}^{2}] 8 [2, 2]  E16
E16  [1, 1, c_{2}^{2}, c_{2}^{2}]  12 [2, 2]  E17
E17  [1, 1, 1, 1]  20 []

Other cases

E18  [1, c_{1}, c_{2}, c_{3}]  1 [2]  E1, E19
E19  [1, c_{1}, c_{2}, c_{3}^{2}]  1 [2, 2]  {E2, E20, E21, E22, \atop E25}
E20  [1, c_{1}, c_{2}, 1]  1 [4, 4]  E5, E23
E21  [1, c_{1}, c_{2}^{2}, c_{3}^{2}]  1 [4, 2, 2]  E3, E23, E26
E22  [1, c_{1}, c_{2}, c_{2}^{2}]  1 [4, 2]  E4, E23, E26
E23  [1, c_{1}, c_{2}^{2}, 1]  1 [4, 4, 2, 2]  E6, E24, E27
E24  [1, c_{1}, 1, 1]  1 [4, 4, 4, 4, 4, 2] E7
E25  [1, c_{1}, c_{2}, c_{2}^{2}c_{1}^{2}] 3 []  E5, E26
E26  [1, c_{1}^{2}, c_{2}, c_{2}^{2}]  3 [2, 2]  E6, E27
E27  [1, c_{1}, 1, c_{1}^{2}]  5 [2, 2]  E7

Appendix B

Locally soluble surfaces with no small solutions

This appendix lists the surfaces of the form

ax^{4} + by^{4} = cu^{4} + dv^{4}

where a, b, c and d are no greater than 15, the equation is locally soluble, but no integer solution of height less than 10^{4} exists. These data are substantially the same as those obtained by Pinch [31]; the only difference is that there only solutions of height up to 300 were sought, or 1000 in some specific cases.

The surfaces are listed as (a, b, c, d) and are grouped according to their case, as classified in the tables in Appendix A. Before each case, we give the structure of Br_{1} / Br_{0} for that case.

Case A8 [2]

(7, 8, 9, 14)

Case A19 [2, 2, 2]

(4, 9, 8, 8)

Case A76 [8]

(2, 9, 6, 12)

Case A109 [2]

(4, 12, 10, 15)

Case A112 [2]

(4, 14, 7, 9) (4, 7, 9, 14)

Case A118 [4, 2, 2]

(1, 4, 8, 9)

Case A121 [2]

(1, 1, 6, 12) (2, 4, 11, 11) (3, 6, 11, 11) (3, 6, 8, 8)

(6, 12, 11, 11)

Case A123 [2, 2]

(2, 4, 9, 9)

Case A127 []

(3, 14, 7, 12) (3, 7, 12, 14)

Case A222 [2]

(1, 14, 6, 11) (1, 15, 10, 11) (1, 15, 11, 13) (1, 15, 11, 14)
(1, 15, 7, 11) (1, 7, 5, 12) (1, 7, 6, 11) (11, 14, 12, 15)
(2, 10, 7, 11) (2, 11, 12, 15) (2, 11, 3, 15) (2, 11, 6, 10)
(2, 12, 7, 11) (2, 13, 5, 11) (2, 13, 6, 10) (2, 14, 10, 15)
(2, 14, 5, 12) (2, 14, 6, 13) (2, 15, 7, 11) (2, 3, 7, 11)
(2, 6, 5, 11) (2, 6, 5, 13) (2, 6, 7, 11) (2, 6, 7, 15)
(2, 7, 10, 15) (2, 7, 11, 12) (3, 13, 10, 15) (3, 13, 7, 8)
(3, 13, 8, 14) (3, 7, 4, 15) (3, 7, 8, 11) (3, 8, 13, 14)
(4, 11, 7, 13) (4, 12, 11, 13) (4, 12, 11, 14) (4, 12, 5, 14)
(4, 12, 7, 10) (4, 15, 6, 14) (4, 5, 11, 14) (4, 5, 12, 13)
(4, 5, 6, 14) (4, 6, 11, 15) (4, 7, 6, 10) (5, 11, 6, 7)
(5, 11, 6, 8) (5, 11, 7, 13) (6, 10, 7, 11) (7, 10, 11, 12)

APPENDIX B. LOCALLY SOLUBLE SURFACES 156

(7, 13, 8, 15) (7, 8, 11, 13) (7, 9, 10, 15) (7, 9, 11, 13)
(7, 9, 12, 13) (7, 9, 12, 15) (8, 12, 11, 13) (8, 15, 11, 13)
(9, 10, 11, 14) (9, 10, 12, 13) (9, 11, 12, 15) (9, 11, 13, 14)

(9, 13, 10, 15)

Case A223 [2, 2]

(4, 9, 11, 14)

Case A224 [2]

(4, 13, 7, 9) (4, 13, 9, 11) (4, 15, 7, 9)

Case A225 [2]

(1, 2, 5, 12) (1, 2, 7, 10) (1, 8, 10, 15) (1, 8, 11, 14)
(1, 8, 6, 11) (2, 14, 3, 6) (2, 5, 6, 12) (2, 7, 6, 12)
(2, 9, 12, 15) (2, 9, 13, 14) (3, 6, 11, 14) (3, 6, 8, 14)
(4, 8, 11, 13) (4, 8, 13, 15) (4, 8, 7, 13) (6, 12, 7, 15)
(7, 14, 11, 12) (8, 9, 10, 14) (8, 9, 11, 13) (8, 9, 11, 14)

Case A226 [2]

(1, 14, 7, 10) (1, 15, 2, 11) (1, 15, 8, 11) (1, 7, 3, 14)
(2, 10, 7, 9) (2, 11, 4, 12) (2, 11, 7, 9) (2, 14, 3, 7)
(2, 14, 5, 7) (2, 14, 6, 7) (2, 14, 7, 10) (2, 14, 7, 12)
(2, 14, 7, 13) (2, 14, 9, 10) (2, 14, 9, 13) (2, 7, 11, 14)
(3, 13, 4, 6) (3, 4, 8, 11) (4, 12, 6, 7) (4, 14, 7, 13)
(4, 7, 6, 14) (5, 14, 7, 9) (6, 10, 12, 13) (6, 14, 7, 15)
(6, 7, 8, 14) (7, 11, 13, 14) (7, 13, 14, 15) (7, 8, 10, 14)
(7, 9, 10, 14) (7, 9, 13, 14) (7, 9, 8, 11)

Case A227 [2, 2]

(2, 2, 7, 11) (4, 11, 10, 10) (4, 4, 11, 13) (4, 4, 7, 13)
(4, 6, 11, 11) (7, 7, 11, 12) (8, 8, 10, 15) (8, 8, 11, 14)

Case A228 [2] APPENDIX B. LOCALLY SOLUBLE SURFACES 157

(1, 12, 3, 14) (1, 14, 4, 13) (2, 3, 8, 11) (3, 13, 12, 14)
(3, 14, 4, 12) (3, 4, 11, 12) (3, 7, 11, 12)
Case A229 [4, 2]
(1, 4, 3, 14) (1, 4, 6, 11) (2, 8, 11, 13) (2, 8, 11, 15)
(2, 8, 3, 15) (2, 8, 5, 11) (2, 8, 5, 13) (2, 8, 7, 11)
Case A234 [2, 2]

(2, 14, 4, 9)

Case A235 [2]

(1, 14, 8, 9) (2, 9, 4, 14) (4, 8, 7, 9)  (4, 8, 9, 11)

(4, 8, 9, 15)

Case A238 [2, 2, 2]

(4, 4, 7, 9)  (4, 4, 9, 11)

Case A240 [4, 2]

(1, 4, 8, 13) (2, 8, 7, 9)  (3, 12, 6, 10) (3, 12, 6, 13)

Case A241 [2]

(4, 12, 9, 10)

Case A248 [2, 2]

(2, 2, 7, 9)  (2, 2, 9, 11) (7, 12, 14, 14) (7, 9, 14, 14)
(8, 8, 9, 11) (8, 8, 9, 13)

Case A249 [4]

(4, 6, 9, 13)

Case A251 [4]

(1, 8, 4, 13) (2, 14, 8, 9)

\begin{aligned}& Case A257 [2, 2] \\ (4, 9, 10, 15) \end{aligned}

Appendix C

Surfaces with trivial Brauer group and no small solutions

This appendix lists surfaces belonging to case A127 of the table in Appendix A. Such surfaces have no Brauer–Manin obstruction arising from the arithmetic part of the Brauer group. Those listed here are everywhere locally soluble and have no rational point of height less than 10^{4}.

The table lists (a, b, c, d), where the surface is of the form

ax^{4} + by^{4} = cu^{4} + dv^{4}.
(7, 54, 28, 27)  (3, 7, 12, 14)  (3, 14, 12, 7)  (21, 2, 84, 1)
(11, 1, 44, 2)  (11, 24, 44, 3)  (11, 9, 44, 72)  (11, 27, 44, 54)
(11, 54, 44, 27) (11, 216, 44, 27) (3, 88, 12, 11)  (33, 1, 132, 2)
(33, 2, 132, 1) (33, 8, 132, 1) (11, 56, 44, 7) (7, 88, 28, 11)
(11, 21, 44, 42) (11, 42, 44, 21) (11, 21, 44, 168) (7, 33, 28, 264)
(33, 7, 132, 56) (21, 22, 84, 11)  (17, 1, 68, 2)  (17, 8, 68, 1)
(17, 3, 68, 6)  (17, 6, 68, 3)  (17, 3, 68, 24)  (17, 72, 68, 9)
(17, 27, 68, 216) (3, 17, 12, 136) (9, 136, 36, 17)  (17, 7, 68, 56)
(17, 168, 68, 21) (21, 34, 84, 17) (17, 11, 68, 88) (11, 17, 44, 34)
(11, 34, 44, 17) (11, 17, 44, 136) (17, 33, 68, 66) (33, 17, 132, 34)
(33, 34, 132, 17) (33, 136, 132, 17) (1, 19, 4, 38)  (19, 1, 76, 8)

APPENDIX C. SURFACES WITH TRIVIAL BRAUER GROUP 160

\begin{matrix}(19, 24, 76, 3) & (19, 9, 76, 18) & (19, 18, 76, 9) & (19, 9, 76, 72) \\ (19, 216, 76, 27) & (3, 19, 12, 38) & (3, 152, 12, 19) & (19, 7, 76, 14) \\ (19, 14, 76, 7) & (19, 56, 76, 7) & (19, 21, 76, 168) & (21, 19, 84, 152) \\ (19, 88, 76, 11) & (11, 152, 44, 19) & (19, 33, 76, 264) & (33, 19, 132, 38) \\ (33, 38, 132, 19) & (33, 19, 132, 152) & (17, 19, 68, 38) & (17, 38, 68, 19) \\ (17, 19, 68, 152) & (1, 23, 4, 184) & (23, 27, 92, 54) & (23, 54, 92, 27) \\ (23, 216, 92, 27) & (9, 23, 36, 46) & (9, 23, 36, 184) & (23, 7, 92, 14) \\ (7, 23, 28, 46) & (7, 184, 28, 23) & (23, 42, 92, 21) & (23, 22, 92, 11) \\ (11, 184, 44, 23) & (33, 23, 132, 184) & (17, 23, 68, 184) & (23, 152, 92, 19) \\ (19, 23, 76, 46) & (19, 46, 76, 23) & (19, 184, 76, 23) & (1, 62, 4, 31) \\ (31, 2, 124, 1) & (31, 24, 124, 3) & (31, 9, 124, 72) & (3, 248, 12, 31) \\ (9, 31, 36, 248) & (31, 56, 124, 7) & (7, 248, 28, 31) & (31, 21, 124, 168) \\ (21, 31, 84, 62) & (21, 62, 84, 31) & (21, 31, 84, 248) & (31, 88, 124, 11) \\ (11, 31, 44, 62) & (11, 62, 44, 31) & (11, 248, 44, 31) & (33, 31, 132, 62) \\ (33, 62, 132, 31) & (33, 31, 132, 248) & (31, 17, 124, 136) & (17, 31, 68, 62) \\ (17, 31, 68, 248) & (31, 19, 124, 38) & (19, 248, 76, 31) & (31, 184, 124, 23) \end{matrix}

Appendix D

Implementation of algorithms

This appendix contains the implementations, in GP, of the algorithms described in the text.

D.1 Subgroups of Γ

A subgroup H of Γ is represented by a 2-element row vector. The first element is a 4 × 4 matrix whose columns generate H ∩ Γ′; the second element is either the empty vector, or else a column vector representing an element g of Γ′ such that τg is the fifth generator for H.

D.1.1 Algorithm 3.9

\\ Given a subgroup H in HNF and an element h, reduce h to a
\\ canonical coset representative by subtracting elements of H
coset_reduce(H,h) = forstep(i = 4, 1, -1,
  h -= (h[i] \ H[i,i]) * H[,i]~); h;
\\ Put a subgroup into a canonical form
subgroup_reduce(H) =
{
  for(i = 1, 3, H[1][i,] %= 4; if(H[2] != [], H[2][i] %= 4));
  H[1][4,] %= 2; if(H[2] != [], H[2][4] %= 2);

  H[1] = mathnf(concat(H[1],matdiagonal([4,4,4,2])));
  if(H[2] != [], H[2] = coset_reduce(H[1],H[2]));
  H;
}

D.1.2 Algorithm 3.10

\\ Return a canonical representative of the conjugacy class
\\ of H
conj_reduce(H) =
{
  for(i = 1, 3, H[1][i,] %= 4; if(H[2] != [], H[2][i] %= 4));
  H[1][4,] %= 2; if(H[2] != [], H[2][4] %= 2);
  H[1] = mathnf(concat(H[1],matdiagonal([4,4,4,2])));
  if(H[2] != [],
    H[2] = coset_reduce(mathnf(concat(H[1],2*matid(4))),H[2]));
  H;
}

D.1.3 Finding canonical subgroups

\\ Swap two rows of the subgroup
swap_rows(H,i,j) =
{
  local(t);
  t = H[1][i,]; H[1][i,] = H[1][j,]; H[1][j,] = t;
  if(H[2] != [], t=H[2][i]; H[2][i] = H[2][j]; H[2][j] = t);
  H;
}
\\ Return the element of t which comes first in the lex
\\ ordering
lexmin(t) =
{

local(n, j);
n = matsize(t)[2];
j = 1;
for(i = 2, n, if(lex(t[i], t[j]) == -1, j = i));
t[j];
}
\\ Return a canonical subgroup equivalent to H by
\\ conjugation and permutation of the first three rows.
make_unique1(H) =
{
local(t);
t = vector(6);
t[1] = conj_reduce(H);
t[2] = conj_reduce(swap_rows(t[1],1,2));
t[3] = conj_reduce(swap_rows(t[2],2,3));
t[4] = conj_reduce(swap_rows(t[3],1,2));
t[5] = conj_reduce(swap_rows(t[4],2,3));
t[6] = conj_reduce(swap_rows(t[5],1,2));
lexmin(t);
}
\\ Return a canonical subgroup equivalent to H by conjugation
\\ and projective permutation of variables
make_unique(H) =
{
local(A, B, C, t);
A = [-1,0,0,0;-1,1,0,0;-1,0,1,0;0,0,0,1];
B = [1,-1,0,0;0,-1,0,0;0,-1,1,0;0,0,0,1];
C = [1,0,-1,0;0,1,-1,0;0,0,-1,0;0,0,0,1];
t = vector(4);
t[1] = make_unique1(H);
if(H[2] == [],

  t[2] = make_unique1([A*H[1], []]);
  t[3] = make_unique1([B*H[1], []]);
  t[4] = make_unique1([C*H[1], []]);
,
  t[2] = make_unique1([A*H[1], (A*H[2]~)~]);
  t[3] = make_unique1([B*H[1], (B*H[2]~)~]);
  t[4] = make_unique1([C*H[1], (C*H[2]~)~]);
);
lexmin(t);
}

D.2 Relating surfaces to subgroups

D.2.1 Algorithm 3.11

find_group(a0, a1, a2, a3) =
{
local(f1, f2, f3, S, n, Q, Qe, K1, K2, K3,
  v1, v2, v3, M, v, w1, w2, w3);
f1 = factor(a1/a0);
f2 = factor(a2/a0);
f3 = factor(a3/a0);
\\ First find out which primes interest us
S = Set([-1, 2]);
S = setunion(S, Set(f1[,1]));
S = setunion(S, Set(f2[,1]));
S = setunion(S, Set(f3[,1]));
n = matsize(S)[2];
for(i = 1, n, S[i] = eval(S[i]));
S = vecsort(S);
\\ Now find the subgroups of Q* containing fourth

\\ powers of the various fields
Q = 4 * matid(n); Q[1,1] = 2;
Qe = Q; Qe[1,2] = 1; Qe[2,2] = 2; Qe[2,1] = 2;
v3 = factor2basis(f3, S); K3 = concat(v3, Qe);
v2 = factor2basis(f2, S); K2 = concat(v2, K3);
v1 = factor2basis(f1, S); K1 = concat(v1, K2);
\\ Work out the degrees of the relevant extensions
M = matrix(4,4); v = vector(4);
v[1] = matdet(mathnf(K1));
v[2] = matdet(mathnf(K2));
v[3] = matdet(mathnf(K3));
v[4] = matdet(Qe);
M[1,1] = 4 * v[1] / v[2];
M[2,2] = 4 * v[2] / v[3];
M[3,3] = 4 * v[3] / v[4];
M[4,4] = 1;
\\ Find out how the various extensions are related to
\\ each other
w1 = matsolvemod(K2, 4, (4 / M[1,1]) * v1);
w2 = matsolvemod(K3, 4, (4 / M[2,2]) * v2);
w3 = matsolvemod(Qe, 4, (4 / M[3,3]) * v3);
\\ Work out the Galois group
M[1,2] = M[2,2] * w1[1] * M[1,1] / 4;
M[2,3] = M[3,3] * w2[1] * M[2,2] / 4;
M[1,3] = (M[2,3] * w1[1] + M[3,3] * w1[2]) * M[1,1] / 4;
M[3,4] = 2 * w3[1] * M[3,3] / 4;
M[2,4] = (M[3,4] * w2[1] + 2 * w2[2]) * M[2,2] / 4;
M[1,4] = (M[2,4] * w1[1] + M[3,4] * w1[2] + 2 * w1[3])
  * M[1,1] / 4;
v[1] = v1[1] % 2;

  v[2] = v2[1] % 2;
  v[3] = v3[1] % 2;
  v[4] = 0;
  [mathnf(M), v];
}

D.2.2 Finding the constant field

\\ Return a code for the constant field fixed by H:
\\   1  Q
\\   2  Q(i)
\\   3  Q(\sqrt(2))
\\   4  Q(\sqrt(-2))
\\   5  Q(i,\sqrt(2))
constant_field(H) =
{
  if(H[2] == [], \\ is i fixed?
    if((H[1][4,] % 2) == [0,0,0,0], 5, 2),
    if((H[1][4,] % 2) == [0,0,0,0],
      if((H[2][4] % 2) == 0, 3, 4),
      1
    )
  )
}

D.2.3 Algorithm 3.17

X4 = [[1,1,1,1],[1,I,-1,-I],[1,-1,1,-1],[1,-I,-1,I],[1,1,1,1]];
X2 = [[1,1],[1,-1],[1,1]];
\\ Return a 128-element vector which contains the effect of
\\ applying the group element x to each of the 128 basis
\\ vectors

doelt(x) =
{
local(A,B,C,D);
A = X4[x[1]+1];
B = X4[x[2]+1];
C = X4[x[3]+1];
D = X2[x[4]+1];
tensor(A, tensor(B, tensor(C, D)))[1,];
}
\\ Return a matrix which gives a set of generators for
\\ the subring of B fixed by H, on the (multiplicative) basis
\\ ((1+i), \sqrt(2), t_1, t_2, t_3)
fixed_field(H) =
{
local(M, n, b, c, N);
\\ First find a basis for the fixed part as a module
n = matsize(H[1])[2];
M = matrix(n, 128);
\\ Do the generators which fix i first
for(i = 1, n, M[i,] = doelt(H[1][,i]));
\\ Find out which basis elements were fixed by all of these
b = vector(128);
for(j = 1, 128,
  b[j] = 1;
  for(i = 1, n, if(M[i,j] != 1, b[j] = 0; break));
);
\\ Now see what happens when we apply the generator which
\\ takes i to -i
if(H[2] != [],
  c = doelt(H[2]);
  for(j = 1, 128,
    if(b[j] == 1,

      if(c[j] == I, b[j] = 1-I,
        if(c[j] == -1, b[j] = I,
          if(c[j] == -I, b[j] = 1+I)
        )
      )
    )
  )
);
\\ Next express our basis elements multiplicatively in terms
\\ of (1+i), \sqrt(2), t_1, t_2, t_3
N = matrix(5,129);
for(i = 1, 128,
  if(b[i] != 0,
    N[3,i] = (i - 1) % 4;
    N[4,i] = ((i - 1) \ 4) % 4;
    N[5,i] = ((i - 1) \ 16) % 4;
    N[2,i] = (i - 1) \ 64;
    if(b[i] == 1+I, N[1,i] = 1,
      if(b[i] == I, N[1,i] = 2,
        if(b[i] == 1-I, N[1,i] = 3)
      )
    )
  )
);
\\ If i is fixed, stick that in too
if(H[2] == [], N[1,129] = 2);
\\ Now find a set of multiplicative generators
N = mathnf(concat(N, matdiagonal([4,2,4,4,4])));
}

D.2.4 Algorithm 3.18

\\ Take powers of 4 out of an integer
remove4s(n) =
{
local(f);
f = factor(n); f[,2] %= 4; factorback(f);
}
\\ Give general expressions for surface (1, a_1, a_2, a_3) to
\\ give subgroup H
example(H) =
{
local(N, v, n, c);
N = concat([0,0,0;0,0,0;1,0,0;0,1,0;0,0,1],fixed_field(H)~)~;
for(i=3,5,
  N[,i] *= (4 / N[i+3,i]);
  for(j=3, i-1, N[,i] -= (N[j+3,i]/N[j+3,j])*N[,j]);
);
v = vector(4);
v[1] = 1;
for(i=3, 5,
  v[i-1] = 'c1^(N[1,i] % 4) * 'c2^(N[2,i] % 4)
    * 'c3^(N[3,i] % 4);
  n = (1+I)^N[4,i] * ('r2^N[5,i] % ('r2^2-2));
  c = gcd(real(content(n)), imag(content(n)));
  n /= (c / remove4s(c));
  v[i-1] *= n;
);
v;
}

D.3 Computing with G-modules

D.3.1 Algorithm 3.19

\\ Find H^0(G, M)
H0(M) =
{
local(n,r,U);
n = matsize(M)[2];
r = matsize(M[1])[1];
U = matker(M[1] - matid(r));
for(i=2,n,
  U = matintersect(U, matker(M[i] - matid(r)));
);
matrixqz(U,-2);
}

D.3.2 Working modulo m

\\ Find H^0(G, M \otimes Z/mZ)
H0torsion(M, m) =
{
local(n,r,U);
n = matsize(M)[2];
r = matsize(M[1])[1];
U = kermod(M[1] - matid(r), m);
for(i=2,n,
  U = latticeintersect(U, kermod(M[i] - matid(r), m));
);
U;
}

D.3.3 Algorithm 3.20

\\ Find a set of generators for H^1(G, M)[m], given as elements
\\ of M \otimes Q/Z
H1torsion(M, m) =
{
local(V,W,A,d,U);
W = H0torsion(G,m);
V = mathnf(concat(H0(G),
  matdiagonal(vector(matsize(M)[1],i,m))));
\\ Find the quotient group and put it into Smith normal form
A = matsnf(M^-1*N,1);
d = matsnf(A[3],4);
\\ Extract the generators
if(d == [], U = [;],
  U = vecextract(M*A[1]^-1, Str("1.." matsize(d)[2])));
[d, U/m];
}

D.3.4 Algorithm 3.22

\\ Represent the element g on the basis
\\   { \epsilon^n t_j/t_i | n=1,3,5,7 }
small_rep(g,i,j) =
{
local(b);
b = 2*g[4];
if(j > 0, b += g[j]);
if(i > 0, b -= g[i]);
b %= 4;
if(b == 0, matid(4),
  if(b == 1, [0,0,0,1;1,0,0,0;0,1,0,0;0,0,1,0],
    if(b == 2, [0,0,1,0;0,0,0,1;1,0,0,0;0,1,0,0],
      [0,1,0,0;0,0,1,0;0,0,0,1;1,0,0,0]

      )
    )
  );
}
small_rep_conj(g,i,j) \
  = [0,0,0,1;0,0,1,0;0,1,0,0;1,0,0,0] * small_rep(g,i,j);
\\ Represent the element g on the basis
\\  { L^{pqr}_{mn} | m,n=1,3,5,7 }
big_rep(g,p) = tensor(small_rep(g,(p%3)+1,((p+1)%3)+1), \
                      small_rep(g,0,p));
big_rep_conj(g,p) = tensor(\
  small_rep_conj(g,(p%3)+1,((p+1)%3)+1), \
  small_rep_conj(g,0,p));
\\ Put three 16-row matrices together into a 48-row one
concat48(M1,M2,M3) =
{
  M1 = concat(M1~,matrix(matsize(M1)[2],32))~;
  M2 = concat(matrix(matsize(M2)[2],16),
    concat(M2~,matrix(matsize(M2)[2],16)))~;
  M3 = concat(matrix(matsize(M3)[2],32),M3~)~;
  concat(M1,concat(M2,M3));
}
rep48(g) = concat48(big_rep(g,1), big_rep(g,2), big_rep(g,3));
rep48_conj(g) = concat48(big_rep_conj(g,1), \
  big_rep_conj(g,2), big_rep_conj(g,3));

D.3.5 Calculating a basis for Pic V̄

intersect(a,m,n,b,i,j) =
{
  if(a == b,
    if((m == i) && (n == j),
      -2,
      if((m == i) || (n == j), 1, 0)
    ),
    if(Mod(b,3) == Mod(a+1,3),
      if(Mod(m-n,8) == Mod(i+j,8),1,0),
      if(Mod(i-j,8) == Mod(m+n,8),1,0)
    )
  )
}
Q = matrix(48,48);
{
  for(a=0,2,
    for(m=0,3,
      for(n=0,3,
        for(b=0,2,
          for(i=0,3,
            for(j=0,3,
              Q[1+a*16+m*4+n,1+b*16+i*4+j] =
        intersect(a,2*m+1,2*n+1,b,2*i+1,2*j+1)
            )
          )
        )
      )
    )
  )
}
\\ A0 = Z-basis for kernel of Q

A0 = matrixqz(matker(Q),-2);
\\ P = plane section ( = Z-basis for positive definite part)
P = [1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,\
      0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]~;
\\ B = Q-basis for negative definite part
B = matintersect(vecextract(matsupplement(A0),"29..48"),\
                 matker(Mat(P~*Q)));
\\ C = matrix which projects onto the subspace we want
C1 = concat(P,concat(B,A0));
C2 = matdiagonal([1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,\
   0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]);
C = C1*C2*C1^-1;
\\ Find a Z-basis for the lattice generated by the
\\ columns of C; this is a basis for NS(V)
D = mathnf(C);
\\ E = projection onto NS(V) (a left inverse for D)
E = matleftinverse(D) * C;

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