ON THE BRAUER–MANIN OBSTRUCTION FOR
CUBIC SURFACES^{∗}

Andreas-Stephan Elsenhans^{#}and Jörg Jahnel^{�}

#Math. Inst., Universität Bayreuth, D-95440 Bayreuth, Germany �FB6 Mathematik, Walter-Flex-Str. 3, D-57068 Siegen, Germany

Received: July 7, 2009; Accepted: October 27, 2009

Abstract

We describe a method to compute the Brauer-Manin obstruction for smooth cubic surfaces over ℚ such that Br( S) / Br( ℚ) is of order two or four. This covers the vast majority of the cases when this group is non-zero. Our approach is to associate a Brauer class with every Galois invariant double-six. We show that all order two Brauer classes may be obtained in this way. We also recover Sir Peter Swinnerton-Dyer’s result that Br( S) / Br( ℚ) may take only �ve values.

2000 Mathematics Subject Classi�cation: Primary 11D25; Secondary 11D85, 11G35.

1 Introduction

1.1. –––– For cubic surfaces, weak approximation and even the Hasse principle are not always ful�lled. The �rst example of a cubic surface violating the Hasse principle was constructed by Sir Peter Swinnerton-Dyer [SD1]. A series of examples generalizing that of Swinnerton-Dyer is due to L. J. Mordell [Mo]. An example of a different sort was given by J. W. S. Cassels and M. J. T. Guy [CG].

A way to explain these examples in a uni�ed manner was provided by Yu. I. Manin in his book [Ma]. This is what today is called the Brauer-Manin obstruction. Manin’s idea is that a non-trivial Brauer class may be responsible for the failure of weak approximation. We will recall the Brauer-Manin obstruction in some detail in section 2.

An important point is that only the factor group Br( S) / Br( ℚ) of the Grothendieck-Brauer group of the cubic surface S is relevant. That is isomorphic to the Galois cohomology group H ^{1}( Gal( \overline{ℚ}/ ℚ) , Pic( S\overline{ℚ} ) ) . A theorem of Sir Peter Swinnerton-Dyer [SD2] states that, for this group, there are only �ve possibilities. It may be isomorphic to 0, ℤ/ 2ℤ, ℤ/ 2ℤ × ℤ/ 2ℤ, ℤ/ 3ℤ, or ℤ/ 3ℤ × ℤ/ 3ℤ. We observed that, today, Swinnerton-Dyer’s theorem from 1993 may easily be established by a script in ���.

∗The computer part of this work was executed on the Sun Fire V20z Servers of the Gauß Laboratory for Scienti�c Computing at the Göttingen Mathematical Institute. Both authors are grateful to Prof. Y. Tschinkel for the permission to use these machines as well as to the system administrators for their support.

#E-mail: Stephan.Elsenhans@uni-bayreuth.de; Website: ���������������������������������∼ ������

�E-mail: ~jjahnel@mathematik.uni-siegen.de; Website: ����������������������������������

The effect of the Brauer-Manin obstruction has been studied by several authors. For example, for diagonal cubic surfaces, the computations were carried out by J.-L. Colliot-Thélène and his coworkers in [CKS]. In this case, Br( S) / Br( ℚ) = ℤ/ 3ℤ. The same applies to the examples of Mordell or Cassels-Guy which were explained by the Brauer-Manin obstruction in [Ma].

1.2. –––– It seems that, for the cases that H ^{1}( Gal( \overline{ℚ}/ ℚ) , Pic( S\overline{ℚ} ) ) ≅ ℤ/ 2ℤ or ℤ/ 2ℤ × ℤ/ 2ℤ, no computations have been done up to now. The goal of the present paper is to �ll this gap.

Our starting point is a somewhat surprising observation. It turns out that H ^{1}( Gal( \overline{ℚ}/ ℚ) , Pic( S\overline{ℚ} ) ) is of order two or four only in cases when, on S, there is a Galois invariant double-six. This reduces the possibilities for the action of Gal( \overline{ℚ}/ ℚ) on the 27 lines. In general, the automorphism group of the con�guration of the 27 lines is the Weyl group W( E_{6}) [Ma, Theorem 23.9]. Among the 350 conjugacy classes of subgroups in W( E_{6}) , exactly 158 stabilize a double-six.

In a previous paper [EJ2], we described a method to construct smooth cubic surfaces with a Galois invariant double-six. Our method is based on the hexahedral form of L. Cremona and Th. Reye and an explicit Galois descent. It is able to produce examples for each of the 158 conjugacy classes.

Among them, however, there are 56 which even stabilize a sixer. Those may be constructed by blowing up six points in P^{2} and, thus, certainly ful�ll weak approximation. There are 26 further conjugacy classes which lead to H ^{1}( Gal( \overline{ℚ}/ ℚ) , Pic( S\overline{ℚ} ) ) = 0.

1.3. –––– In this article, we compute the Brauer-Manin obstruction for each of the 76 cases such that H ^{1}( Gal( \overline{ℚ}/ ℚ) , Pic( S\overline{ℚ} ) ) ≅ ℤ/ 2ℤ or ℤ/ 2ℤ × ℤ/ 2ℤ. We start with two “model cases” for the Brauer groups ℤ/ 2ℤ and ℤ/ 2ℤ × ℤ/ 2ℤ. These are the maximal subgroup U_{1} ⊂ W( E_{6}) stabilizing a double-six and the maximal subgroup U_{3} ⊂ W( E_{6}) stabilizing a triple of azygetic double sixes [Ko].

In both cases, we compute the Brauer group explicitly. This means, we produce representatives which we describe as Azumaya algebras. We then show that every subgroup H ⊂ W( E_{6}) which leads to a Brauer group of order four is actually contained in U_{3}. Recall that every subgroup H ⊂ W( E_{6}) leading to a Brauer group of order two is contained in U_{1}. Finally, we prove the main result that the restriction map is bijective in each of the cases.

1.4. –––– The article is concluded by examples showing the effect of the Brauer-Manin obstruction. It turns out that, unlike the situation described in [CKS] where a Brauer class of order three typically excludes two thirds of the adelic points, various fractions are possible.

2 The Brauer-Manin obstruction – Generalities

2.1. –––– For cubic surfaces, all known counterexamples to the Hasse principle or weak approximation are explained by the following observation.

2.2. De�nition. –––– Let X be a projective variety over ℚ and Br( X ) its Grothendieck-Brauer group. Then, we will call

ev_{ν} : Br( X ) × X ( ℚ_{ν} ) − → ℚ/ ℤ, ( α , ξ ) ↦ inv_{ν} ( α | _{ξ} )

the local evaluation map. Here, inv_{ν} : Br( ℚ_{ν} ) → ℚ/ ℤ (and inv_{∞} : Br( ℝ) → \frac{1}{2}ℤ/ ℤ) denote the canonical isomorphisms.

2.3. Observation (Manin). —– Let π : X → Spec( ℚ) be a projective variety over ℚ. Choose an element α ∈ Br( X ) . Then, every ℚ-rational point x ∈ X ( ℚ) gives rise to an adelic point ( x_{ν} ) _{ν} ∈ X ( A_{ℚ} ) satisfying the condition

∑_{ν ∈ Val( ℚ)} ev_{ν} ( α , x_{ν} ) = 0.

2.4. Remarks. –––– i) It is obvious that altering α ∈ Br( X ) by some Brauer class π ^{∗} ρ for ρ ∈ Br( ℚ) does not change the obstruction de�ned by α . Consequently, it is only the factor group Br( X ) / π ^{∗} Br( ℚ) which is relevant for the Brauer-Manin obstruction. ii) The local evaluation map ev_{ν} : Br( X ) × X ( ℚ_{ν} ) → ℚ/ ℤ is continuous in the second variable. iii) Further, for every projective variety X over ℚ and every α ∈ Br( X ) , there exists a �nite set S ⊂ Val( ℚ) such that ev( α , ξ ) = 0 for every ν ∉ S and ξ ∈ X ( ℚ_{ν} ) . These facts imply that the Brauer-Manin obstruction, if present, is an obstruction to the principle of weak approximation.

2.5. Lemma. –––– Let π : S → Specℚ be a non-singular cubic surface. Then, there is a canonical isomorphism

δ : H ^{1}( Gal( \overline{ℚ}/ ℚ) , Pic( S\overline{ℚ} ) ) − → Br( S) / π ^{∗} Br( ℚ)

making the diagram

[figure: H ^{1}( Gal( \overline{ℚ}/ ℚ) , Pic( S\overline{ℚ} ) ); δ; d; Br( S) / π ^{∗} Br( ℚ); res; H ^{2}( Gal( \overline{ℚ}/ ℚ) , \overline{ℚ}( S) ^{∗} / \overline{ℚ}^{∗} ); H ^{2}( Gal( \overline{ℚ}/ ℚ) , \overline{ℚ}( S) ^{∗} ) / π ^{∗} Br( ℚ); inf; Br( ℚ( S) ) / π ^{∗} Br( ℚ)]

commute. Here, d is induced by the short exact sequence

0 → \overline{ℚ}( S) ^{∗} / \overline{ℚ}^{∗} → Div( S\overline{ℚ} ) → Pic( S\overline{ℚ} ) → 0

and the other morphisms are the canonical ones. Proof. The equality at the lower left corner comes from the fact [Ta, section 11.4] that H ^{3}( Gal( \overline{ℚ}/ ℚ) , \overline{ℚ}^{∗} ) = 0. The main assertion is [Ma, Lemma 43.1.1]. □

2.6. Remark. –––– The group H ^{1}( Gal( \overline{ℚ}/ ℚ) , Pic( S\overline{ℚ} ) ) is always �nite. Hence, by Remark 2.4.iii), we know that only �nitely many primes are relevant for the Brauer-Manin obstruction.

3 One double-six

3.1. –––– Recall that, on a non-singular cubic surface, there are exactly 27 lines. Their con�guration was studied intensively by the geometers of the 19th century. They showed that the 27 lines contain 72 sixers of lines which are mutually skew. For each sixer, there is a complementary one formed by the lines meeting exactly �ve lines of the sixer. A pair of complementary sixers is called a double-six.

The automorphism group W( E_{6}) acts transitively on double-sixes. In the blown-up model, one double-six may easily be written down explicitly. For this, we refer to [Ha, Remark V.4.9.1] from which we also adopt the notation for the lines.

3.2. Lemma. –––– Let S be a non-singular cubic surface over ℚ. Suppose that, under the operation of Gal( \overline{ℚ}/ ℚ) , the 27 lines on S decompose into orbits one of which is of size 15. Then, the complementary twelve lines form a double-six. Proof. The Galois group Gal( \overline{ℚ}/ ℚ) operates via a certain subgroup G ⊆ W( E_{6}) . Our assumption implies that 5| #G. I.e., G contains the 5-Sylow subgroup of W( E_{6}) .

The operation of this is easily described in the blown-up model. The cyclic group ⟨ ( 12345) ⟩ ⊂ S_{6} acts on the indices. The two lines E_{0} and G_{0} are stationary while the others form �ve orbits of size �ve each. These are E := { E_{1}, …, E_{5}} , G := { G_{1}, …, G_{5}} , F _{0} := { F _{01}, …, F _{05}} , F _{1} := { F _{12}, F _{23}, F _{34}, F _{45}, F _{15}} , and, �nally, F _{2} := { F _{13}, F _{24}, F _{35}, F _{14}, F _{25}} .

The intersection matrix of the �ve latter orbits turns out to be

( \begin{matrix}− 5 & 20 & 5 & 10 & 10 \\ 20 & − 5 & 5 & 10 & 10 \\ 5 & 5 & − 5 & 15 & 15 \\ 10 & 10 & 15 & 5 & 5 \\ 10 & 10 & 15 & 5 & 5 \end{matrix} ) .

We assert that a size 15 orbit must be formed by F _{0}, F _{1}, and F _{2}.

Indeed, three orbits of size �ve may be put together to form an orbit only if, for the corresponding divisors, D( D+ D′ + D′′) = D′( D+ D′ + D′′) = D′′( D+ D′ + D′′) . This excludes all combinations, except for E ∪ G ∪ F _{1}, (E ∪ G ∪ F _{2}, and F _{0}∪ F _{1}∪ F _{2}). The set E ∪ G ∪ F _{1} contains, however, only two �vers of skew lines, namely E and G. As the lines in F _{1} are missing, this is a contradiction. □

3.3. Remark. –––– U_{1}, the largest subgroup of W( E_{6}) stabilizing a double-six is isomorphic to S_{6}× ℤ/ 2ℤ of order 1440.

3.4. Notation. –––– Let S be a non-singular cubic surface. Assume that twelve of the 27 lines on S form a double-six which is Gal( \overline{ℚ}/ ℚ) -invariant. Choose such a double-six. Then, there are two kinds of tritangent planes. We have 15 tritangent planes which meet S only within the 15 complementary lines. The other 30 tritangent planes contain one of the 15 lines and two from the double-six.

We write F _{30} for a product over the linear forms de�ning the 30 tritangent planes and F _{15} for a product over the linear forms de�ning the 15 others. Note that F _{30}/ F _{15}^{2} ∈ ℚ( S) .

3.5. Theorem. –––– Let π : S → Specℚ be a non-singular cubic surface such that the 27 lines have orbit structure [ 12, 15] under the operation of Gal( \overline{ℚ}/ ℚ) .

i) Then, Br( S) / π ^{∗} Br( ℚ) = ℤ/ 2ℤ. ii) For 0 ≠ c ∈ Br( S) / π ^{∗} Br( ℚ) , a representative \underline{c} of res( c) ∈ Br( ℚ( S) ) / π ^{∗} Br( ℚ) is given as follows. Consider the quadratic number �eld ℚ( \sqrt{D}) splitting the double-six into two sixers. Then, apply to the class

( F _{30}/ F _{15}^{2}) ∈ Ĥ ^{0}( Gal( ℚ( \sqrt{D}) / ℚ) , ℚ( \sqrt{D}) ( S) ^{∗} ) = ℚ( S) ^{∗} / N ℚ( \sqrt{D}) ( S) ^{∗}

the periodicity isomorphism to H ^{2} and the in�ation map.

Proof. First step. In�ation. We have the isomorphism δ : H ^{1}( Gal( \overline{ℚ}/ ℚ) , Pic( S\overline{ℚ} ) ) → Br( S) / π ^{∗} Br( ℚ) and will work with the group on the left.

An element of the group Gal( \overline{ℚ}/ ℚ) may either �ip the two sixers forming the twelve lines or not. Therefore, there is an index two subgroup stabilizing the sixers. This group corresponds to the quadratic number �eld ℚ( \sqrt{D}) . By Fact 3.6 below, we know H ^{1}( Gal( \overline{ℚ}/ ℚ( \sqrt{D}) ) , Pic( S\overline{ℚ} ) ) = 0. The in�ation-restriction-sequence yields that

inf: H ^{1}( Gal( ℚ( \sqrt{D}) / ℚ) , Pic( S\overline{ℚ} ) ^{Gal( \overline{ℚ} / ℚ( \sqrt{D}) )} ) − → H ^{1}( Gal( \overline{ℚ}/ ℚ) , Pic( S\overline{ℚ} ) )

is an isomorphism.

Second step. Divisors. The orbit structure of the 27 lines under the operation of Gal( \overline{ℚ}/ ℚ( \sqrt{D}) ) is [ 6, 6, 15] . Indeed, when going over to an index two subgroup an orbit of odd size must not split. Denote by E, G, and F the divisors formed by summing over the �rst, second, and third orbit, respectively. E, G, and F clearly de�ne elements in Pic( S\overline{ℚ} ) ^{Gal( \overline{ℚ} / ℚ( \sqrt{D}) )} . Write P for the subgroup generated by these three divisors.

P is of �nite index in Pic( S\overline{ℚ} ) ^{Gal( \overline{ℚ} / ℚ( \sqrt{D}) )} . Indeed, every element of Pic( S\overline{ℚ} ) is an integral linear combination of the divisors given by the 27 lines. Therefore, every element in Pic( S\overline{ℚ} ) ^{Gal( \overline{ℚ} / ℚ( \sqrt{D}) )} is a ℚ-linear combination of E, G, and F and the denominators are at most six or 15.

We claim that the index of P is a divisor of 15. In fact, we have the relation 5E + 5G − 4F ∼ 0. Further, the discriminant of the lattice spanned by E and F is

| \begin{matrix}− 6 & 30 \\ 30 & 75 \end{matrix} | = − 1350 = ( − 6) · 15^{2}.

Consequently, P is of odd index in Pic( S\overline{ℚ} ) ^{Gal( \overline{ℚ} / ℚ( \sqrt{D}) )} . This implies that the natural homomorphism

H ^{1}( Gal( ℚ( \sqrt{D}) / ℚ) , P ) − → H ^{1}( Gal( ℚ( \sqrt{D}) / ℚ) , Pic( S\overline{ℚ} ) ^{Gal( \overline{ℚ} / ℚ( \sqrt{D}) )} )

is a bijection.

Third step. The fundamental class. As Gal( ℚ( \sqrt{D}) / ℚ) is a cyclic group of order two, we have

H ^{2}( Gal( ℚ( \sqrt{D}) / ℚ) , ℤ) ≅ ℤ/ 2ℤ.

Write u for the non-zero element. Then, the periodicity isomorphism is given by

∪ u: Ĥ ^{− 1}( Gal( ℚ( \sqrt{D}) / ℚ) , P ) − → H ^{1}( Gal( ℚ( \sqrt{D}) / ℚ) , P ) .

Observe that, for a cyclic group of order two, this isomorphism is canonical as there is no ambiguity in the choice of u.

Fourth step. Computing Ĥ ^{− 1}. We have P = S/ S_{0} for S := ℤE ⊕ ℤG ⊕ ℤF and S_{0} the group of the principal divisors contained in S. The relation Ĥ ^{− 1}( Gal( ℚ( \sqrt{D}) / ℚ) , S) = 0 follows immediately from the de�- nition. Hence, the short exact sequence

0 → S_{0} → S → P → 0

yields

Ĥ ^{− 1}( Gal( ℚ( \sqrt{D}) / ℚ) , P ) =
= ker( Ĥ ^{0}( Gal( ℚ( \sqrt{D}) / ℚ) , S_{0}) → Ĥ ^{0}( Gal( ℚ( \sqrt{D}) / ℚ) , S) )
= ker( S_{0}^{Gal( ℚ( \sqrt{D}) / ℚ)} / N S_{0} → S^{Gal( ℚ( \sqrt{D}) / ℚ)} / N S)
= ( S_{0}^{Gal( ℚ( \sqrt{D}) / ℚ)} ∩ N S) / N S_{0}.

Here, the norm map acts by the rule N : aE + bG + cF ↦ ( a + b) E + ( a + b) G + 2cF . Hence, N S = ⟨ E + G, 2F ⟩ . Principal divisors are characterized by the property that all intersection numbers are zero. A direct calculation shows

S_{0}^{Gal( ℚ( \sqrt{D}) / ℚ)} ∩ N S = ⟨ 5E + 5G − 4F ⟩ .

The generator is the norm of any divisor of the form aE + ( 5 − a) G − 2F . None of these is principal. Indeed, the intersection number with E is equal to − 6a + 30( 5 − a) − 60 = − 36a + 90 and this terms does not vanish for a ∈ ℤ. Assertion i) is proven.

Fifth step. The representative. We actually constructed a non-zero element c′ ∈ H ^{1}( Gal( \overline{ℚ}/ ℚ) , Pic( S\overline{ℚ} ) ) . The calculations given above show that

d( c′) = ( F _{30}/ F _{15}^{2}) ∪ u.

Indeed, it is easy to see that div( F _{30}/ F _{15}^{2}) = 5E + 5G− 4F . The assertion now follows from the commutative diagram given in Lemma 2.5. □

3.6. Fact. –––– Let S be a non-singular cubic surface over a �eld K obtained by blowing up P_{K}^{2} in six \overline{K} -rational points which form a Galois invariant set. Then, H ^{1}( Gal( \overline{K} / K ) , Pic( S\overline{K} ) ) = 0.

Proof. This is, of course, a particular case of the very general [Ma, Theorem 29.1]. Let us give an elementary proof here.

According to Shapiro’s lemma, we may replace Gal( \overline{K} / K ) by a �nite quotient G. We have Pic( S\overline{K} ) = ℤH ⊕ ℤE_{1}⊕ … ⊕ ℤE_{6} for H the hyperplane section of P^{2} and E_{1}, …, E_{6} the exceptional divisors. Therefore, as a G-module,

Pic( S\overline{K} ) = ℤ ⊕ ℤ[ G/ H _{1}] ⊕ … ⊕ ℤ[ G/ H _{l}]

for l the number of Galois orbits and certain subgroups H _{1}, …, H _{l}. Clearly, we have H ^{1}( G, ℤ) = Hom( G, ℤ) = 0.

On the other hand, for any subgroup H , the G-module ℤ[ G/ H ] is equipped with a nondegenerate pairing. Hence, ℤ[ G/ H ] ≅ Hom( ℤ[ G/ H ] , ℤ) and

H ^{1}( G, ℤ[ G/ H ] ) ≅ H ^{1}( G, Hom( ℤ[ G/ H ] , ℤ) )
≅ Ĥ ^{0}( G, Hom( ℤ[ G/ H ] , ℚ/ ℤ) )
≅ Hom( Ĥ ^{− 1}( G, ℤ[ G/ H ] ) , ℚ/ ℤ)

by the duality theorem for cohomology of �nite groups [CE, Chap. XII, Corollary 6.5]. Finally, Ĥ ^{− 1}( G, ℤ[ G/ H ] ) vanishes as is seen immediately from the de�nition. □

4 Triples of azygetic double-sixes

4.1. De�nition. –––– Let π : S → Specℚ be a smooth cubic surface and 𝒟 be a Galois invariant double-six. This induces a group homomorphism Gal( \overline{ℚ}/ ℚ) → U_{1}, given by the operation on the 27 lines.

i) Then, the image of the non-zero element under the natural homomorphism

ℤ/ 2ℤ ≅ H ^{1}( U_{1}, Pic( S\overline{ℚ} ) ) − → H ^{1}( Gal( \overline{ℚ}/ ℚ) , Pic( S\overline{ℚ} ) ) − →^{δ} Br( S) / π ^{∗} Br( ℚ)

is called the Brauer class associated with the double-six 𝒟. We denote it by cl( 𝒟) . ii) This de�nes a map

cl: Φ _{S}^{Gal( \overline{ℚ} / ℚ)} − → Br( S) / π ^{∗} Br( ℚ)

from the set of all Galois invariant double-sixes which we will call the class map.

4.2. –––– Two double-sixes may have either four or six lines in common. In the former case, the two are called syzygetic, in the latter azygetic. A pair of azygetic double-sixes is built as follows.

( \begin{matrix}E_{0} & E_{1} & E_{2} & E_{3} & E_{4} & E_{5} \\ G_{0} & G_{1} & G_{2} & G_{3} & G_{4} & G_{5} \end{matrix} ) , ( \begin{matrix}E_{0} & E_{1} & E_{2} & F _{45} & F _{35} & F _{34} \\ F _{12} & F _{02} & F _{01} & G_{3} & G_{4} & G_{5} \end{matrix} ) .

The twelve lines which appear only once form a third double-six

( \begin{matrix}F _{12} & F _{02} & F _{01} & E_{3} & E_{4} & E_{5} \\ G_{0} & G_{1} & G_{2} & F _{45} & F _{35} & F _{34} \end{matrix} )

azygetic to the other two.

4.3. –––– The largest group U_{3} stabilizing a triple of azygetic double-sixes is isomorphic to ( S_{3} × S_{3}) ⋉ ℤ/ 2ℤ of order 72. The induced orbit structure is [ 6, 6, 6, 9] . The orbits themselves are, in the notation above, { E_{0}, E_{1}, E_{2}, G_{3}, G_{4}, G_{5}} , { G_{0}, G_{1}, G_{2}, E_{3}, E_{4}, E_{5}} , { F _{01}, F _{02}, F _{12}, F _{34}, F _{35}, F _{45}} , and { F _{03}, F _{04}, F _{05}, F _{13}, F _{14}, F _{15}, F _{23}, F _{24}, F _{25}} .

The quadratic extension ℚ( \sqrt{D}) splitting one of the three double-sixes into two sixers automatically splits the others, too. The operation of Gal( \overline{ℚ}/ ℚ( \sqrt{D}) ) yields the orbits { E_{0}, E_{1}, E_{2}} , { E_{3}, E_{4}, E_{5}} , { G_{0}, G_{1}, G_{2}} , { G_{3}, G_{4}, G_{5}} , { F _{01}, F _{02}, F _{12}} , { F _{34}, F _{35}, F _{45}} , and { F _{03}, F _{04}, F _{05}, F _{13}, F _{14}, F _{15}, F _{23}, F _{24}, F _{25}} .

4.4. Theorem. –––– Let π : S → Specℚ be a non-singular cubic surface. Assume that Gal( \overline{ℚ}/ ℚ) stabilizes a triple \{ 𝒟_{1}, 𝒟_{2}, 𝒟_{3}\} of azygetic double-sixes and that the 27 lines have orbit structure [ 6, 6, 6, 9] .

i) Then, Br( S) / π ^{∗} Br( ℚ) = ℤ/ 2ℤ × ℤ/ 2ℤ. ii) The three non-zero elements are cl( 𝒟_{1}) , cl( 𝒟_{2}) , and cl( 𝒟_{3}) .

Proof. First step. In�ation. We will follow the same strategy as in the proof of Theorem 3.5. In particular, we will work with the group H ^{1}( Gal( ℚ( \sqrt{D}) / ℚ) , Pic( S\overline{ℚ} ) ^{Gal( \overline{ℚ} / ℚ( \sqrt{D}) )} ) .

Second step. Divisors. The orbit structure of the 27 lines under the operation of Gal( \overline{ℚ}/ ℚ( \sqrt{D}) ) is [ 3, 3, 3, 3, 3, 3, 9] . For the lines and double-sixes, we use the notation introduced in 4.2. Further, denote by

E^{( 1)} , E^{( 2)} , G^{( 1)} , G^{( 2)} , F ^{( 1)} , F ^{( 2)} , F ^{( 3)}

the divisors formed by summing over the orbits. These clearly de�ne elements in Pic( S\overline{ℚ} ) ^{Gal( \overline{ℚ} / ℚ( \sqrt{D}) )} . We write P for the subgroup generated by these seven divisors.

Every element of Pic( S\overline{ℚ} ) is an integral linear combination of the divisors given by the 27 lines. Therefore, every element in Pic( S\overline{ℚ} ) ^{Gal( \overline{ℚ} / ℚ( \sqrt{D}) )} is a ℚ-linear combination of E^{( 1)} , E^{( 2)} , G^{( 1)} , G^{( 2)} , F ^{( 1)} , F ^{( 2)} , and F ^{( 3)} and the denominators are divisors of nine. Consequently, P is of odd index in Pic( S\overline{ℚ} ) ^{Gal( \overline{ℚ} / ℚ( \sqrt{D}) )} . This implies that the natural homomorphism

H ^{1}( Gal( ℚ( \sqrt{D}) / ℚ) , P ) − → H ^{1}( Gal( ℚ( \sqrt{D}) / ℚ) , Pic( S\overline{ℚ} ) ^{Gal( \overline{ℚ} / ℚ( \sqrt{D}) )} )

is a bijection.

Third step. The fundamental class. Again, we write u for the non-zero element in H ^{2}( Gal( ℚ( \sqrt{D}) / ℚ) , ℤ) ≅ ℤ/ 2ℤ. Then, the periodicity isomorphism is given by

∪ u: Ĥ ^{− 1}( Gal( ℚ( \sqrt{D}) / ℚ) , P ) − → H ^{1}( Gal( ℚ( \sqrt{D}) / ℚ) , P ) .

Fourth step. Computing Ĥ ^{− 1}. We have P = S/ S_{0} for S := ℤE^{( 1)} ⊕ ℤE^{( 2)} ⊕ ℤG^{( 1)} ⊕ ℤG^{( 2)} ⊕ ℤF ^{( 1)} ⊕ ℤF ^{( 2)} ⊕ ℤF ^{( 3)} and S_{0} the group of the principal divisors contained in S. To simplify formulas, we will use the notation D^{( 1)} := E^{( 1)} + G^{( 2)} , D^{( 2)} := E^{( 2)} + G^{( 1)} , and D^{( 3)} := F ^{( 1)} + F ^{( 2)} .

The relation Ĥ ^{− 1}( Gal( ℚ( \sqrt{D}) / ℚ) , S) = 0 follows immediately from the de�nition. Hence, the short exact sequence 0 → S_{0} → S → P → 0 yields, as above,

Ĥ ^{− 1}( Gal( ℚ( \sqrt{D}) / ℚ) , P ) = ( S_{0}^{Gal( ℚ( \sqrt{D}) / ℚ)} ∩ N S) / N S_{0}.

Here, the norm map acts by the rule

N : a_{1}E^{( 1)} + a_{2}E^{( 2)} + b_{1}G^{( 1)} + b_{2}G^{( 2)} + c_{1}F ^{( 1)} + c_{2}F ^{( 2)} + c_{3}F ^{( 3)}
↦ ( a_{1}+ b_{2}) D^{( 1)} + ( a_{2}+ b_{1}) D^{( 2)} + ( c_{1}+ c_{2}) D^{( 3)} + 2c_{3}F ^{( 3)} .

Hence, N S = ⟨ D^{( 1)} , D^{( 2)} , D^{( 3)} , 2F ^{( 3)} ⟩ . Principal divisors are characterized by the property that all intersection numbers are zero. A direct calculation shows

S_{0}^{Gal( ℚ( \sqrt{D}) / ℚ)} ∩ N S
= { d_{1}D^{( 1)} + d_{2}D^{( 2)} + d_{3}D^{( 3)} + 2eF ^{( 3)} | d_{1}+ d_{2}+ d_{3}+ 3e = 0}
= ⟨ D^{( 1)} − D^{( 2)} , D^{( 1)} − D^{( 3)} , D^{( 1)} + D^{( 2)} + D^{( 3)} − 2F ^{( 3)} ⟩ .

It is easy to see that E^{( 1)} + G^{( 1)} + F ^{( 1)} − F ^{( 3)} is a principal divisor. Hence,

D^{( 1)} + D^{( 2)} + D^{( 3)} − 2F ^{( 3)} ∈ N S_{0}.

Further, N S_{0} contains all principal divisors which are divisible by 2. As it turns out that these two sorts of elements generate the whole of N S_{0}, assertion i) follows.

Fifth step. The representatives. We actually constructed non-zero elements c_{1}, c_{2}, c_{3} ∈ H ^{1}( Gal( \overline{ℚ}/ ℚ) , Pic( S\overline{ℚ} ) ) , represented by D^{( 1)} − D^{( 2)} , D^{( 1)} − D^{( 3)} , D^{( 2)} − D^{( 3)} ∈ S_{0}^{Gal( ℚ( \sqrt{D}) / ℚ)} ∩ N S. The �rst representative is equivalent to

3( D^{( 1)} − D^{( 2)} ) + 2( D^{( 1)} + D^{( 2)} + D^{( 3)} − 2F ^{( 3)} ) + 6( D^{( 2)} − D^{( 3)} ) = div( F _{30}/ F _{15}^{2}) .

Hence, d( c_{1}) = ( F _{30}/ F _{15}^{2}) ∪ u. The corresponding Brauer class is cl( ( {E_{0}… E_{5} \atop G_{0}… G_{5}}) ) . For the two other classes, the situation is analogous. □

4.5. Remark. –––– In the [ 6, 6, 6, 9] -case, the 45 tritangent planes decompose into �ve orbits.

• [ E_{i}, G_{j}, F _{i j}] for i, j ∈ { 0, 1, 2} or i, j ∈ { 3, 4, 5} , i ≠ j. (twelve planes)

• [ E_{i}, G_{j}, F _{i j}] for i ∈ { 0, 1, 2} and j ∈ { 3, 4, 5} . (nine planes)

• [ E_{i}, G_{j}, F _{i j}] for i ∈ { 3, 4, 5} and j ∈ { 0, 1, 2} . (nine planes)

• [ F _{i_{0}i_{1}}, F _{i_{2}i_{3}}, F _{i_{4}i_{5}}] for { i_{0}, i_{1}, i_{2}, i_{3}, i_{4}, i_{5}} = { 0, 1, 2, 3, 4, 5} , i_{0}, i_{1} ∈ { 0, 1, 2} , and i_{2}, i_{3} ∈ { 3, 4, 5} . (nine planes)

• [ F _{i_{0}i_{1}}, F _{i_{2}i_{3}}, F _{i_{4}i_{5}}] for { i_{0}, i_{2}, i_{4}} = { 0, 1, 2} and { i_{1}, i_{3}, i_{5}} = { 3, 4, 5} . (six planes)

The three forms of type F _{30} are obtained by multiplying the linear forms de�ning the orbit of size twelve together with those for two of the orbits of size nine. Actually, the size twelve orbit is irrelevant. Up to a scalar factor, it is the square of a sextic form.

4.6. Remark. –––– Triples of azygetic double-sixes have been studied by the classical algebraic geometers. See, for example, [Ko, §6]. A result from the 19th century states that there are exactly 120 triples of azygetic double-sixes on a smooth cubic surface. Actually, the automorphism group W( E_{6}) acts transitively on them.

5 The general case of a Galois group stabilizing a double-six

5.1. –––– To explicitly compute H ^{1}( G, Pic( S\overline{ℚ} ) ) as an abstract abelian group, one may use Manin’s formula [Ma, Proposition 31.3]. This means the following.

Pic( S\overline{ℚ} ) is generated by the 27 lines. The group of all permutations of the 27 lines respecting the intersection pairing is isomorphic to the Weyl group W( E_{6}) of order 51840. The group G operates on the 27 lines via a �nite quotient G/ H which is isomorphic to a subgroup of W( E_{6}) . Then,

H ^{1}( G, Pic( S\overline{ℚ} ) ) ≅ Hom( N D ∩ D_{0}/ N D_{0}, ℚ/ ℤ) .

Here, D is the free abelian group generated by the 27 lines and D_{0} is the subgroup of all principal divisors. N : D → D denotes the norm map as a G/ H -module.

5.2. –––– Using Manin’s formula, we computed H ^{1}( G, Pic( S\overline{ℚ} ) ) for each of the 350 conjugacy classes of subgroups of W( E_{6}) . The computations in ��� took approximately 28 seconds of CPU time. Thereby, we recovered the following result of Sir Peter Swinnerton-Dyer [SD2]. (See also P. K. Corn [Co].)

5.3. Theorem (Swinnerton-Dyer). —– Let S be a non-singular cubic surface over ℚ. Then, H ^{1}( Gal( \overline{ℚ}/ ℚ) , Pic( S\overline{ℚ} ) ) may take only �ve values, 0, ℤ/ 2ℤ, ℤ/ 2ℤ × ℤ/ 2ℤ, ℤ/ 3ℤ, and ℤ/ 3ℤ × ℤ/ 3ℤ. □

5.4. Remark. –––– One has H ^{1}( G, Pic( S\overline{ℚ} ) ) = 0 in 257 of the 350 cases.

5.5. –––– More importantly, we make the following observation. Proposition. Let S be a non-singular cubic surface over ℚ.

i) If H ^{1}( Gal( \overline{ℚ}/ ℚ) , Pic( S\overline{ℚ} ) ) = ℤ/ 2ℤthen, on S, there is a Galois invariant double-six. ii) If H ^{1}( Gal( \overline{ℚ}/ ℚ) , Pic( S\overline{ℚ} ) ) = ℤ/ 2ℤ × ℤ/ 2ℤ then, on S, there is a triple of azygetic double-sixes stabilized by Gal( \overline{ℚ}/ ℚ) .

Proof. This is seen by a case-by-case study using ���. □

5.6. Remarks. –––– i) On the other hand, if there is a Galois invariant double-six on S then H ^{1}( Gal( \overline{ℚ}/ ℚ) , Pic( S\overline{ℚ} ) ) is either 0, or ℤ/ 2ℤ or ℤ/ 2ℤ × ℤ/ 2ℤ. ii) Proposition 5.5 immediately provokes the question whether the cohomology classes are always “the same” as in the [ 12, 15] - and [ 6, 6, 6, 9] -cases. I.e., of the type cl( 𝒟) for certain Galois invariant double-sixes. Somewhat surprisingly, this is indeed the case.

5.7. Lemma. –––– Let 𝒮 be a non-singular cubic surface over an algebraically closed �eld, H a group of automorphisms of the con�guration of the 27 lines, and H ′ ⊆ H any subgroup. Each of the criteria below is suf�cient for

res: H ^{1}( H , Pic( 𝒮) ) → H ^{1}( H ′, Pic( 𝒮) )

being an injection.

i) H and H ′ generate the same orbit structure. ii) H ′ is of odd index in H and H ^{1}( H , Pic( 𝒮) ) is a 2-group. iii) H ′ is a normal subgroup in H and rkPic( 𝒮) ^{H} = rkPic( 𝒮) ^{H ′}.

Proof. i) This follows immediately from the formula of Manin [Ma, Proposition 31.3]. ii) Here, cores∘ res: H ^{1}( H , Pic( 𝒮) ) → H ^{1}( H , Pic( 𝒮) ) is the multiplication by an odd number, hence the identity. iii) The assumption ensures that H / H ′ operates trivially on Pic( 𝒮) ^{H ′}. Hence, H ^{1}( H / H ′, Pic( 𝒮) ^{H ′}) = 0. The in�ation-restriction sequence

0 → H ^{1}( H / H ′, Pic( 𝒮) ^{H ′}) → H ^{1}( H , Pic( 𝒮) ) → H ^{1}( H ′, Pic( 𝒮) )

yields the assertion. □

5.8. Proposition. –––– Let 𝒮 be a non-singular cubic surface over an algebraically closed �eld, U_{1} the group of automorphisms of the con�guration of the 27 lines stabilizing a double-six and U_{3} the group stabilizing a triple of azygetic double-sixes.

a) Let H ⊆ U_{1} be such that H ^{1}( H , Pic( 𝒮) ) = ℤ/ 2ℤ. Then, the restriction

res: H ^{1}( U_{1}, Pic( 𝒮) ) − → H ^{1}( H , Pic( 𝒮) )

is a bijection. b) Let H ⊆ U_{3} be a subgroup such that H ^{1}( H , Pic( 𝒮) ) = ℤ/ 2ℤ× ℤ/ 2ℤ. Then, the restriction

res: H ^{1}( U_{3}, Pic( 𝒮) ) − → H ^{1}( H , Pic( 𝒮) )

is a bijection.

Proof. The proof has a computer part. We use the machine to verify that the criteria formulated in Lemma 5.7 suf�ce to establish the result in all cases.

b) Here, the subgroup ( A_{3}× A_{3}) ⋉ℤ/ 2ℤ of order 18, as well as the two intermediate groups of order 36 produce the same orbit structure [ 6, 6, 6, 9] . It turns out that every subgroup H which leads to ℤ/ 2ℤ × ℤ/ 2ℤ is a subgroup of odd (1, 3, or 9)index in one of those.

a) By Lemma 5.7.ii), we may test this on the 2-Sylow subgroups of H and U_{1}. U_{1}^{( 2)} is a group of order 32 such that the Picard rank is equal to two. It turns out that, for 2-groups H ′ such that H ^{1}( H ′, Pic( 𝒮) ) = ℤ/ 2ℤ, the Picard rank may be only two or three.

There is a maximal 2-group such that the Picard rank is three. This is a group of order 16 generating the orbit structure [ 1, 1, 1, 4, 4, 4, 4, 4, 4] . To prove the assertion for this group, one �rst observes that it is of index three in a group of order 48 with orbit structure [ 3, 12, 12] . This group, in turn, is of index two in the maximal group with that orbit structure. That one, being of order 96, is the maximal group stabilizing a double-six and a tritangent plane containing three complementary lines. It is of index 15 in U_{1}. □

5.9. Corollary. –––– Let H ′ ⊆ H ⊆ U_{1} be arbitrary. Then, for the restriction map res: H ^{1}( H , Pic( 𝒮) ) − → H ^{1}( H ′, Pic( 𝒮) ) , there are the following limitations.

i) If H ^{1}( H , Pic( 𝒮) ) = 0 then H ^{1}( H ′, Pic( 𝒮) ) = 0. ii) If H ^{1}( H , Pic( 𝒮) ) ≅ ℤ/ 2ℤ and H ^{1}( H ′, Pic( 𝒮) ) ≠ 0 then res is an injection. iii) If H ^{1}( H , Pic( 𝒮) ) ≅ ℤ/ 2ℤ × ℤ/ 2ℤ then H ^{1}( H ′, Pic( 𝒮) ) ≅ ℤ/ 2ℤ × ℤ/ 2ℤ or 0. In the former case, res is a bijection. The latter is possible only when H ′ stabilizes a sixer.

Proof. We know from Remark 5.6.i) that both groups may be only 0, ℤ/ 2ℤ, or ℤ/ 2ℤ × ℤ/ 2ℤ.

i) If H ^{1}( H ′, Pic( 𝒮) ) were isomorphic to ℤ/ 2ℤ or ℤ/ 2ℤ × ℤ/ 2ℤ then the restriction from U_{1}, respectively U_{3}, to H ′ would be the zero map. ii) is immediate from the computations above. iii) If H ^{1}( H ′, Pic( 𝒮) ) ≅ ℤ/ 2ℤ then, by composition, we could produce the zero map on ℤ/ 2ℤ. The �nal assertion is an experimental observation. □

5.10. Remark (Pairs of syzygetic double-sixes). —– U_{2}, the largest group stabilizing two syzygetic double-sixes is of order 96. In view of Proposition 5.5.ii), this ensures that H ^{1}( U_{2}, Pic( 𝒮) ) ≅ ℤ/ 2ℤ or 0. Actually, it is isomorphic to ℤ/ 2ℤ. Corollary 5.9.ii) implies that the Brauer classes associated with the two double-sixes coincide. Both are equal to the non-zero element.

Actually, the group U_{2} leads to an orbit structure [ 1, 4, 6, 8, 8] . It is easy to compute H ^{1}( U_{2}, Pic( 𝒮) ) directly using the same methods as in the proof of Theorem 4.4.

5.11. Theorem. –––– Let π : S → Specℚ be an arbitrary smooth cubic surface. Then, consider the class map

cl: Φ _{S}^{Gal( \overline{ℚ} / ℚ)} → Br( S) / π ^{∗} Br( ℚ) ,

introduced in De�nition 4.1.ii), from the set of all Galois invariant double-sixes.

a) cl has the properties below.

i) If 𝒟_{1}, 𝒟_{2} are syzygetic double-sixes then cl( 𝒟_{1}) = cl( 𝒟_{2}) . ii) If 𝒟_{1}, 𝒟_{2} are azygetic then cl( 𝒟_{1}) + cl( 𝒟_{2}) + cl( 𝒟_{3}) = 0 for 𝒟_{3} the third double-six of the corresponding triple.

b) If Br( S) / π ^{∗} Br( ℚ) ≠ 0 then

i) cl( 𝒟) ≠ 0 for every Galois invariant double-six. Further, cl( Φ _{S}^{Gal( \overline{ℚ} / ℚ)} ) contains exactly the elements of order two. ii) Two double-sixes 𝒟_{1}, 𝒟_{2} are syzygetic if and only if cl( 𝒟_{1}) = cl( 𝒟_{2}) and azygetic if and only if cl( 𝒟_{1}) ≠ cl( 𝒟_{2}) . □

6 Explicit Brauer-Manin obstruction

6.1. –––– Let S be a non-singular cubic surface with a Galois invariant double-six 𝒟. This determines a class c := cl( 𝒟) ∈ Br( S) / π ^{∗} Br( ℚ) . Choose a representative \underline{c} ∈ Br( S) and the corresponding rational function F _{30}/ F _{15}^{2} ∈ ℚ( S) . Finally, let ℚ( \sqrt{D}) be the quadratic extension splitting the double-six into two sixers.

6.2. Fact. –––– The quaternion algebra over ℚ( S) corresponding to \underline{c} is

Q := ℚ( S) { X , Y} / ( X Y+ YX , X ^{2}− D, Y^{2}− F _{30}/ F _{15}^{2}) .

6.3. Remark. –––– It is well known that a class in Br( S) is uniquely determined by its restriction to Br( ℚ( S) ) . The corresponding quaternion algebra over the whole of S may be described as follows.

Let x ∈ S. We know that div( F _{30}/ F _{15}^{2}) is the norm of a divisor on S_{ℚ( \sqrt{D})} . That one is necessarily locally principal. I.e., we have a rational function f_{x} = a_{x}+ b_{x}\sqrt{D} such that div( N f_{x}) = div( F _{30}/ F _{15}^{2}) on a Zariski neighbourhood of x. Over the maximal such neighbourhood U_{x}, we de�ne a quaternion algebra by

Q_{x} := 𝒪_{U_{x}}\{ X , Y_{x}\} / ( X Y_{x}+ Y_{x}X , X ^{2}− D, Y_{x}^{2}− \frac{F _{30}}{F _{15}^{2}N f_{x}}) .

In particular, in a neighbourhood U_{η} of the generic point, we have the quaternion algebra Q_{η} := 𝒪_{U_{η}} { X , Y} / ( X Y+ YX , X ^{2}− D, Y^{2}− F _{30}/ F _{15}^{2}) .

Over U_{η} ∩ U_{x}, there is the isomorphism ι _{η , x}: Q_{η} |_{U_{η} ∩ U_{x}} → Q_{x}|_{U_{η} ∩ U_{x}}, given by

X ↦ X , Y ↦ ( a_{x}+ b_{x}X ) Y_{x}.

For two points x, y ∈ S, the isomorphism ι _{η , y}∘ ι _{η , x}^{− 1}: Q_{x}|_{U_{η} ∩ U_{x}∩ U_{y}} → Q_{y}|_{U_{η} ∩ U_{x}∩ U_{y}} extends to U_{x}∩ U_{y}.

Hence, the quaternion algebras Q_{x} may be glued together along these isomorphisms. This yields a quaternion algebra 𝒬 over S.

6.4. Corollary. –––– Let π : S → Specℚ be a non-singular cubic surface with a Galois invariant double-six 𝒟. Further, let p be a prime number and \underline{c} ∈ Br( S) a representative of the class cl( 𝒟) ∈ Br( S) / π ^{∗} Br( ℚ) . Then, the local evaluation map ev_{p}( \underline{c}, . ) : S( ℚ_{p}) → ℚ/ ℤ is given as follows.

i) Let x ∈ S( ℚ_{p}) . Choose a rational function f_{x} such that div( N f_{x}) = div( F _{30}/ F _{15}^{2}) . Then,

ev_{p}( \underline{c}, x) = \{ \begin{matrix}0 & if \frac{F _{30}}{F _{15}^{2}N f_{x}}( x) ∈ ℚ_{p}^{∗} is in the image of N : ℚ_{p}( \sqrt{D}) − → ℚ_{p}, \\ \frac{1}{2} & otherwise. \end{matrix}

Here, F _{30}/ F _{15}^{2} ∈ ℚ( S) is the rational function corresponding to the representative \underline{c}. ℚ( \sqrt{D}) is the quadratic �eld splitting the double-six into two sixers. ii) If x is not contained in any of the 27 lines then f_{x} ≡ 1 is allowed.

Proof. Assertion i) immediately follows from the above. For ii), recall that div( F _{30}/ F _{15}^{2}) is a linear combination of the 27 lines. □

6.5. –––– As already noticed in Remark 2.6, the local evaluation map carries information only at �nitely many primes. To exclude a particular prime, the following elementary criteria are highly practical.

6.6. Lemma (The local H ^{1}-criterion). —– Let S be a non-singular cubic surface over ℚ. Suppose that, for the decomposition group G_{p} ≅ Gal( \overline{ℚ}_{p}/ ℚ_{p}) at a prime number p,

H ^{1}( G_{p}, Pic( S\overline{ℚ} ) ) = 0.

Then, for every α ∈ Br( S) , the value of ev_{p}( α , x) is independent of x ∈ S( ℚ_{p}) . Proof. The local evaluation map ev_{p} factors via Br( S × _{Specℚ} Specℚ_{p}) . By [Ma, Lemma 43.1.1], we have that

Br( S × _{Specℚ} Specℚ_{p}) / Br( ℚ_{p}) ≅ H ^{1}( Gal( \overline{ℚ}_{p}/ ℚ_{p}) , Pic( S_{\overline{ℚ} _{p}}) ) .

Together with the assumption, this yields Br( S× _{Specℚ} Specℚ_{p}) = Br( ℚ_{p}) = ℚ/ ℤ. The assertion follows. □

6.7. Remark. –––– Recall from Remark 5.4 that we have H ^{1}( G, Pic( S\overline{ℚ} ) ) = 0 for 257 of the 350 possible conjugacy classes of subgroups.

6.8. Corollary. –––– Let π : S → Specℚ be a non-singular cubic surface with a Galois invariant double-six 𝒟. Further, let \underline{c} ∈ Br( S) be a representative of the class cl( 𝒟) ∈ Br( S) / π ^{∗} Br( ℚ) . If a prime number p splits in the quadratic number �eld ℚ( \sqrt{D}) splitting the two sixers then the value of ev_{p}( \underline{c}, x) is independent of x ∈ S( ℚ_{p}) . Proof. This criterion is, of course, an immediate consequence of Corollary 6.4. In view of Fact 3.6, it is also a particular case of the local H ^{1}-criterion. □

6.9. Proposition. –––– Let π : S → Specℚ be a non-singular cubic surface with a Galois invariant double-six 𝒟. Further, let \underline{c} ∈ Br( S) be a representative of the class cl( 𝒟) ∈ Br( S) / π ^{∗} Br( ℚ) .

Then, for a prime number p such that

• the �eld extension ℚ( \sqrt{D}) / ℚ splitting the double-six is unrami�ed at p,

• the reduction S_{p} is geometrically irreducible and no ℚ_{p}-rational point on S reduces to a singularity of S_{p}, the value of ev_{p}( \underline{c}, x) is independent of x ∈ S( ℚ_{p}) .

Proof. If p splits in the quadratic extension ℚ( \sqrt{D}) then the assertion is true, trivially. Thus, we may assume that p remains prime in ℚ( \sqrt{D}) . The requirement that z ∈ ℚ_{p}^{∗} is a norm from ℚ_{p}( \sqrt{D}) then simply means that ν _{p}( z) is even.

Further, we may restrict our considerations to points x ∈ S( ℚ_{p}) which are not contained in any of the 27 lines on S. Indeed, the local evaluation map is p-adically continuous and the complement of the 27 lines is dense in S( ℚ_{p}) according to Hensel’s lemma. In particular, we may work with F _{30}/ F _{15}^{2} itself.

By assumption, we have a model 𝒮 of S over ℤ_{p} such that the special �ber of 𝒮 × _{Specℤ_{p}} Spec𝒪_{ℚ_{p}( \sqrt{D})} is irreducible. We delete its singularities to obtain a model \underline{𝒮} , smooth over 𝒪_{ℚ_{p}( \sqrt{D})} . According to the last assumption, every x ∈ S( ℚ_{p}) determines a unique extension \underline{x}∈ \underline{𝒮} ( 𝒪_{ℚ_{p}( \sqrt{D})} ) .

It will suf�ce to construct a Zariski neighbourhood of \underline{x} such that ev_{p}( \underline{c}, . ) is constant. We have, on the geometric generic �ber,

div( F _{30}/ F _{15}^{2}) = 5E + 5G − 4F .

Here, the divisors E := E_{1} + … + E_{6}, G := G_{1} + … + G_{6}, and F := F _{12} + … + F _{56} are Gal( \overline{ℚ}/ ℚ( \sqrt{D}) ) -invariant, and, therefore, de�ned over S × _{Specℚ_{p}} Specℚ_{p}( \sqrt{D}) . \underline{𝒮} is a regular model of that variety. Hence, every divisor on \underline{𝒮} is locally principal. This yields, in a Zariski neighbourhood 𝒰\underline{x},

F _{30}/ F _{15}^{2} = Cp^{k}e^{5}g^{5}/ f^{4}

for e, g, and f rational functions corresponding to the divisors E, G, and F , respectively, k ∈ ℤ, and a certain C ∈ Γ ( 𝒰\underline{x}, 𝒪_{𝒰\underline{x}}^{∗} ) . Note that we get by with one power of p since the special �ber is irreducible.

The scheme \underline{𝒮} is acted upon by the conjugation σ ∈ Gal( ℚ( \sqrt{D}) / ℚ) . Restricting to an open subscheme, if necessary, we may assume that 𝒰\underline{x} is invariant under σ . The two sixers are interchanged by σ . Consequently,

σ ( e) = cp^{l}g, σ ( f) = c′p^{l′} f

for l, l′ ∈ ℤ and regular functions c, c′, invertible on 𝒰\underline{x}. This yields

F _{30}/ F _{15}^{2} = Cc′^{2}C^{− 5}p^{k+ 2l′− 5l}N ( e^{5}/ f^{2}) .

For x_{0} ∈ S( ℚ_{p}) specializing to 𝒰\underline{x}, we therefore see

ν _{p}( ( F _{30}/ F _{15}^{2}) ( x_{0}) ) ≡ k + l ( mod 2) .

In particular, the local evaluation map ev_{p}( \underline{c}, . ) : S( ℚ_{p}) → ℚ/ ℤ is constant on the set of all points specializing to 𝒰\underline{x}. As the point x ∈ S( ℚ_{p}) de�ning the open subset 𝒰\underline{x} is arbitrary and the special �ber 𝒮_{p} is irreducible, this implies the assertion. □

6.10. Remarks. –––– a) Assuming resolution of singularities in unequal characteristic, there is a proper model 𝒮 of S being a regular scheme. Then, for p a prime unrami�ed in ℚ( \sqrt{D}) , the evaluation ev_{p}( \underline{c}, x) depends only on the component of 𝒮 × _{Specℤ} Spec𝔽_{p}_{2}, the point x specializes to. If p is rami�ed and p ≠ 2 then we have at least that ev_{p}( \underline{c}, x) is determined by the reduction of x modulo p.

b) As one might expect from the proof given, Proposition 6.9 is true in more generality. The reader might consult [B, Theorem 1].

7 Explicit Galois descent

7.1. –––– Recall that in [EJ2], we described a method to construct non-singular cubic surfaces over ℚ with a Galois invariant double-six. The idea was to start with cubic surfaces in hexahedral form. For these, we developed an explicit version of Galois descent.

7.2. –––– More concretely, given a starting polynomial f ∈ ℚ[ T ] of degree six without multiple zeroes, we construct a cubic surface S_{( a_{0}, …, a_{5})} over ℚ such that

S_{( a_{0}, …, a_{5})} × _{Specℚ} Spec\overline{ℚ}

is isomorphic to the surface S^{( a_{0}, …, a_{5})} in P^{5} given by

X _{0}^{3}+ X _{1}^{3}+ X _{2}^{3}+ X _{3}^{3}+ X _{4}^{3}+ X _{5}^{3} = 0,
X _{0} + X _{1} + X _{2} + X _{3} + X _{4} + X _{5} = 0,
a_{0}X _{0} + a_{1}X _{1} + a_{2}X _{2} + a_{3}X _{3} + a_{4}X _{4} + a_{5}X _{5} = 0.

Here, a_{0}, …, a_{5} ∈ \overline{ℚ} are the zeroes of f.

The operation of an element σ ∈ Gal( \overline{ℚ}/ ℚ) on S_{( a_{0}, …, a_{5})} × _{Specℚ} Spec\overline{ℚ} goes over into the automorphism π _{σ} ∘ t_{σ} : S^{( a_{0}, …, a_{5})} → S^{( a_{0}, …, a_{5})} . Here, π _{σ} permutes the coordinates according to the rule a_{π _{σ} ( i)} = σ ( a_{i}) while t_{σ} is the naive operation of σ on S_{( a_{0}, …, a_{5})} as a morphism of schemes twisted by σ .

7.3. Remarks. –––– i) More details on the theory are given in [EJ2, Theorem 6.6]. ii) On S_{( a_{0}, …, a_{5})} , there are the 15 obvious lines given by

X _{i_{0}} + X _{i_{1}} = X _{i_{2}} + X _{i_{3}} = X _{i_{4}} + X _{i_{5}} = 0

for { i_{0}, i_{1}, i_{2}, i_{3}, i_{4}, i_{5}} = { 0, 1, 2, 3, 4, 5} . They clearly form a Galois invariant set. The complement is a double-six. Correspondingly, there are the 15 obvious tritangent planes given by X _{i}+ X _{j} = 0 for i ≠ j. There are formulas for the 30 non-obvious tritangent planes, too [EJ2, Proposition 7.1.ii)]. What is important is that they are de�ned over ℚ( a_{0}, …, a_{5}, \sqrt{d_{4}}) for

d_{4}( a_{0}, …, a_{5}) := σ _{2}^{2}− 4σ _{4}+ σ _{1}( 2σ _{3}− \frac{3}{2}σ _{1}σ _{2}+ \frac{5}{16}σ _{1}^{3})

the Coble quartic. Here, σ _{i} is the i-th elementary symmetric function in a_{0}, …, a_{5}.

Further, an element σ ∈ Gal( \overline{ℚ}/ ℚ) �ips the double-six if and only if it de�nes the conjugation of ℚ( \sqrt{D}) for D := d_{4}· Δ, the second factor denoting the discriminant of a_{0}, …, a_{5} [EJ2, Proposition 7.4]. iii) The smooth manifold S( ℝ) has two connected components if and only if exactly four of the a_{0}, …, a_{5} are real and d_{4}( a_{0}, …, a_{5}) > 0. Otherwise, S( ℝ) is connected [EJ2, Corollary 8.4]. iv) The descent variety S_{( a_{0}, …, a_{5})} may easily be computed completely explicitly. In fact, [EJ2, Algorithm 6.7] yields a quaternary cubic form with 20 rational coef�cients.

7.4. –––– Using the criteria provided in section 6, we have the following strategy to compute the Brauer-Manin obstruction on S_{( a_{0}, …, a_{5})} .

Strategy (to explicitly compute the Brauer-Manin obstruction on S_{( a_{0}, …, a_{5})} ).

i) Compute D := d_{4}( a_{0}, …, a_{5}) · Δ( a_{0}, …, a_{5}) . Determine the list L_{1} of all primes at which ℚ( \sqrt{D}) is rami�ed. ii) By a Gröbner basis calculation, determine all the primes outside L_{1} at which S_{( a_{0}, …, a_{5})} has bad reduction. Write them into a list L_{2}. iii) From L_{2}, delete all primes which split in ℚ( \sqrt{D}) . Further, erase all those primes from L_{2} for which the singular points on the reduction modulo p are not de�ned over 𝔽_{p} or do not lift to S_{( a_{0}, …, a_{5})} × _{Specℤ} Specℤ/ p^{k}ℤ for k large. iv) Put L := L_{1}∪ L_{2}. If D < 0 and S_{( a_{0}, …, a_{5})} ( ℝ) is not connected then the in�nite place has to be added to this list of critical primes.

v) Delete all the primes from L for which the local H ^{1}-criterion works successfully. vi) Scale the form F _{30} by a constant factor such that the local evaluation maps are zero for all the primes outside L. vii) For the primes p that remained in L, the form F _{30} has to be evaluated. For that, cover S( ℚ_{p}) by �nitely many open subsets which are suf�ciently small to ensure that the �rst p-adic digit of F _{30} does not change. If p = 2 then the �rst three digits have to be taken into account. In the case that we have a Galois invariant triple of azygetic double-sixes, the last step has to be executed three times, once for each of the corresponding forms of type F _{30}.

8 Application: Manin’s conjecture

8.1. –––– Recall that a conjecture, due to Yu. I. Manin, asserts that the number of ℚ-ra-tional points of anticanonical height ≤ B on a Fano variety S is asymptotically equal to τ Blog^{rkPic( S) − 1}B, for B → ∞ . Further, the coef�cient τ ∈ ℝ is conjectured to be the Tamagawa-type number τ ( S) introduced by E. Peyre in [Pe]. In the particular case of a cubic surface, the anticanonical height is the same as the naive height.

8.2. –––– E. Peyre’s Tamagawa-type number is de�ned in [PT, De�nition 2.4] as

τ ( S) := α ( S) · β ( S) · lim_{s→ 1}( s − 1) ^{t}L( s, χ _{Pic( S\overline{ℚ} )} ) · τ _{H} ( S( 𝔸_{ℚ} ) ^{Br})

for t = rkPic( S) .

Here, the factor β ( S) is simply de�ned as β ( S) := #H ^{1}( Gal( \overline{ℚ}/ ℚ) , Pic( S\overline{ℚ} ) ) . α ( S) is given as follows [Pe, Dé�nition 2.4]. Let Λ _{eff}( S) ⊂ Pic( S) ⊗ _{ℤ} ℝ be the cone generated by the effective divisors. Identify Pic( S) ⊗ _{ℤ} ℝ with ℝ^{t} via a mapping induced by an isomorphism Pic( S) − →^{≅} ℤ^{t}. Consider the dual cone Λ _{eff}^{∨}( S) ⊂ ( ℝ^{t}) ^{∨} . Then, α ( S) := t · vol{ x ∈ Λ _{eff}^{∨} | ⟨ x, − K ⟩ ≤ 1} .

L( · , χ _{Pic( S\overline{ℚ} )} ) denotes the Artin L-function of the Gal( \overline{ℚ}/ ℚ) -representation Pic( S\overline{ℚ} ) ⊗ _{ℤ} ℂ which contains the trivial representation t times as a direct summand. Therefore, L( s, χ _{Pic( S\overline{ℚ} )} ) = ζ ( s) ^{t} · L( s, χ _{P} ) and

lim_{s→ 1}( s − 1) ^{t}L( s, χ _{Pic( S\overline{ℚ} )} ) = L( 1, χ _{P} )

where ζ denotes the Riemann zeta function and P is a representation which does not contain trivial components. [Mu, Corollary 11.5 and Corollary 11.4] show that L( s, χ _{P} ) has neither a pole nor a zero at s = 1.

Finally, τ _{H} is the Tamagawa measure on the set S( 𝔸_{ℚ} ) of adelic points on S and S( 𝔸_{ℚ} ) ^{Br} ⊆ S( 𝔸_{ℚ} ) denotes the part which is not affected by the Brauer-Manin obstruction.

8.3. –––– As S is projective, we have S( 𝔸_{ℚ} ) = ∏ _{ν ∈ Val( ℚ)} S( ℚ_{ν} ) . Then, the Tamagawa measure τ _{H} is de�ned to be the product measure τ _{H} := ∏ _{ν ∈ Val( ℚ)} τ _{ν} .

Here, for a prime number p, the local measure τ _{p} on S( ℚ_{p}) is given as follows. Let a ∈ S( ℤ/ p^{k}ℤ) and put 𝔘_{a}^{( k)} := { x ∈ S( ℚ_{p}) | x ≡ a ( mod p^{k}) } . Then,

τ _{p}( 𝔘_{a}^{( k)} ) := det( 1 − p^{− 1}Frob_{p} | Pic( S\overline{ℚ} ) ^{I _{p}}) · lim_{m→ ∞} \frac{#\{ y ∈ S( ℤ/ p^{m}ℤ) | y ≡ a ( mod p^{k}) \}}{p^{mdimS}} .

Pic( S\overline{ℚ} ) ^{I _{p}} denotes the �xed module under the inertia group.

τ _{∞} is described in [Pe, Lemme 5.4.7]. In the case of a cubic surface, de�ned by the equation f = 0, this yields

τ _{∞} ( U) = \frac{1}{2} ∫_{\begin{matrix}C & U \\ | x_{0}| , …, | x_{3}| ≤ 1 \end{matrix}} ω _{Leray}

for U ⊂ S( ℝ) . Here, ω _{Leray} is the Leray measure on the cone CS( ℝ) . It is related to the usual hypersurface measure by the formula ω _{Leray} = \frac{1}{∥ grad f∥} ω _{hyp}.

8.4. –––– Using [EJ2, Algorithm 6.7], we constructed many examples of smooth cubic surfaces over ℚ with a Galois invariant double-six. For each of them, one may apply Strategy 7.4 to compute the effect of the Brauer-Manin obstruction. Then, the method described in [EJ1] applies for the computation of Peyre’s constant.

From the ample supply, the examples below were chosen in the hope that they indicate the main phenomena. The Brauer-Manin obstruction may work at many primes simultaneously but examples where few primes are involved are the most interesting. We show that the fraction of the Tamagawa measure excluded by the obstruction can vary greatly. We also show that there may be an obstruction at the in�nite prime.

8.5. Example. –––– The polynomial

f := T ^{6}− 390T ^{4}− 10180T ^{3}+ 10800T ^{2}+ 2164296T + 13361180 ∈ ℚ[ T ]

yields the cubic surface S given by the equation

− x^{2}z− x^{2}w − 3xy^{2}+ xz^{2}+ 14xzw + 8xw^{2}− 2y^{3}− 11y^{2}z

+ y^{2}w + 4yz^{2}+ 4yzw + 10yw^{2}+ 4z^{3}− 11z^{2}w + 9zw^{2}− 6w^{3} = 0.

S has bad reduction at 2, 3, 5, 11, and 9265613761. The Galois group operating on the 27 lines is S_{6} acting in such a way that the orbit structure is [ 12, 15] . Therefore, we have H ^{1}( Gal( \overline{ℚ}/ ℚ) , Pic( S\overline{ℚ} ) ) ≅ ℤ/ 2ℤ. The quadratic �eld splitting the double-six is ℚ( \sqrt{10}) .

The primes 3 and 9265613761 split in ℚ( \sqrt{10}) . The local H ^{1}-criterion excludes the prime 5. Further, it turns out that the local evaluation map at 11 is constant. Hence, the Brauer-Manin obstruction works only at the prime 2. From the whole of S( ℚ_{2}) which is of measure 4 only a subset of measure \frac{9}{4} is allowed.

Using this, for Peyre’s constant, we �nd τ ( S) ≈ 1. 7005. There are actually 6641 ℚrational points of height at most 4000 in comparison with a prediction of 6802.

8.6. Example. –––– The polynomial

f := T ^{6}+ 60T ^{4}− 40T ^{3}− 900T ^{2}+ 15072T − 27860 ∈ ℚ[ T ]

yields the cubic surface S given by the equation

5x^{3}− 9x^{2}y + x^{2}z + 6x^{2}w + 3xy^{2}+ xyz + 6xyw − 2xz^{2}
− 4xzw − y^{3}− 3y^{2}z + 2yz^{2}+ 2yzw + 4z^{3}+ 2z^{2}w + 2zw^{2} = 0.

S has bad reduction at 2, 3, 5, and 73. The Galois group operating on the 27 lines is A_{6}× ℤ/ 2ℤ and the orbit structure is [ 12, 15] . The quadratic �eld ℚ( \sqrt{2}) splits the doublesix.

The prime 73 splits in ℚ( \sqrt{2}) . Further, the local H ^{1}-criterion excludes the prime 5. Hence, the Brauer-Manin obstruction works only at the primes 2 and 3. At 2, the local evaluation map decomposes S( ℚ_{2}) into two sets of measures 1 and \frac{1}{2}, respectively. At 3, the corresponding measures are \frac{70}{81} and \frac{28}{81}. An easy calculation shows τ _{H} ( S( 𝔸_{ℚ} ) ^{Br}) = \frac{4}{7}τ _{H} ( S( 𝔸_{ℚ} ) ) .

Using this, for Peyre’s constant, we �nd τ ( S) ≈ 5. 0879. Up to a search bound of 4000, there are actually 19302 ℚ-rational points in comparison with a prediction of 20352.

8.7. Example. –––– The polynomial f := T ( T ^{5}− 5T − 2) ∈ ℚ[ T ] yields the cubic surface S given by the equation

2x^{3}+ x^{2}y − 4x^{2}z − x^{2}w + 2xy^{2}+ 2xyz + 2xyw − 2xz^{2}− 4xzw

− 2xw^{2}+ 2y^{2}z − y^{2}w + yz^{2}+ 2yzw − 5yw^{2}− 3z^{2}w + 6zw^{2}+ 9w^{3} = 0.

S has bad reduction at 2, 3, and 5. Further, S( ℝ) consists of two connected components. The Galois group operating on the 27 lines is isomorphic to S_{5}× ℤ/ 2ℤ and the orbit structure is [ 12, 15] . ℚ( \sqrt{− 15}) is the �eld splitting the double-six.

The prime 2 splits in ℚ( \sqrt{− 15}) . Further, the local H ^{1}-criterion excludes the prime 5. Hence, the Brauer-Manin obstruction works only at 3 and the in�nite prime. At 3, the local evaluation map decomposes S( ℚ_{2}) into two sets of measures \frac{2}{3} and \frac{4}{9}, respectively. At the in�nite prime, the corresponding measures are approximately 1. 9179 and 1. 1673. An easy calculation shows τ _{H} ( S( 𝔸_{ℚ} ) ^{Br}) ≈ 0. 5243· τ _{H} ( S( 𝔸_{ℚ} ) ) .

Using this, for Peyre’s constant, we �nd τ ( S) ≈ 3. 7217. Up to a search bound of 4000, there are actually 14249 ℚ-rational points in comparison with a prediction of 14887.

8.8. Example. –––– The polynomial

f := T ( T ^{5}− 60T ^{3}− 90T ^{2}+ 675T + 810) ∈ ℚ[ T ]

yields the cubic surface S given by the equation

3x^{3}+ 2x^{2}z + xy^{2}− 2xyz − 2xyw − xzw + 2xw^{2}− yzw − yw^{2}− z^{3}+ z^{2}w = 0.

S has bad reduction at 2, 3, and 5. The Galois group operating on the 27 lines is isomorphic to S_{5} and the orbit structure is [ 12, 15] . The quadratic �eld ℚ( \sqrt{− 3}) splits the double-six.

The local H ^{1}-criterion excludes the prime 5. Further, the local evaluation maps turn out to be constant on S( ℚ_{2}) and S( ℝ) . At the real prime, the reason is simply that S( ℝ) is connected. Consequently, the Brauer-Manin obstruction works only at the prime 3. From the whole of S( ℚ_{3}) measuring \frac{2}{3} a subset of measure \frac{4}{9} is allowed.

Using this, for Peyre’s constant, we �nd τ ( S) ≈ 2. 2647. Up to a search bound of 4000, there are actually 8886 ℚ-rational points in comparison with a prediction of 9059.

8.9. Example. –––– The polynomial f := T ( T ^{5}+ 20T + 16) ∈ ℚ[ T ] yields the cubic surface S given by the equation

− 3x^{3}− 7x^{2}y − 4x^{2}z + 5x^{2}w + 4xy^{2}+ 10xyz − 4xyw − 2xz^{2}

+ 2xzw + xw^{2}− 4y^{2}z + yz^{2}− 4yzw − 16yw^{2}+ z^{2}w − 5zw^{2} = 0.

S has bad reduction at 2 and 5. The Galois group operating on the 27 lines is isomorphic to A_{5}× ℤ/ 2ℤ. The orbit structure is [ 12, 15] . The quadratic �eld ℚ( \sqrt{− 5}) splits the doublesix.

The local H ^{1}-criterion excludes the prime 5. At the in�nite prime, the local evaluation map is constant since S( ℝ) is connected. Hence, the Brauer-Manin obstruction works only at the prime 2. It allows a subset of measure \frac{17}{16} out of S( ℚ_{2}) measuring \frac{5}{4}.

Using this, for Peyre’s constant, we �nd τ ( S) ≈ 2. 4545. Up to a search bound of 4000, there are actually 9736 ℚ-rational points in comparison with a prediction of 9818.

8.10. Example. –––– The polynomial

f := T ^{6}− 456T ^{4}− 904T ^{3}+ 102609T ^{2}+ 1041060T + 2935300 ∈ ℚ[ T ] yields the cubic surface S given by the equation

− 2x^{3}+ 3x^{2}z + 9x^{2}w − 4xy^{2}− 8xyz − 10xzw + 4xw^{2}− 4y^{3}− 3y^{2}z
− 4y^{2}w − 2yz^{2}− 2yzw + 8yw^{2}− z^{3}+ z^{2}w − 6zw^{2}− 2w^{3} = 0.

S has bad reduction at 2, 3, 5, 31, and 11071. The Galois group operating on the 27 lines is isomorphic to ( S_{3} × S_{3}) ⋉ ℤ/ 2ℤ of order 72. The orbit structure is [ 6, 6, 6, 9] . There is a triple of Galois invariant double-sixes. Therefore, we have that H ^{1}( Gal( \overline{ℚ}/ ℚ) , Pic( S\overline{ℚ} ) ) ≅ ℤ/ 2ℤ × ℤ/ 2ℤ. The quadratic �eld splitting the doublesixes is ℚ( \sqrt{2}) .

The primes 31 and 11071 split in ℚ( \sqrt{2}) . Further, the local evaluation maps turn out to be constant on S( ℚ_{3}) . Consequently, the Brauer-Manin obstruction works only at the primes 2 and 5.

The local evaluation maps decompose S( ℚ_{2}) into four sets of measures \frac{7}{16}, \frac{5}{16}, \frac{1}{4}, and \frac{1}{4}, respectively. At the prime 5, the corresponding measures are \frac{516}{625}, 0, \frac{96}{625}, and 0. Observe, for one of the three non-zero Brauer classes, the local evaluation map is constant on S( ℚ_{5}) .

A simple calculation shows that τ _{H} ( S( 𝔸_{ℚ} ) ^{Br}) = \frac{111}{340}τ _{H} ( S( 𝔸_{ℚ} ) ) . Using this, for Peyre’s constant, we �nd τ ( S) ≈ 1. 8532. Up to a search bound of 4000, there are actually 6994 ℚrational points in comparison with a prediction of 7413.

References

[B] M. Bright: Ef�cient evaluation of the Brauer-Manin obstruction, Math. Proc. Cambridge Philos. Soc. 142 (2007), 13–23

[CE] H. Cartan and S. Eilenberg: Homological Algebra, Princeton Univ. Press, Princeton 1956

[CG] J. W. S. Cassels and M. J. T. Guy: On the Hasse principle for cubic surfaces, Mathematika 13 (1966), 111–120

[CKS] J.-L. Colliot-Thélène, D. Kanevsky, and J.-J. Sansuc: Arithmétique des surfaces cubiques diagonales, in: Diophantine approximation and transcendence theory (Bonn 1985), Lecture Notes in Math. 1290, Springer, Berlin 1987, 1–108

[Co] P. K. Corn: Del Pezzo surfaces and the Brauer-Manin obstruction, Ph.D. thesis, Harvard 2005

[EJ1] Elsenhans, A.-S. and Jahnel, J.: Experiments with general cubic surfaces, in: Algebra, Arithmetic, and Geometry, In Honor of Yu. I. Manin, Edited by Y. Tschinkel and Y. Zarhin, Volume I, Progress in Mathematics 269, Birkhäuser, Boston 2007, 637–654

[EJ2] Elsenhans, A.-S. and Jahnel, J.: Cubic surfaces with a Galois invariant double-six, Central European Journal of Mathematics 8 (2010), 646–661 [Ha] R. Hartshorne: Algebraic Geometry, Graduate Texts in Mathematics 52, Springer,

New York 1977

[Ko] G. Kohn: Über einige Eigenschaften der allgemeinen Fläche dritter Ordnung, Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften Wien 117 (1908), 53–73

[Ma] Yu. I. Manin: Cubic forms, algebra, geometry, arithmetic, North-Holland Publishing Co. and American Elsevier Publishing Co., Amsterdam, London, and New York 1974

[Mo] L. J. Mordell: On the conjecture for the rational points on a cubic surface, J. London Math. Soc. 40 (1965), 149–158

[Mu] M. R. Murty: Applications of symmetric power L-functions, in: Lectures on automorphic L-functions, Fields Inst. Monogr. 20, Amer. Math. Soc., Providence 2004, 203–283

[Pe] E. Peyre: Hauteurs et mesures de Tamagawa sur les variétés de Fano, Duke Math. J. 79 (1995), 101–218

[PT] E. Peyre and Y. Tschinkel: Tamagawa numbers of diagonal cubic surfaces, numerical evidence, Math. Comp. 70 (2001), 367–387

[SD1] Sir Peter Swinnerton-Dyer: Two special cubic surfaces, Mathematika 9 (1962), 54–56

[SD2] Sir Peter Swinnerton-Dyer: The Brauer group of cubic surfaces, Math. Proc. Cambridge Philos. Soc. 113 (1993), 449–460

[Ta] J. Tate: Global class �eld theory, in: Algebraic number theory, Edited by J. W. S. Cassels and A. Fröhlich, Academic Press and Thompson Book Co., London and Washington 1967