TAMAGAWA NUMBERS OF DIAGONAL CUBIC SURFACES OF HIGHER RANK

EMMANUEL PEYRE AND YURI TSCHINKEL

Abstract. We consider diagonal cubic surfaces defined by an equation of the form ax^{3} + by^{3} + cz^{3} + dt^{3} = 0. Numerically, one can find all rational points of height ⩽ B for B in the range of up to 10^{5}, thanks to a program due to D. J. Bernstein. On the other hand, there are precise conjectures concerning the constants in the asymptotics of rational points of bounded height due to Manin, Batyrev and the authors. Changing the coefficients one can obtain cubic surfaces with rank of the Picard group varying between 1 and 4. We check that numerical data are compatible with the above conjectures. In a previous paper we considered cubic surfaces with Picard groups of rank one with or without Brauer-Manin obstruction to weak approximation. In this paper, we test the conjectures for diagonal cubic surfaces with Picard groups of higher rank.

Contents

1. Introduction 1 2. Description of the conjectural constant 3 3. The Galois module Pic(\overline{V}) 5 4. Euler product for the good places 10 5. Density at the bad places 11 6. The constant α(V) 13 7. Some statistical formulae 18 8. Presentation of the results 19 References 25

1. Introduction

This paper is devoted to numerical tests of a refined version of a conjecture of Manin about the number of points of bounded height on Fano varieties (see [BM], [FMT], [Pe], or [BT] for a description of the conjectures). The choice of diagonal cubic surfaces to test these conjectures was motivated by the work of Heath-Brown [H-B] in which he treated the cases

X^{3} + Y^{3} + Z^{3} + aT^{3} = 0

for a = 2 or 3. The results he obtained were used as a benchmark for the subsequent attempts to interpret the asymptotic constants (see, in particular, [S-D], [Pe] and [PT]).

Date: January 18, 2001.

More precisely, we consider a diagonal cubic surface V ⊂ P_{Q}^{3} given by an equation of the form

aX^{3} + bY^{3} + cZ^{3} + dT^{3} = 0.

Let H be the height function on P^{3}(Q) defined by the formula: for any Q = (x : y : z : t) in P^{3}(Q), one has

H(Q) = max\{|x|, |y|, |z|, |t|\} if \begin{cases}(x, y, z, t) ∈ Z^{4}, \\ gcd(x, y, z, t) = 1. \end{cases}

Let U be the complement in V to the 27 lines. We are interested in the asymptotic behavior of the cardinal

N_{U,H}(B) = #{ Q ∈ U(Q) | H(Q) ⩽ B }

as B goes to infinity.

Assume that V (Q) is Zariski dense, which by a result of Segre (see [Man2, §29,§30]) is equivalent to V (Q) ≠ ∅. It is expected that

N_{U,H}(B) = BP (log(B)) + o(B)

as B goes to +∞, where P is a polynomial of degree rk Pic(V) − 1, with leading coefficient θ_{H}(V). This constant has a conjectural description. The goal is to compute θ_{H}(V) explicitly in the examples at hand and to compare it with the constant obtained from numerical data. Our previous paper [PT] was devoted to surfaces with Picard groups of rank one with or without Brauer-Manin obstruction to weak approximation. In this paper, we consider examples with Picard groups of higher rank.

Note that in these examples the relative error term

(N_{U,H}(B) − θ_{H}(V)B(log B)^{rk Pic(V)−1})/B(log B)^{rk Pic(V)−1}

is expected to decrease more slowly. Indeed, if rk Pic(V) = 1 this error term is expected to decrease as 1/B^{ϵ} for some ϵ > 0, whereas for higher ranks it should be comparable to 1/ log B. Thus we decided not only to compare the conjectural constant θ_{H}(V) with the quotient

N_{U,H}(B)/B(log B)^{rk Pic(V)−1}

with B = 10^{5}, but also to take into account that a polynomial P of degree rk Pic(V) − 1 should appear in the asymptotics and use a naive statistical formula to estimate its leading coefficient θ_{H}^{stat}(V) from the data. We observe a quite good accordance: the difference between θ_{H}(V) and θ_{H}^{stat}(V) is less than 6% in the examples. Moreover the fact that θ_{H}^{stat}(V) is nearer to θ_{H}(V) than the above quotient is in itself a point in favor of the conjecture: indeed there is no obvious purely statistical reason for which this should be true in general.

The paper is organized as follows: in section 2 we define θ_{H}(V). Section 3 contains the description of the Galois action on the geometric Picard group Pic(\overline{V}). In section 4 we compute the Euler product corresponding to good reduction places. In section 5 we explain how to compute the local densities at the places of bad reduction. In section 6 we determine in each case the value of the geometric constant α(V) defined in §2. Section 7 is devoted to the description of statistical tools we used to analyze the numerical data. In section 8 we present the results.

We would like to thank the referee for the improvements he suggested.

2. Description of the conjectural constant

In this section we give a short description of the conjectural asymptotic constant for heights defined by an adelic metrization of the anticanonical line bundle (see [Pe] for more details and [BT] for a discussion in a more general setting).

Notations 2.1. For any field E, we denote by \overline{E} an algebraic closure of E. If X is a variety over E, then X(E) denotes the set of rational points of X and \overline{X} the product X ×_{Spec(E)} Spec \overline{E}. The cohomological Brauer group Br(X) is defined as the étale cohomology group H_{ét}^{2}(X, G_{m}). For any A in Br(X), any extension E′ of E and any P in V (E′), we denote by A(P) the evaluation of A at P.

For a number field F we denote by Val(F) the set of places of F and by Val_{f}(F) the set of finite places. The absolute discriminant of F is denoted by d_{F}. For any place v of F, let F_{v} be the v-adic completion of F. If v is finite, then 𝒪 _{v} is the ring of v-adic integers and F_{v} the residue field. By global class field theory we have an exact sequence

(2.1) 0 → Br(F) → ⨁_{v∈Val(F)}Br(F_{v}) \underrightarrow{∑ inv_{v}} Q/Z → 0.

In the following, V is a smooth projective geometrically integral variety over a number field F satisfying the conditions:

(i) H^{i}(V, 𝒪 _{V}) = 0 for i = 1 or 2, (ii) Pic(\overline{V}) has no torsion, (iii) the anticanonical line bundle ω_{V}^{−1} belongs to the interior of the cone of classes of

effective divisors Λ_{eff}(V) ⊂ Pic(V) ⊗_{Z} R. The adelic space V (A_{F}) of V is the product ∏_{v∈Val(F)} V (F_{v}). By [CT, lemma 1], for any class A in Br(V), one has a map ρ_{A} defined as the composition

V (A_{F}) → ⨁_{v∈Val(F)} Br(F_{v}) \underrightarrow{∑ inv_{v}} Q/Z
(P_{v})_{v∈Val(F)} ↦ (A(P_{v}))_{v∈Val(F)}.

Then one defines

V (A_{F})^{Br} = ⋂_{A∈Br(V)} ker(ρ_{A}) ⊂ V (A_{F}).

By the exact sequence (2.1), one has the inclusion

\overline{V (F)} ⊂ V (A_{F})^{Br}

where \overline{V (F)} denotes the topological closure of the set of rational points. Conjecturally both sets coincide for cubic surfaces. (See also the text of Swinnerton-Dyer in this volume). There is a Brauer-Manin obstruction to weak approximation, as described by Manin in [Man1] and by Colliot-Thélène and Sansuc in [CTS], if one has

V (A_{F})^{Br} ≠ V (A_{F}).

Let us assume that the height H on V is defined by an adelic metric (∥ · ∥_{v})_{v∈Val(F)} on ω_{V}^{−1}. By definition, this means that we consider ω_{V}^{−1} as a line bundle, that the functions ∥·∥_{v} are v-adically continuous metrics on ω_{V}^{−1}(F_{v}) which for almost all places v are given by a smooth model of V, and that the height of a rational point x of V is given by the formula

∀y ∈ ω_{V}^{−1}(x) − {0}, H(x) = ∏_{v∈Val(F)} ∥y∥_{v}^{−1}

where ω_{V}^{−1}(x) is the fiber of ω_{V}^{−1} at x.

If v ∈ Val(F) the Haar measure dx_{v} on F_{v} is normalized as follows:

- ∫_{𝒪 _{v}} dx_{v} = 1 if v is finite,

- dx_{v} is the usual Lebesgue measure if F_{v} ⟶ R,

- dx_{v} = dz d\overline{z} = 2dx dy if F_{v} ⟶ C. The metric ∥ · ∥_{v} defines a measure ω_{H,v} on the locally compact space V (F_{v}). In local v-adic analytic coordinates x_{1,v} … x_{n,v} on V (F_{v}) this measure is given by the formula

ω_{H,v} = ∥\frac{∂}{∂x_{1,v}} ∧ ⋯ ∧ \frac{∂}{∂x_{n,v}} ∥_{v} dx_{1,v} … dx_{n,v} .

If M is a discrete representation of Gal(\overline{F} /F) over Q, then for any finite place 𝔭 of F, the local term of the corresponding Artin L-function is defined as follows: we choose an algebraic closure \overline{F}_{𝔭} of F_{𝔭} containing \overline{F}. We get an exact sequence

1 → I_{𝔭} → D_{𝔭} → Gal(\overline{F}_{𝔭}/F_{𝔭}) → 1

where D_{𝔭} is the decomposition group and I_{𝔭} the inertia. We denote by F̃r_{𝔭} a lifting of the Frobenius map to D_{𝔭} ⊂ Gal(\overline{F} /F) (which up to conjugation depends only on 𝔭), and put

L_{𝔭}(s, M) = \frac{1}{Det(1 − (#F_{𝔭})^{−s}F̃r_{𝔭} | M^{I_{𝔭}})}.

We fix a finite set S of bad places containing the archimedean ones so that V admits a smooth projective model 𝒱 over the ring of S-integers 𝒪 _{S}. For any 𝔭 in Val(F) − S we consider

L_{𝔭}(s, Pic(\overline{V})) = L_{𝔭}(s, Pic(\overline{V}) ⊗_{Z} Q).

The corresponding global L-function is given by the Euler product

L_{S}(s, Pic(\overline{V})) = ∏_{𝔭∈Val(F)−S} L_{𝔭}(s, Pic(\overline{V}))

which converges for Re s > 1 and has a meromorphic continuation to C with a pole of order t = rk Pic(V) at 1. One introduces local convergence factors λ_{v} given by

λ_{v} = \begin{cases}L_{v}(1, Pic(\overline{V})) if v ∈ Val(F) − S, \\ 1 otherwise. \end{cases}

The Weil conjectures (proved by Deligne) imply that the Tamagawa measure

∏_{v∈Val(F)} λ_{v}^{−1}ω_{H,v}

converges on V (A_{F}) (see [Pe, proposition 2.2.2]).

Definition 2.2. The Tamagawa measure on V (A_{F}) corresponding to the adelic metric (∥ · ∥_{v})_{v∈Val(F)} is defined by

ω_{H} = \frac{1}{\sqrt{d_{F}} ^{dim V}} lim_{s→1}(s − 1)^{t}L_{S}(s, Pic(\overline{V})) ∏_{v∈Val(F)} λ_{v}^{−1}ω_{H,v}.

From the arithmetic standpoint, it seems more natural to integrate ω_{H} over the closure \overline{V (F)} ⊂ V (A_{F}) (as in the original approach to the Tamagawa number). However, computationally, it is easier to work with V (A_{F})^{Br}. Therefore, following a suggestion of Salberger, we define here

Definition 2.3.

τ_{H}(V) = ω_{H}(V (A)^{Br}). (2.2)

Let Pic(V)^{∨} be the dual lattice to Pic(V). We denote by dy the corresponding Lebesgue measure on Pic(V)^{∨} ⊗_{Z} R and by

Λ_{eff}(V)^{∨} = { x ∈ Pic(V)^{∨} ⊗_{Z} R | ∀y ∈ Λ_{eff}(V), ⟨x, y⟩ ⩾ 0 }

the dual cone of Λ_{eff}(V).

Definition 2.4. We define

α(V) = \frac{1}{(t − 1)!} ∫_{Λ_{eff}(V)^{∨}} e^{−⟨ω_{V}^{−1},y⟩}dy

and

β(V) = #H^{1}(k, Pic(\overline{V})).

The theoretical constant attached to V and H is defined as

θ_{H}(V) = α(V)β(V)τ_{H}(V). (2.3)

In the following sections we compute θ_{H}(V) for various diagonal cubic surfaces.

3. The Galois module Pic(\overline{V})

The description of this Galois module is based upon the study of the 27 lines of the cubic. We fix notations for these lines which are slightly different from those given by Colliot-Thélène, Kanevsky and Sansuc in [CTKS, p. 9].

Notations 3.1. From now on V is a diagonal cubic surface V given by an equation of the form

aX^{3} + bY^{3} + cZ^{3} + dT^{3} = 0 (3.1)

where a, b, c and d are strictly positive integers with gcd(a, b, c, d) = 1. Let

S = {∞, 3} ∪ { p | p|abcd}

We fix a cubic root α (resp. α′, α′′) of b/a (resp. c/a, d/a) (which is assumed to be in Q if b/a (resp. c/a, d/a) is a cube in Q) and we put

β = \frac{α′′}{α′} = \sqrt[3]{\frac{d}{c}},  β′ = \frac{α}{α′′} = \sqrt[3]{\frac{b}{d}}  and  β′′ = \frac{α′}{α} = \sqrt[3]{\frac{c}{b}}.

We also consider

γ = \frac{α}{α′α′′} = \sqrt[3]{\frac{ab}{cd}},  γ′ = \frac{α′}{α′′α} = \sqrt[3]{\frac{ac}{bd}}  and  γ′′ = \frac{α′′}{αα′} = \sqrt[3]{\frac{ad}{bc}}.

We denote by θ a primitive third root of one. The 27 lines of the cubic surface (3.1) are given by the following equations, where i belongs to Z/3Z:

\begin{aligned}L(i) & : & \begin{cases}X+θ^{i}αY = 0, \\ Z+θ^{i}βT = 0. \end{cases} & L′(i) & : & \begin{cases}X+θ^{i}αY = 0, \\ Z+θ^{i+1}βT = 0. \end{cases} & L′′(i) & : & \begin{cases}X+θ^{i}αY = 0, \\ Z+θ^{i+2}βT = 0. \end{cases} \\ M(i) & : & \begin{cases}X+θ^{i}α′Z = 0, \\ T +θ^{i}β′Y = 0. \end{cases} & M′(i) & : & \begin{cases}X+θ^{i}α′Z = 0, \\ T +θ^{i+1}β′Y = 0. \end{cases} & M′′(i) & : & \begin{cases}X+θ^{i}α′Z = 0, \\ T +θ^{i+2}β′Y = 0. \end{cases} \\ N(i) & : & \begin{cases}X+θ^{i}α′′T = 0, \\ Y +θ^{i}β′′Z = 0. \end{cases} & N′(i) & : & \begin{cases}X+θ^{i}α′′T = 0, \\ Y +θ^{i+1}β′′Z = 0. \end{cases} & N′′(i) & : & \begin{cases}X+θ^{i}α′′T = 0, \\ Y +θ^{i+2}β′′Z = 0. \end{cases} \end{aligned}

Let K be the field Q(θ, α, α′, α′′). It is a Galois extension of Q. In the generic case, K is an extension of degree 54 with a Galois group isomorphic to

(Z/3Z)^{3} ⋊ Z/2Z.

It is generated by the elements c, τ, τ′ and τ′′ characterized by their action on θ, α, α′ and α′′.

θ α α′ α′′
c θ^{2} α α′ α′′
τ θ θα α′ α′′
τ′ θ α θα′ α′′
τ′′ θ α α′ θα′′

Their action on the 27 lines is given as follows: for τ we have

(3.2) \begin{matrix}L(i) & ⟶ & L′′(i + 1) \\ ↖ & ↙ \\ L′(i + 2) \end{matrix}  \begin{matrix}M(i) & ⟶ & M′(i) \\ ↖ & ↙ \\ M′′(i) \end{matrix}  and \begin{matrix}N(i) & ⟶ & N′′(i) \\ ↖ & ↙ \\ N′(i) \end{matrix}

for τ′:

(3.3) \begin{matrix}L(i) & ⟶ & L′′(i) \\ ↖ & ↙ \\ L′(i) \end{matrix}  \begin{matrix}M(i) & ⟶ & M′′(i + 1) \\ ↖ & ↙ \\ M′(i + 2) \end{matrix} and \begin{matrix}N(i) & ⟶ & N′(i) \\ ↖ & ↙ \\ N′′(i) \end{matrix}

for τ′′:

(3.4) \begin{matrix}L(i) & ⟶ & L′(i) \\ ↖ & ↙ \\ L′′(i) \end{matrix}  \begin{matrix}M(i) & ⟶ & M′′(i) \\ ↖ & ↙ \\ M′(i) \end{matrix}  and \begin{matrix}N(i) & ⟶ & N′′(i + 1) \\ ↖ & ↙ \\ N′(i + 2) \end{matrix}

for c:

L(0) L′(0) L′′(0) L(1) L′(1) L′′(1) L(2) L′(2) L′′(2) M(0) M′(0) M′′(0) M(1) M′(1) M′′(1) M(2) M′(2) M′′(2) N(0) N′(0) N′′(0) N(1) N′(1) N′′(1) N(2) N′(2) N′′(2). (3.5)

To describe the relations between the classes of these divisors in Pic(\overline{V}), which shall be useful for the computation of α(V), we consider \overline{V} as the blow-up of a plane P2 \overline{Q} in six points P_{1}, P_{2}, P_{3}, P_{4}, P_{5} and P_{6}. The 27 lines may then be described as the 6 exceptional divisors E_{1}, …, E_{6}, the 15 strict transforms L_{i,j} of the projective lines (P_{i}P_{j}) for 1 ⩽ i < j ⩽ 6 and the 6 strict transforms of the conics Q_{i} going through all points except P_{i}. Let Λ be the preimage of a line of P2 \overline{Q} which does not contain any of the points P_{1}, …, P_{6}. Then

([Λ], [E_{1}], [E_{2}], [E_{3}], [E_{4}], [E_{5}], [E_{6}])

is a basis of Pic(\overline{V}) and we have the following relations in Pic(\overline{V}):

\begin{aligned}[L_{i,j}] & = [Λ] − [E_{i}] − [E_{j}] for 1 ⩽ i < j ⩽ 6, \\ [Q_{i}] & = 2[Λ] − ∑_{j≠i}[E_{j}]. \end{aligned} (3.6)

In the following, we choose the projection of \overline{V} to P2 \overline{Q} so that we have the equalities:

\begin{aligned}E_{1} & = L(0), & E_{2} & = L(1), & E_{3} & = L(2), \\ E_{4} & = M(1), & E_{5} & = M′(2), & E_{6} & = M′′(0), \\ Q_{1} & = L′(1), & Q_{2} & = L′(2), & Q_{3} & = L′(0), \\ Q_{4} & = M(0), & Q_{5} & = M′(1), & Q_{6} & = M′′(2), \\ L_{1,2} & = L′′(1), & L_{2,3} & = L′′(2), & L_{3,1} & = L′′(0), \\ L_{4,5} & = M′′(1), & L_{5,6} & = M(2), & L_{6,4} & = M′(0), \\ L_{1,4} & = N(0), & L_{1,5} & = N(1), & L_{1,6} & = N(2), \\ L_{2,4} & = N′(1), & L_{2,5} & = N′(2), & L_{2,6} & = N′(0), \\ L_{3,4} & = N′′(2), & L_{3,5} & = N′′(0), & L_{3,6} & = N′′(1). \end{aligned} (3.7)

Notations 3.2. We consider the étale algebra E_{1} over Q defined as Q(γ) if ab/cd is not a cube in Q and as Q(θ) × Q otherwise. Similarly, we define the algebra E_{2} (resp. E_{3}) corresponding to γ′ (resp. γ′′) and we put

E = E_{1} × E_{2} × E_{3}.

We also consider the following elements of Pic(\overline{V})

e_{0}^{1} = [M(0)] + [M(1)] + [M(2)],  e_{1}^{1} = [M′(0)] + [M′(1)] + [M′(2)],
e_{2}^{1} = [M′′(0)] + [M′′(1)] + [M′′(2)], e_{0}^{2} = [N(0)] + [N(1)] + [N(2)],
e_{1}^{2} = [N′(0)] + [N′(1)] + [N′(2)],  e_{2}^{2} = [N′′(0)] + [N′′(1)] + [N′′(2)],
e_{0}^{3} = [L(0)] + [L(1)] + [L(2)],  e_{1}^{3} = [L′(0)] + [L′(1)] + [L′(2)],
e_{2}^{3} = [L′′(0)] + [L′′(1)] + [L′′(2)]

and the sets

ℰ_{1} = {e_{0}^{1}, e_{1}^{1}, e_{2}^{1}}, ℰ_{2} = {e_{0}^{2}, e_{1}^{2}, e_{2}^{2}}, ℰ_{3} = {e_{0}^{3}, e_{1}^{3}, e_{2}^{3}},

and ℰ = ℰ_{1} ∪ ℰ_{2} ∪ ℰ_{3}.

Lemma 3.3. The sets ℰ_{1}, ℰ_{2} and ℰ_{3} are globally invariant under the action of Gal(K/Q) and the étale algebra corresponding to the set ℰ_{i} is isomorphic to E_{i}.

Proof. The fact that the sets ℰ_{1}, ℰ_{2} and ℰ_{3} are globally invariant follows immediately from the descriptions (3.2)–(3.5). Theétale algebra F corresponding to a finite Gal(K/Q)- set ℱ may be defined as the algebra

(K[ℱ ])^{Gal(K/Q)}

where K[ℱ ] is the algebra K^{ℱ} and where Gal(K/Q) acts simultaneously on K and ℱ . In the generic case, let us consider

σ = τ′τ′′, σ′ = τ′′τ and σ′′ = ττ′.

Then σ sends γ on θγ and acts trivially on γ′, γ′′ and θ. We may describe similarly the actions of σ′ and σ′′. The action of Gal(K/Q) on ℰ_{1} in the generic case is given by the table

e_{0}^{1} e_{1}^{1} e_{2}^{1}
c e_{0}^{1} e_{2}^{1} e_{1}^{1}
σ e_{1}^{1} e_{2}^{1} e_{0}^{1}
σ′ e_{0}^{1} e_{1}^{1} e_{2}^{1}
σ′′ e_{0}^{1} e_{1}^{1} e_{2}^{1}

This implies that if ab/cd is not a cube in Q, then ℰ_{1} is isomorphic to

Gal(K/Q)/ Gal(K/Q(γ))

as a Gal(K/Q)-set. Then the corresponding étale algebra is

(K[Gal(K/Q)/ Gal(K/Q(γ))])^{Gal(K/Q)} ⟶ K^{Gal(K/Q(γ))} = Q(γ) = E_{1}.

Similarly if ab/cd is a cube in Q, then we may decompose ℰ_{1} into two orbits and we see that the corresponding étale algebra is Q(θ) × Q = E_{1}. The proofs for ℰ_{2} and ℰ_{3} are similar.

Lemma 3.4. There exists an exact sequence of Gal(K/Q) modules

0 → Q^{2} → Q[ℰ] → Pic(\overline{V}) ⊗_{Z} Q → 0.

Proof. By (3.6) and (3.7), we have in Pic(\overline{V}) the relations

e_{0}^{1} = 3[Λ] − [E_{1}] − [E_{2}] − [E_{3}] + [E_{4}] − 2[E_{5}] − 2[E_{6}],
e_{1}^{1} = 3[Λ] − [E_{1}] − [E_{2}] − [E_{3}] − 2[E_{4}] + [E_{5}] − 2[E_{6}],
e_{2}^{1} = 3[Λ] − [E_{1}] − [E_{2}] − [E_{3}] − 2[E_{4}] − 2[E_{5}] + [E_{6}],
e_{0}^{2} = 3[Λ] − 3[E_{1}] − [E_{4}] − [E_{5}] − [E_{6}],
e_{1}^{2} = 3[Λ] − 3[E_{2}] − [E_{4}] − [E_{5}] − [E_{6}],
e_{2}^{2} = 3[Λ] − 3[E_{3}] − [E_{4}] − [E_{5}] − [E_{6}],
e_{0}^{3} = [E_{1}] + [E_{2}] + [E_{3}],
e_{1}^{3} = 6[Λ] − 2[E_{1}] − 2[E_{2}] − 2[E_{3}] − 3[E_{4}] − 3[E_{5}] − 3[E_{6}],
e_{2}^{3} = 3[Λ] − 2[E_{1}] − 2[E_{2}] − 2[E_{3}]

which proves that the natural projection from Q[ℰ] to Pic(\overline{V})⊗_{Z}Q is surjective. Moreover one has the relations

3ω_{V}^{−1} = ∑_{x∈ℰ _{1}} x = ∑_{x∈ℰ _{2}} x = ∑_{x∈ℰ _{3}} x,

which gives a homomorphism of Gal(K/Q)-modules

Q^{2} → Q[ℰ]

and the exact sequence of the lemma.

Notations 3.5. For any prime p and any finite field extension F of Q, we consider the local factor ζ_{F,p} of the function ζ_{F} at p which is defined by

ζ_{F,p}(s) = ∏_{{ v∈Val(F)|v|p }}(1 − #F_{v}^{−s})^{−1}.

Let F be an étale algebra over Q and F = ∏_{i∈I} F_{i} its decomposition in fields. Put

ζ_{F} (s) = ∏_{i∈I} ζ_{F_{i}}(s) and ζ_{F,p}(s) = ∏_{i∈I} ζ_{F_{i},p}(s).

For any prime p, we denote by ν_{F} (p) the number of components of F ⊗_{Q} Q_{p} of degree one over Q_{p}.

Proposition 3.6. With notation as above, for any prime p not in S, one has

(i) L_{p}(s, Pic(\overline{V})) = \frac{ζ_{E,p}(s)}{ζ_{Q,p}(s)^{2}} ,

(ii) Tr(F̃r_{p} | Pic(\overline{V})) = ν_{E}(p) − 2.

Proof. By lemma 3.4, we have

L_{p}(s, Pic(\overline{V})) = \frac{L_{p}(s, Q[ℰ])}{L_{p}(s, Q)^{2}}.

Thus it is enough to prove that if E is an arbitrary étale algebra over Q corresponding to a Gal(\overline{Q}/Q)-set ℰ and if p is a prime such that E/Q is not ramified at p, then

ζ_{E,p}(s) = L_{p}(s, Q[ℰ]).

This well-known assertion follows from the fact that the components of E ⊗ Q_{p} are in bijection with the orbits of F̃r_{p} in ℰ, and the degree of each component is the length of the corresponding orbit. This proves (i).

But this also shows that

Tr(F̃r_{p} | Q[ℰ]) = ν_{E}(p)

which implies (ii).

Remark 3.7. Thus the factor λ′_{p} which was defined in proposition 5.1 in [PT] coincides with L_{p}(1, Pic(\overline{V})) at the good places (as suggested by the referee of that paper).

4. Euler product for the good places

We need to compute the number of solutions of (3.1) modulo p for all primes not in S.

Proposition 4.1. For any prime p not in S, one has

\frac{#V (F_{p})}{p^{2}} = 1 + \frac{ν_{E}(p) − 2}{p} + \frac{1}{p^{2}}

where E is the étale algebra defined in §3.

Proof. By a result of Weil (see [Man2, theorem 23.1]),

#V (F_{p}) = 1 + Tr(Fr_{p} | Pic(\overline{V}))p + p^{2}.

Proposition 3.6 implies that

Tr(Fr_{p} | Pic(\overline{V})) = ν_{E}(p) − 2.

Remark 4.2. We could have proved this result directly as in [PT]. Let N(p) be the number of solutions of (3.1) in F_{p}^{4}. We have

#V (F_{p}) = \frac{N(p) − 1}{p − 1}.

By [IR, §8.7 theorem 5], one has

N(p) = p^{3} + ∑ \overline{χ}_{1}(a)\overline{χ}_{2}(b)\overline{χ}_{3}(c)\overline{χ}_{4}(d)J_{0}(χ_{1}, χ_{2}, χ_{3}, χ_{4}),

where the sum is taken over all quadruples (χ_{1}, …, χ_{4}) of nontrivial cubic characters from F_{p}^{∗} to C^{∗} such that χ_{1}χ_{2}χ_{3}χ_{4} = 1 and where

J_{0}(χ_{1}, χ_{2}, χ_{3}, χ_{4}) = ∑_{t_{1}+···+t_{4}=0} ∏_{i=1}^{4} χ_{i}(t_{i}),

the characters being extended by χ_{i}(0) = 0. For p ≡ 2 mod 3 there are no nontrivial characters and the formula is obvious. Otherwise there are exactly two nontrivial conjugated characters χ and \overline{χ}. By [PT, proof of prop. 4.1], we have

J_{0}(χ_{1}, χ_{2}, χ_{3}, χ_{4}) = p(p − 1)

and

#V (F_{p}) = 1 + p(1 + ∑ χ_{1}(a)χ_{2}(b)χ_{3}(c)χ_{4}(d)) + p^{2}

where the sum is taken over the same quadruples as above. The formula

∑ χ_{1}(a)χ_{2}(b)χ_{3}(c)χ_{4}(d) =

χ(\frac{ab}{cd}) + \overline{χ}(\frac{ab}{cd}) + χ(\frac{ac}{bd}) + \overline{χ}(\frac{ac}{bd}) + χ(\frac{ad}{bc}) + \overline{χ}(\frac{ad}{bc})

implies the result.

Notations 4.3. For any place v of Q, we put

λ_{v} = \begin{cases}\frac{ζ_{E,v}(s)}{ζ_{Q,v}(s)^{2}} if v is finite, \\ 1 otherwise. \end{cases}

TAMAGAWA NUMBERS OF CUBIC SURFACES 11

Remark 4.4. By proposition 3.6, λ_{p} = L_{p}(1, Pic(\overline{V})) if p ∈ Val(Q) − S. Thus the Tamagawa measure ω_{H} is given by the formula

ω_{H} = lim_{s→1}(s − 1)^{rk Pic(V)}(\frac{ζ_{E}(s)}{ζ_{Q}(s)^{2}} ) × ∏_{v∈Val(Q)} λ_{v}^{−1}ω_{H,v}.

By lemmata 3.2 and 3.4 in [PT] and lemma 5.4.6 in [Pe], for any p in Val(Q) − S one has

ω_{H,p}(V (Q_{p})) = \frac{#V (F_{p})}{p^{2}}

(see also [Pe, lemma 2.2.1] and [PT, remark 5.2]). Therefore, the local factor at a good place p is given by

(1 − \frac{1}{p})^{7}(1 + \frac{7}{p} + \frac{1}{p^{2}} )  if p ≡ 1 mod 3 and ν_{E}(p) = 9
(1 − \frac{1}{p})^{4}(1 − \frac{1}{p^{3}} )(1 + \frac{4}{p} + \frac{1}{p^{2}} ) if p ≡ 1 mod 3 and ν_{E}(p) = 6
(1 − \frac{1}{p})(1 − \frac{1}{p^{3}} )^{2}(1 + \frac{1}{p} + \frac{1}{p^{2}} ) if p ≡ 1 mod 3 and ν_{E}(p) = 3
(1 − \frac{1}{p})^{−2}(1 − \frac{1}{p^{3}} )^{3}(1 − \frac{2}{p} + \frac{1}{p^{2}} ) if p ≡ 1 mod 3 and ν_{E}(p) = 0
(1 − \frac{1}{p})(1 − \frac{1}{p^{2}} )^{3}(1 + \frac{1}{p} + \frac{1}{p^{2}} ) if p ≡ 2 mod 3.

We get (for the good places) the factors C_{0}, C_{1}, C_{2} and C_{3} where

C_{0} = ∏_{\begin{matrix}p/|3abcd, \\ p≡2 mod 3. \end{matrix}} (1 − \frac{1}{p^{3}} )(1 − \frac{1}{p^{2}} )^{3},
C_{1} = ∏_{\begin{matrix}p/|3abcd, \\ p≡1 mod 3, \\ ν_{E}(p)=9. \end{matrix}} (1 − \frac{1}{p})^{7}(1 + \frac{7}{p} + \frac{1}{p^{2}} ),

C_{2} = ∏_{\begin{matrix}p/|3abcd, \\ p≡1 mod 3, \\ ν_{E}(p)=6. \end{matrix}} (1 − \frac{1}{p^{3}} )(1 − \frac{1}{p})^{4}(1 + \frac{4}{p} + \frac{1}{p^{2}} ),

C_{3} = ∏_{\begin{matrix}p/|3abcd, \\ p≡1 mod 3, \\ ν_{E}(p)=0 or 3. \end{matrix}} (1 − \frac{1}{p^{3}} )^{3}.

These products converge rapidly and are easily approximated.

5. Density at the bad places

In this section we restrict to cubic surfaces with equations of the form X^{3} + Y^{3} + qZ^{3} + q^{2}T^{3} = 0 (5.1) with q prime and aX^{3} + aY^{3} + qZ^{3} + qT^{3} = 0 (5.2) with q prime and a an integer coprime to q.

Notations 5.1. If V is defined by the equation (3.1), and p is a prime, then we consider

N^{∗}(p^{r}) = #{(x, y, z, t) ∈ (Z/p^{r}Z)^{4}−(pZ/p^{r}Z)^{4} | ax^{3}+by^{3}+cz^{3}+dt^{3}=0 in Z/p^{r}Z}

Remark 5.2. By [PT, lemmata 3.2 and 3.4], there is an explicit integer r_{0} such that

ω_{H,p}(V (Q_{p})) = \frac{1}{1 − p^{−1}} × \frac{N^{∗}(p^{r_{0}})}{p^{3r_{0}}}.

If p = 3 and 3 /| abcd, then a direct computation in (Z/9Z)^{4} gives the value of N^{∗}(9) and thus of ω_{H,p}(V (Q_{p})). Thus, in the following lemma we restrict to the case when V is given by (5.1) or (5.2) and p = q.

Lemma 5.3. If V is given by the equation

X^{3} + Y^{3} + pZ^{3} + p^{2}T^{3} = 0

then for r ⩾ 2,

\frac{N^{∗}(p^{r})}{p^{3r}} = \begin{cases}1 − \frac{1}{p} & if p ≡ 2 mod 3, \\ 3(1 − \frac{1}{p}) & if p ≡ 1 mod 3, \\ \frac{2}{3} & if p = 3. \end{cases}

If V is given by the equation

aX^{3} + aY^{3} + pZ^{3} + pT^{3} = 0,

with p /| a, then for r ⩾ 3,

\frac{N^{∗}(p^{r})}{p^{3r}} = \begin{cases}1 − \frac{1}{p^{2}} & if p ≡ 2 mod 3, \\ 3(1 − \frac{1}{p^{2}} ) & if p ≡ 1 mod 3, \\ \frac{4}{3} & if p = 3. \end{cases}

Remark 5.4. This lemma implies that if V is given by the first equation then the local factor at p is given by

λ_{p}ω_{H,p}(V (Q_{p})) = \begin{cases}(1 − \frac{1}{p^{2}} )(1 − \frac{1}{p}) & if p ≡ 2 mod 3, \\ 3(1 − \frac{1}{p})^{3} & if p ≡ 1 mod 3, \\ \frac{4}{9} & if p = 3, \end{cases}

and if V is given by the second equation then this factor is

λ_{p}ω_{H,p}(V (Q_{p})) = \begin{cases}(1 − \frac{1}{p^{2}} )^{3} & if p ≡ 2 mod 3, \\ 3(1 − \frac{1}{p})^{4}(1 − \frac{1}{p^{2}} ) & if p ≡ 1 mod 3, \\ \frac{16}{27} & if p = 3. \end{cases}

Proof. Let us consider the set of quadruples (x, y, z, t) in (Z/p^{r}Z)^{4} − (pZ/p^{r}Z)^{4} such that

x^{3} + y^{3} + pz^{3} + p^{2}t^{3} = 0 in Z/p^{r}Z. (5.3)

If p | x then p | y, p | z and p | t. Therefore, for any (x, y, z, t) as above, p /| x and p /| y. But for any triple (y, z, t) in (Z/p^{r}Z − pZ/p^{r}Z) × (Z/p^{r}Z)^{2}, there exists exactly one x verifying (5.3) if p ≡ 2 mod 3 and exactly three of them if p ≡ 1 mod 3. If p = 3 and y belongs to Z/3^{r}Z − 3Z/3^{r}Z then (5.3) implies that 3|z. For any triple (y, z, t) with y in (Z/3^{r}Z) − (3Z/3^{r}Z), z in (3Z/3^{r}Z) and t in (Z/3^{r}Z) there exist exactly three x in Z/3^{r}Z which satisfy (5.3). We get that

\frac{N^{∗}(p^{r})}{p^{3r}} = \begin{cases}\frac{(p−1)p^{r−1}×p^{r}×p^{r}}{p^{3r}} = 1 − \frac{1}{p} & if p ≡ 2 mod 3, \\ 3\frac{(p−1)p^{r−1}×p^{r}×p^{r}}{p^{3r}} = 3(1 − \frac{1}{p}) & if p ≡ 1 mod 3, \\ 3\frac{2×3^{r−1}×3^{r−1}×3^{r}}{3^{3r}} = \frac{2}{3} & if p = 3. \end{cases}

Let us now turn to the set of (x, y, z, t) in (Z/p^{r}Z)^{4} − (pZ/p^{r}Z)^{4} such that

ax^{3} + ay^{3} + pz^{3} + pt^{3} = 0.

We decompose this set as follows

N_{1}^{∗}(p^{r}) = # \{ (x, y, z, t) ∈ (Z/p^{r}Z)^{4}−(pZ/p^{r}Z)^{4} |\begin{cases}p /| x, \\ ax^{3} + ay^{3} + pz^{3} + pt^{3} = 0. \end{cases} \}

N_{2}^{∗}(p^{r}) = # \{ (x, y, z, t) ∈ (Z/p^{r}Z)^{4}−(pZ/p^{r}Z)^{4} |\begin{cases}p|x, p /| z, \\ ax^{3} + ay^{3} + pz^{3} + pt^{3} = 0. \end{cases} \}

As above we have the formula

\frac{N_{1}^{∗}(p^{r})}{p^{3r}} = \begin{cases}\frac{(p−1)p^{r−1}×p^{r}×p^{r}}{p^{3r}} = 1 − \frac{1}{p} & if p ≡ 2 mod 3, \\ 3 × \frac{(p−1)p^{r−1}×p^{r}×p^{r}}{p^{3r}} = 3(1 − \frac{1}{p}) & if p ≡ 1 mod 3, \\ 3\frac{2×3^{r−1}×3^{r−1}×3^{r}}{3^{3r}} = \frac{2}{3} & if p = 3, \end{cases}

where for p = 3 we use the equality

3^{r−1} × 3^{r} = #{ (z, t) ∈ (Z/3^{r}Z)^{2} | z^{3} ≡ t^{3} mod 3 }.

On the other hand,

N_{2}^{∗}(p^{r}) = p^{2} \{ (x, y, z, t) ∈ (Z/p^{r−1}Z)^{4} |\begin{cases}p /| z \\ ap^{2}x^{3} + ap^{2}y^{3} + z^{3} + t^{3} = 0. \end{cases} \}

and

\frac{N_{2}^{∗}(p^{r})}{p^{3r}} = \frac{p^{2}}{p^{3}} × \begin{cases}\frac{(p−1)p^{r−2}×p^{r−1}×p^{r−1}}{p^{3(r−1)}} = 1 − \frac{1}{p} & if p ≡ 2 mod 3, \\ 3\frac{(p−1)p^{r−2}×p^{r−1}×p^{r−1}}{p^{3(r−1)}} = 3(1 − \frac{1}{p}) & if p ≡ 2 mod 3, \\ 3\frac{2×3^{r−2}×3^{r−1}×3^{r−1}}{3^{3(r−1)}} = 2 & if p = 3. \end{cases}

We conclude:

\frac{N^{∗}(p^{r})}{p^{3r}} = \begin{cases}1 − \frac{1}{p} + \frac{1}{p} − \frac{1}{p^{2}} = 1 − \frac{1}{p^{2}} & if p ≡ 2 mod 3, \\ 3(1 − \frac{1}{p^{2}} ) & if p ≡ 1 mod 3, \\ \frac{2}{3} + \frac{2}{3} = \frac{4}{3} & if p = 3. \end{cases}

6. The constant α(V)

Since the cubic surfaces we consider in this paper are Q-rational (which implies that β(V) = 1), it remains to compute the rank t of the Picard group and the value of α(V).

Proposition 6.1. If V is given by the equation

X^{3} + Y^{3} + aZ^{3} + a^{2}T^{3} = 0, (6.1)

where a is not a cube in Q, then rk Pic(V) = 2 and α(V) = 2.

If V is given by the equation

aX^{3} + aY^{3} + bZ^{3} + bT^{3} = 0, (6.2)

where a and b are strictly positive integers and b/a is not a cube in Q, then rk Pic(V) = 3 and α(V) = 1.

If V is given by the equation

X^{3} + Y^{3} + Z^{3} + T^{3} = 0 (6.3)

then rk Pic(V) = 4 and α(V) = 7/18.

Proof. To compute α(V) we shall use its original definition [Pe, §2]:

α(V) = Vol{ x ∈ Λ_{eff}(V)^{∨} | ⟨ω_{V}^{−1}, x⟩ = 1 }

where the Lebesgue measure on the affine hyperplane

ℋ (λ) = { x ∈ Pic(V)^{∨} ⊗_{Z} R | ⟨ω_{V}^{−1}, x⟩ = λ }

is defined by the (t − 1)-form dx which is characterized by the relation

dx ∧ dω_{V}^{−1} = dy

(where dω_{V}^{−1} is the linear form defined by ω_{V}^{−1} on Pic(V)^{∨} and dy is the form corresponding to the natural Lebesgue measure on Pic(V)^{∨} ⊗_{Z} R). More explicitely, let (e_{1}, …, e_{t}) be a basis of Pic(V) and (e_{1}^{∨}, …, e_{t}^{∨}) be the dual basis. Write

ω_{V}^{−1} = ∑_{i=1}^{t} λ_{i}e_{i}

with λ_{t} ≠ 0. Let f_{1}, …, f_{t−1} be the projection of e_{1}^{∨}, …, e_{t−1}^{∨} on ℋ (0) along e_{t}^{∨}. Then

dx = \frac{1}{λ_{t}} df_{1}^{∨} ∧ ⋯ ∧ df_{t−1}^{∨} .

By [SK, pages 14 and 55], if O_{1}, …, O_{m} are the orbits of the action of Gal(K/Q) on the 27 lines, then Λ_{eff}(V) is generated by the classes [O_{i}] = ∑_{x∈O_{i}}[x].

When V is given by the equation (6.1) the Galois group Gal(K/Q) is

Z/3Z ⋊ Z/2Z

and the orbits of its action on the 27 lines are

O_{1} = {L(0), L′(0), L′′(0)},
O_{2} = {L(1), L(2), L′(1), L′(2), L′′(1), L′′(2)},
O_{3} = {M(0), M(1), M(2)},
O_{4} = {M′(0), M′(1), M′(2), M′′(0), M′′(1), M′′(2)},
O_{5} = {N(0), N′(1), N′′(2)},
O_{6} = {N(1), N(2), N′(0), N′(2), N′′(0), N′′(1)}.

In the basis ([Λ], [E_{1}], …, [E_{6}]), a basis of Pic(V) = (Pic \overline{V})^{Gal(K:Q)} is given by

e_{1} = ω_{V}^{−1}, e_{2} = −2[E_{4}] + [E_{5}] + [E_{6}].

In the basis (e_{0}, e_{1}), the effective cone Λ_{eff}(V) is generated by the classes

[O_{1}] = e_{1},  [O_{2}] = 2e_{1},  [O_{3}] = e_{1} − e_{2},
[O_{4}] = 2e_{1} + e_{2},  [O_{5}] = e_{1} + e_{2},  [O_{6}] = 2e_{1} − e_{2}.

Therefore, this cone is generated by the elements e_{1} − e_{2} and e_{1} + e_{2} and α(V) is given as the volume of the domain

x = 1, x + y > 0 and x − y > 0,

that is, as the volume of the segment [−1, 1] and α(V) = 2.

If V is given by the equation (6.2) then Gal(K/Q) is isomorphic to

Z/3Z ⋊ Z/2Z

and the orbits of the Galois action on the 27 lines are

O_{1} = {L(0)},
O_{2} = {L(1), L(2)},
O_{3} = {L′(0), L′′(0)},
O_{4} = {L′(1), L′′(2)},
O_{5} = {L′(2), L′′(1)},
O_{6} = {M(0), M′(1), M′′(2)},
O_{7} = {M(1), M(2), M′(0), M′(2), M′′(0), M′′(1)},
O_{8} = {N(0), N(1), N(2)},
O_{9} = {N′(0), N′(1), N′(2), N′′(0), N′′(1), N′′(2)}.

A basis of Pic(V) is given by

e_{1} = ω_{V}^{−1}, e_{2} = [E_{1}], e_{3} = [E_{2}] + [E_{3}],

and the cone Λ_{eff}(V) is generated by

[O_{1}] = e_{2},  [O_{2}] = e_{3},  [O_{3}] = e_{1} − e_{2},
[O_{4}] = e_{1} + e_{2} − e_{3},  [O_{5}] = e_{1} − e_{2},  [O_{6}] = 2e_{1} − e_{2} − e_{3},
[O_{7}] = e_{1} + e_{2} + e_{3},  [O_{8}] = e_{1} − 2e_{2} + e_{3},  [O_{9}] = 2e_{1} + 2e_{2} − e_{3},

that is, by

e_{2}, e_{3}, e_{1} + e_{2} − e_{3}, 2e_{1} − e_{2} − e_{3}, e_{1} − 2e_{2} + e_{3}

(since 3[O_{3}] = [O_{6}] + [O_{8}]). Thus α(V) is the volume of the domain given by

\begin{cases}x = 1, y > 0, z > 0, \\ x + y − z > 0, \\ 2x − y − z > 0, \\ x − 2y + z > 0. \end{cases}

Using the description above, α(V) is the volume of

\begin{cases}0 < y, 0 < z, \\ z − y < 1, \\ y + z < 2, \\ 2y − z < 1. \end{cases}  z  y

Therefore α(V) = 1.

If V is given by the equation (6.3), then Gal(K/Q) = Z/2Z and the orbits of the Galois action on the 27 lines are given by

O_{1} = {L(0)},  O_{2} = {L(1), L(2)},  O_{3} = {L′(0), L′′(0)},
O_{4} = {L′(1), L′′(2)},  O_{5} = {L′(2), L′′(1)},
O_{6} = {M(0)},  O_{7} = {M(1), M(2)}, O_{8} = {M′(0), M′′(0)},
O_{9} = {M′(1), M′′(2)}, O_{10} = {M′(2), M′′(1)},
O_{11} = {N(0)},  O_{12} = {N(1), N(2)}, O_{13} = {N′(0), N′′(0)},
O_{14} = {N′(1), N′′(2)}, O_{15} = {N′(2), N′′(1)}.

A basis of the Picard group is given by

e_{1} = [Λ] − [E_{4}], e_{2} = [E_{1}], e_{3} = [E_{2}] + [E_{3}], e_{4} = −2[E_{4}] + [E_{5}] + [E_{6}].

The effective cone Λ_{eff}(V) is generated by

[O_{1}] = e_{2},  [O_{2}] = e_{3},
[O_{3}] = 3e_{1} − 2e_{2} − e_{3} − e_{4},  [O_{4}] = 3e_{1} − 2e_{3} − e_{4},
[O_{5}] = 3e_{1} − 2e_{2} − e_{3} − e_{4},  [O_{6}] = 2e_{1} − e_{2} − e_{3} − e_{4},
[O_{7}] = e_{1} − e_{4},  [O_{8}] = e_{1},
[O_{9}] = 4e_{1} − 2e_{2} − 2e_{3} − e_{4},  [O_{10}] = e_{1},
[O_{11}] = e_{1} − e_{2},  [O_{12}] = 2e_{1} − 2e_{2} − e_{4},
[O_{13}] = 2e_{1} − e_{3} − e_{4},  [O_{14}] = 2e_{1} − e_{3},
[O_{15}] = 2e_{1} − e_{3} − e_{4}.

Since [O_{3}] = [O_{5}] = [O_{6}] + [O_{11}] and [O_{13}] = [O_{15}] = [O_{6}] + [O_{2}], we get that Λ_{eff}(V) is generated by

\begin{matrix}e_{2}, e_{3}, 3e_{1} − 2e_{3} − e_{4}, 2e_{1} − e_{2} − e_{3} − e_{4}, e_{1} − e_{4}, \\ 4e_{1} − 2e_{2} − 2e_{3} − e_{4}, e_{1} − e_{2}, 2e_{1} − 2e_{2} − e_{4}, 2e_{1} − e_{3}. \end{matrix}

The anticanonical class is given by

ω_{V}^{−1} = 3e_{1} − e_{2} − e_{3} − e_{4}.

Thus α(V) is the volume of the domain

\begin{cases}3x − y − z − t = 1, \\ y > 0, z > 0, \\ x − y > 0, \\ 2x − z > 0, \\ x − t > 0, \\ 3x − 2z − t > 0, \\ 2x − y − z − t > 0, \\ 4x − 2y − 2z − t > 0, \\ 2x − 2y − t > 0, \end{cases}

that is, of the domain P in R^{3} given by

\begin{cases}y > 0, z > 0, \\ x − y > 0, \\ 2x − z > 0, \\ 1 − 2x + y + z > 0, \\ 1 + y − z > 0, \\ 1 − x > 0, \\ 1 + x − y − z > 0, \\ 1 − x − y + z > 0. \end{cases}

We compute its volume as follows: decompose P into cones with apex (0, 0, 0) and supported by the faces not containing this point. Thus we consider the following faces of P:

F_{1} : 1 − x = 0,  F_{2} : 1 − 2x + y + z = 0,
F_{3} : 1 + y − z = 0,  F_{4} : 1 + x − y − z = 0,
F_{5} : 1 − x − y + z = 0.

One has

α(V) = Vol(P) = \frac{1}{3} ∑_{i=1}^{5} Area(F_{i}).

The area of F_{1} is the volume of the domain

\begin{cases}y > 0, z > 0, \\ 1 − y > 0, \\ 2 − z > 0, \\ −1 + y + z > 0, \\ 1 + y − z > 0, \\ 2 − y − z > 0, \\ z − y > 0, \end{cases}  z  y

and we get Area(F_{1}) = \frac{1}{2}. For F_{2} we have the equations

\begin{cases}y > 0, \\ −1 + 2x − y > 0, \\ x − y > 0, \\ 1 + y > 0, \\ 2 − 2x + 2y > 0, \\ 1 − x > 0, \\ 2 − x > 0, \\ x − 2y > 0. \end{cases}  y  x

We get Area(F_{2}) = \frac{1}{6}. For F_{3} we have the same equations and the same area. For F_{4} we have the equations

\begin{cases}y > 0, \\ 1 + x − y > 0, \\ x − y > 0, \\ −1 + x + y > 0, \\ 2 − x > 0, \\ −x + 2y > 0, \\ 1 − x > 0, \\ 2 − 2y > 0. \end{cases}  y  x

We find Area(F_{4}) = 1/8 + 1/24 = 1/6. The face F_{5} is given by the same equations and Area(F_{5}) = 1/6. Finally

α(V) = \frac{1}{3}(\frac{1}{2} + \frac{4}{6}) = \frac{7}{18}.

7. Some statistical formulae

The most naive way to test the conjecture is to compute the quotient

N_{U,H}(B)/θ_{H}(V)B(log B)^{t−1} (7.1)

for large values of B. However, as explained in the introduction, the relative error term is expected to decrease slowly. Therefore it is natural to use the fact that we expect an asymptotic of the form

N_{U,H}(B) = BP (log(B)) + o(B),

where P is a polynomial of degree t−1 with a dominant coefficient equal to θ_{H}(V). With the program of D. J. Bernstein [Be], we can get a family of pairs (B_{i}, N_{U,H}(B_{i}))_{1⩽i⩽N}. In the examples below we took for B_{i} successive powers of 6/5 between 200 and 10^{5}. For any i between 1 and N, let

x_{i} = log(B_{i}) and y_{i} = N_{U,H}(B_{i})/B_{i}.

The simplest statistical tool in this setting is to look for a polynomial Q of degree t − 1 such that

∑_{i=1}^{N}(Q(x_{i}) − y_{i})^{2}

is minimal and to compute its leading coefficient A_{t−1}. We then test the conjecture using the quotient

A_{t−1}/θ_{H}(V) (7.2)

The advantage of this method is that, if the expected formula is correct, and if we take for B_{i} successive powers of a fixed real number λ between B_{1} and B_{N}, then the relative error term for (7.2) should at least decrease as C/(log(B_{N}) − log(B_{1}))^{t−1} for B_{N}/B_{1} large enough with a constant C going to 0 as B_{1} goes to infinity.

Of course, due to the arithmetic nature of N_{U,H}(B), the errors are not as independent as one would need for a clean statistical treatment of the data. Also, since we do not have a good understanding of the difference

N_{U,H}(B) − BP (log(B)),

and in order to limit the number of arbitrary parameters involved in the statistical computation, we prefered not to weight the points.

Notations 7.1. Let R(X, Y) be a polynomial in Q[X, Y] and denote by ⟨R(X, Y)⟩ the mean value of (R(x_{i}, y_{i}))_{1⩽i⩽N}, that is,

⟨R(X, Y)⟩ = \frac{1}{N} ∑_{i=1}^{N} R(x_{i}, y_{i}).

If t = 2 the leading coefficient of Q (if it is uniquely defined) is given by

A_{1} = \frac{⟨XY ⟩ − ⟨Y ⟩⟨X⟩}{⟨X^{2}⟩ − ⟨X⟩^{2}}.

If t = 3 the leading coefficient is

A_{2} = \frac{⟨Y X^{2}⟩ − ⟨Y ⟩⟨X^{2}⟩ − \frac{(⟨X^{3}⟩−⟨X⟩⟨X^{2}⟩)(⟨Y X⟩−⟨Y ⟩⟨X⟩)}{⟨X^{2}⟩−⟨X⟩^{2}}}{⟨X^{4}⟩ − ⟨X^{2}⟩^{2} − \frac{(⟨X^{3}⟩−⟨X⟩⟨X^{2}⟩)^{2}}{⟨X^{2}⟩−⟨X⟩^{2}}} .

If t = 4, the leading coefficient is

A_{3} = \frac{⟨Y X^{3}⟩ − ⟨Y ⟩⟨X^{3}⟩ − \frac{(⟨X^{4}⟩−⟨X⟩⟨X^{3}⟩)(⟨Y X⟩−⟨Y ⟩⟨X⟩)}{⟨X^{2}⟩−⟨X⟩^{2}} − \frac{βδ}{γ}}{⟨X^{6}⟩ − ⟨X^{3}⟩^{2} − \frac{(⟨X^{4}⟩−⟨X⟩⟨X^{3}⟩)^{2}}{⟨X^{2}⟩−⟨X⟩^{2}} − \frac{β^{2}}{γ}} ,

with

β = ⟨X^{5}⟩ − ⟨X^{3}⟩⟨X^{2}⟩ − \frac{⟨X^{3}⟩ − ⟨X⟩⟨X^{2}⟩}{⟨X^{2}⟩ − ⟨X⟩^{2}}(⟨X^{4}⟩ − ⟨X^{3}⟩⟨X⟩),
γ = ⟨X^{4}⟩ − ⟨X^{2}⟩^{2} − \frac{(⟨X^{3}⟩ − ⟨X⟩⟨X^{2}⟩)^{2}}{⟨X^{2}⟩ − ⟨X⟩^{2}} ,
δ = ⟨Y X^{2}⟩ − ⟨Y ⟩⟨X^{2}⟩ − \frac{⟨X^{3}⟩ − ⟨X⟩⟨X^{2}⟩}{⟨X^{2}⟩ − ⟨X⟩^{2}}(⟨Y X⟩ − ⟨Y ⟩⟨X⟩).

In the next section, we denote by θ_{H}^{stat}(V) the leading coefficient A_{t−1}.

8. Presentation of the results

We consider only cubic surfaces of the form (6.1), (6.2), or (6.3). By [CTKS, Lemme 1], the corresponding surface V is Q-rational and, in particular, Br(V) = Br(Q). Thus the Brauer-Manin obstruction to weak approximation is void and

V (A_{Q})^{Br} = V (A_{Q}) = ∏_{v∈Val(Q)} V (Q_{v}).

Moreover,

β(V) = #H^{1}(Q, Pic(\overline{V})) = 1.

By (2.2) and (2.3), the constant θ_{H}(V) may be written as

θ_{H}(V) = α(V)ω_{H}(V (A_{Q}))

Using remark 4.4 we get

θ_{H}(V) = α(V) lim_{s→1}(s − 1)^{t+2}ζ_{E}(s) × ω_{H,∞}(V (R))

× ∏_{p|3abcd} λ_{p}ω_{H,p}(V (Q_{p})) × ∏_{i=0}^{3} C_{i},

where E is the étale algebra defined in 3.2. The residue of the zeta function could have been computed directly (see, for example, [Co, chapter 4]), but instead we used PARI. The volume at the real place is given by the formula

\frac{1}{2} ∫_{\{(x,y,z,t)|\begin{cases}ax^{3}+by^{3}+cz^{3}+dt^{3}=0 \\ sup(|x|,|y|,|z|,|t|)⩽1 \end{cases} \}} ω_{L}(x, y, z, t),

where ω_{L} is the Leray form

ω_{L}(x, y, z, t) = \frac{\sqrt[3]{d}^{−1}}{3(ax^{3} + by^{3} + cz^{3})^{2/3}} dx dy dz .

Decomposing the domain of integration (and using the various expressions of the Leray form) it is possible to remove the singularities of this integral which is then easily estimated on a computer. The factors corresponding to the bad places have been described in section 5 and the constants C_{0}, C_{1}, C_{2}, and C_{3} may be computed directly as in section 4.

We considered the following examples: for the cubic surfaces with a Picard group of rank 2 we used

(S_{1})  X^{3} + Y^{3} + 2Z^{3} + 4T^{3} = 0,
(S_{2})  X^{3} + Y^{3} + 5Z^{3} + 25T^{3} = 0,
(S_{3})  X^{3} + Y^{3} + 3Z^{3} + 9T^{3} = 0.

For the rank 3 case:

(S_{4})  X^{3} + Y^{3} + 2Z^{3} + 2T^{3} = 0,
(S_{5})  X^{3} + Y^{3} + 5Z^{3} + 5T^{3} = 0,
(S_{6})  X^{3} + Y^{3} + 7Z^{3} + 7T^{3} = 0,
(S_{7})  2X^{3} + 2Y^{3} + 3Z^{3} + 3T^{3} = 0,

and for rank 4:

(S_{8})  X^{3} + Y^{3} + Z^{3} + T^{3} = 0.

We draw below the corresponding experimental curves in which we compare the value of N_{U,H}(B)/(B(log B)^{t−1}) with θ_{H}(V). On each drawing, only the points on the right of the vertical line have been used for the computation of θ_{H}^{stat}(V).

\begin{aligned}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & log(B) & 0 & θ_{H}(V) \\ & & & & & & S_{1} \end{aligned} \begin{aligned}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & log(B) & 0 & θ_{H}(V) \\ & & & & & & S_{2} \end{aligned}

\begin{aligned}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & log(B) & 0 & θ_{H}(V) \\ & & & & & & S_{3} \end{aligned} \begin{aligned}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & log(B) & 0 & θ_{H}(V) \\ & & & & & & S_{4} \end{aligned}

\begin{aligned}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & log(B) & 0 & θ_{H}(V) \\ & & & & & & S_{5} \end{aligned} \begin{aligned}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & log(B) & 0 & θ_{H}(V) \\ & & & & & & S_{6} \end{aligned}

\begin{aligned}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & log(B) & 0 & θ_{H}(V) \\ & & & & & & S_{7} \end{aligned} \begin{aligned}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & log(B) & 0 & θ_{H}(V) \\ & & & & & & S_{8} \end{aligned}

We finish with tables of numerical results. The value of θ_{H}^{stat}(V) is obtained from the pairs (B_{i}, N_{U,H}(B_{i})) as described in section 7. We denote by ζ_{E_{i}}^{∗}(1) the limit

ζ_{E_{i}}^{∗}(1) = lim_{s→1}(s − 1)^{t_{i}}ζ_{E_{i}}^{∗}(s),

where t_{i} is the number of components of E_{i}. Note that for the examples with a Picard group of rank 2, C_{2} is equal to 1.

Surface  S_{1}  S_{2}  S_{3}
B  100000  100000  100000
N_{U,H}(B)  433526  286040  455164
α(V)  2  2  2
ab/cd  1/8  1/125  1/27
ζ_{E_{0}}^{∗}(1)  6.045998 × 10^{−1} 6.045998 × 10^{−1} 6.045998 × 10^{−1}
ac/bd  1/2  1/5  1/3
ζ_{E_{1}}^{∗}(1)  8.146241 × 10^{−1} 1.163730  1.017615
ad/bc  2  5  3
ζ_{E_{2}}^{∗}(1)  8.146241 × 10^{−1} 1.163730  1.017615
λ′_{3}ω_{H}(V (Q_{3}))  4/9  4/9  4/9
p_{0}  2  5
λ′_{p_{0}}ω_{H}(V (Q_{p_{0}}))  3/8  96/125
C_{0}  8.306815 × 10^{−1} 3.493824 × 10^{−1} 3.066383 × 10^{−1}
C_{1}  9.540383 × 10^{−1} 8.704106 × 10^{−1} 9.762028 × 10^{−1}
C_{3}  9.893865 × 10^{−1} 9.906098 × 10^{−1} 9.892790 × 10^{−1}
ω_{H}(V (R))  3.255161  1.360417  2.221359
θ_{H}(V)  3.413500 × 10^{−1} 2.290769 × 10^{−1} 3.660885 × 10^{−1}
N_{U,H}(B)/θ_{H}(V)B log(B) 1.103137  1.084575  1.079931
θ_{H}^{stat}(V)/θ_{H}(V)  0.988687  1.067208  1.051041

24 EMMANUEL PEYRE AND YURI TSCHINKEL

For the examples with a Picard group of rank 3, C_{3} is equal to 1.

Surface  S_{4}  S_{5}  S_{6}  S_{7}
B  100000  100000  100000  100000
N_{U,H}(B)  3051198  1976482  3420784  1966160
α(V)  1  1  1  1
ab/cd  1/4  1/25  1/49  4/9
ζ_{E_{0}}^{∗}(1)  8.146241 × 10^{−1} 1.163730  1.265025  1.028996
ac/bd  1  1  1  1
ζ_{E_{1}}^{∗}(1)  6.045998 × 10^{−1} 6.045998 × 10^{−1} 6.045998 × 10^{−1} 6.045998 × 10^{−1}
ad/bc 1 1 1 1
ζ_{E_{2}}^{∗}(1)  6.045998 × 10^{−1} 6.045998 × 10^{−1} 6.045998 × 10^{−1} 6.045998 × 10^{−1}
λ′_{3}ω_{H}(V (Q_{3}))  16/27  16/27  16/27  16/27
p_{0}  2  5  7  2
λ′_{p_{0}}ω_{H}(V (Q_{p_{0}}))  27/64  13824/15625  186624/117649 27/64
C_{0}  8.306815 × 10^{−1} 3.493824 × 10^{−1} 3.066383 × 10^{−1} 8.306815 × 10^{−1}
C_{1}  9.540383 × 10^{−1} 8.704106 × 10^{−1} 9.297617 × 10^{−1} 8.196347 × 10^{−1}
C_{2}  7.827314 × 10^{−1} 8.112747 × 10^{−1} 9.228033 × 10^{−1} 8.294515 × 10^{−1}
ω_{H}(V (R)) 4.105301 2.347970 1.910125 2.430506
θ_{H}(V)  1.895795 × 10^{−1} 1.291945 × 10^{−1} 2.184437 × 10^{−1} 1.290720 × 10^{−1}
N_{U,H}(B)/θ_{H}(V)B log(B)^{2} 1.214249  1.154191  1.181448  1.149252
θ_{H}^{stat}(V)/θ_{H}(V)  0.981952  1.035070  0.999247  1.063376

For the last example we have C_{2} = C_{3} = 1 and E_{1} = E_{2} = E_{3} and we get

Surface  S_{8}
B  100000
N_{U,H}(B)  12137664
α(V)  7/18
ζ_{E_{i}}^{∗}(1)  6.045998 × 10^{−1}
λ′_{3}ω_{H}(V (Q_{3}))  16/27
C_{0}  3.066383 × 10^{−1}
C_{1}  5.129319 × 10^{−1}
ω_{H}(V (R))  6.121864
θ_{H}(V)  4.904057 × 10^{−2}
N_{U,H}(B)/θ_{H}(V)B log(B)^{3} 1.621894
θ_{H}^{stat}(V)/θ_{H}(V)  1.012304

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Institut Fourier, UFR de Mathématiques, UMR 5582, Université de Grenoble I et CNRS, BP 74, 38402 Saint-Martin d’Hères CEDEX, France

URL: http://www-fourier.ujf-grenoble.fr/~peyre
E-mail address: Emmanuel.Peyre@ujf-grenoble.fr

Department of Mathematics, Princeton University, Washington Rd., Princeton, NJ 08544-1000, U.S.A.

E-mail address: ytschink@math.princeton.edu