Estimates for Tamagawa numbers of diagonal cubic surfaces^{1}

Andreas-Stephan Elsenhans^{a}, Jörg Jahnel^{b}

aUniversität Bayreuth, Mathematisches Institut, Universitätsstraße 30, D-95447 Bayreuth, Germany bFachbereich 6, Mathematik, Universität Siegen, Walter-Flex-Straße 3, D-57068 Siegen, Germany

Abstract

For diagonal cubic surfaces, we give an upper bound for E. Peyre’s Tamagawa type number in terms of the coefficients of the defining equation. This bound shows that the reciprocal \frac{1}{τ(S)} admits a fundamental finiteness property on the set of all diagonal cubic surfaces. As an application, we show that the infinite series of Tamagawa numbers related to the Fano cubic bundles considered by Batyrev and Tschinkel [BT] are indeed convergent.

Key words: Diagonal cubic surface, Diophantine equation, E. Peyre’s Tamagawa-type number 2000 MSC:11G35, 11G50, 11G40, 14J20, 14J26

1. Introduction

1.1. –––– A conjecture, due to Yu. I. Manin, asserts that the number of ℚ-rational points of anticanonical height < B on a del Pezzo surface S is asymptotically equal to τB log^{rkPic(S)−1} B, for B → ∞. Further, the coefficient 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.

1.2. E. Peyre’s constant. –––– E. Peyre’s Tamagawa-type number is defined in [PT, Definition 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 defined as

β(S) := #H^{1}(Gal(\overline{ℚ}/ℚ), Pic(S\overline{ℚ} )).

Email addresses: stephan.elsenhans@uni-bayreuth.de (Andreas-Stephan Elsenhans), ~jjahnel@mathematik.uni-siegen.de (Jörg Jahnel)

URL: http://www.staff.uni-bayreuth.de/∼btm216 (Andreas-Stephan Elsenhans), http://wwwuser.gwdg.de/~jjahnel (Jörg Jahnel)

1The computer part of this work was executed on the Sun Fire V20z Servers of the Gauß Laboratory for Scientific 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.

Preprint submitted to Journal of Number Theory January 26, 2010

α(S) is given as follows [Pe, Définition 2.4]. Let Λ_{eff}(S) ⊂ Pic(S) ⊗_{ℤ} ℝ be the cone generated by the effective divisors. Consider the dual cone Λ_{eff}^{∨}(S) ⊂ (Pic(S) ⊗_{ℤ} ℝ)^{∨}, defined by

Λ_{eff}^{∨}(S) := {μ ∈ (Pic(S) ⊗_{ℤ} ℝ)^{∨} | ⟨μ, λ⟩ ≥ 0 for every λ ∈ Λ_{eff}(S)} .

Then,

α(S) := t · vol{ μ ∈ Λ_{eff}^{∨}(S) | ⟨μ, −K⟩ ≤ 1 } .

Here, vol denotes the Lebesgue measure on (Pic(S) ⊗_{ℤ} ℝ)^{∨}, normalized such that a primitive cell of the lattice Pic(S)^{∨} ⊂ (Pic(S) ⊗_{ℤ} ℝ)^{∨} is of measure one.

Further, L ( · , χ_{Pic(S\overline{ℚ} )}) is 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. Then, L (1, χ_{P}) > 0.

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} ) consists of those adelic points which are orthogonal to the Brauer group Br(S) with respect to the Brauer-Manin pairing

S(𝔸_{ℚ} ) × Br(S) → ℚ/ℤ, ({x_{ν}}, α) ↦ ∑_{ν} inv_{ν}α|_{x_{ν}} .

1.3. –––– As S is projective, we have

S(𝔸_{ℚ} ) = ∏_{ν∈Val(ℚ)}S(ℚ_{ν}).

τ_{H} is defined to be a product measure τ_{H} := ∏ _{ν∈Val(ℚ)} τ_{ν}.

For a prime number p, the local measure τ_{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^{m dimS}}.

Here, Pic(S\overline{ℚ} )^{I_{p}} denotes the fixed module under the inertia group.

The measure τ_{∞} is described in [Pe, Lemme 5.4.7]. In the case of a hypersurface of degree d in P^{n}, defined by the equation f = 0, this yields

τ_{∞} (U) = \frac{n + 1 − d}{2} ∫_{\begin{matrix}CU \\ |x_{0}|, ...,|x_{n}|≤1 \end{matrix}} ω_{Leray}

for every Borel set U ⊂ S(ℝ). Here, ω_{Leray} is the Leray measure on the cone CS(ℝ) ⊂ ℝ^{n+1} associated with the equation f = 0. It is given by the differential form \frac{1}{|∂f /∂x_{0}|} dx_{1}∧ …∧ dx_{n}.

1.4. Remark. –––– There is a “(hyper)surface area” ω_{hyp} typically introduced for hypersurfaces in ℝ^{n+1} in multivariable calculus. That measure is actually the canonical volume associated with the Riemannian metric CS(ℝ) inherits from ℝ^{n+1} [Di, 20.8.6.2]. The Leray measure is related to the hypersurface area by the formula ω_{Leray} = \frac{1}{∥ gradf ∥} ω_{hyp}.

1.5. The main result. –––– At least for diagonal cubic surfaces, the reciprocal \frac{1}{τ(S)} admits a fundamental finiteness property. More precisely, we will prove the following result.

Theorem. For 𝔞 = (a_{0}, …, a_{3}) ∈ (ℤ\{0})^{4} any vector, we denote by S^{𝔞} the cubic surface in P_{ℚ}^{3} given by a_{0}x_{0}^{3} + … + a_{3}x_{3}^{3} = 0. Then, for each ε > 0, there exists a constant C(ε) > 0 such that

\frac{1}{τ(S^{𝔞})} ≥ C(ε) · H_{naive}(\frac{1}{a_{0}} : …: \frac{1}{a_{3}})^{\frac{1}{3}−ε}.

1.6. Corollary (Fundamental finiteness). —– For each T > 0, there are only finitely many diagonal cubic surfaces S^{𝔞}: a_{0}x_{0}^{3} + … + a_{3}x_{3}^{3} = 0 in P_{ℚ}^{3} such that τ(S^{𝔞}) > T.

1.7. Remark. –––– For diagonal quartic threefolds, these results were shown in [EJ]. The case of the classical cubic surfaces is, however, more complicated.

The reason for this is that quartic threefolds are of geometric Picard rank one. Hence, the Gal(\overline{ℚ}/ℚ)-representation considered was always trivial and the L-factor was automatically equal to 1. In the situation of a diagonal cubic surface, the factors lim_{s→1}(s − 1)^{t}L (s, χ_{Pic(S\overline{ℚ} )}) add new difficulty.

There is also a difference concerning the factors α and β. This point is, however, of minor significance. For quartic threefolds, we always had α(S) = β(S) = 1. For cubic surfaces, these factors may vary but it is not at all hard to estimate them.

1.8. An application. –––– For Fano varieties of dimension ≥ 3, the obvious generalization of Manin’s conjecture is known to be wrong. Due to Batyrev and Tschinkel [BT], there are counterexamples of Picard rank 2. These are smooth hypersurfaces X ⊂ P^{n}× P^{3} of bidegree (1, 3). Such a hypersurface is equipped with a fibration into cubic surfaces given by the projection to the first factor. It is assumed that those are diagonal.

Seemingly, many people believe that the actual growth of the number of ℚ-rational points on X is dominated by the fibres of Picard rank 4. This means, the asymptotics is expected to be τB log^{3}B for

τ := ∑_{x∈P^{n}(ℚ) S^{ι(x)}non−singular rkPic(S^{ι(x)})=4} \frac{1}{H_{naive}^{n}(x)} τ(S^{ι(x)}) .  (1)

Here, ι: P^{n}−→ (P^{3})^{∨} is the linear map defined by the fibration.

As an application of Theorem 1.5, we will show that the series (1) are indeed convergent. For this, as will turn out, it is already sufficient that the Tamagawa numbers of diagonal cubic surfaces are uniformly bounded. Details will be given in section 3.

2. Estimates for Peyre’s constant

Consider a general diagonal cubic surface S^{(a_{0},...,a_{3})} ⊂ P_{ℚ}^{3} given by

a_{0}x_{0}^{3} + … + a_{3}x_{3}^{3} = 0 .

Our goal is to establish the estimate for τ^{(a_{0},...,a_{3})} := τ(S^{(a_{0},...,a_{3})}) formulated in Theorem 1.5. For this, in the subsections below, we will give an individual estimate for each of the factors occurring in the definition of τ(S^{(a_{0},...,a_{3})}).

2.1. Estimates for α and β

2.1.1. –––– Recall that on a smooth cubic surface 𝒮 over an algebraically closed field, there are exactly 27 lines. For the Picard group, which is isomorphic to ℤ^{7}, the classes of these lines form a system of generators.

2.1.2. Notation. –––– i) The set ℒ of the 27 lines is equipped with the intersection product ⟨ , ⟩: ℒ ×ℒ → {−1, 0, 1}. The pair (ℒ , ⟨ , ⟩) is the same for all smooth cubic surfaces. It is well known [Ma, Theorem 23.9.ii] that the group of permutations of ℒ respecting ⟨ , ⟩ is isomorphic to W(E_{6}). We fix such an isomorphism. Denote by F ⊂ Div(𝒮) the group generated by the 27 lines and by F_{0} ⊂ F the subgroup of principal divisors. Then, F is equipped with an operation of W(E_{6}) such that F_{0} is a W(E_{6})- submodule. We have Pic(𝒮) F/F_{0}. ii) If S is a smooth cubic surface over ℚ then Gal(\overline{ℚ}/ℚ) operates canonically on the set ℒ_{S} of the 27 lines on S\overline{ℚ} . Fix a bijection i_{S}: ℒ_{S} −→^{}ℒ respecting the intersection pairing. This induces a group homomorphism ι_{S}: Gal(\overline{ℚ}/ℚ) → W(E_{6}). We denote its image by G ⊂ W(E_{6}).

2.1.3. Lemma. –––– There is a constant c such that, for all smooth cubic surfaces S over ℚ,

1 ≤ β(S) ≤ c .

Proof. By definition, β(S) = #H^{1}(Gal(\overline{ℚ}/ℚ), Pic(S\overline{ℚ} )). Using the notation just introduced, we may write H^{1}(Gal(\overline{ℚ}/ℚ), Pic(S\overline{ℚ} )) = H^{1}(G, F/F_{0}).

Note that this cohomology group is always finite. Indeed, since G is a finite group and F/F_{0} is a finite ℤ[G]-module, the description via the standard complex shows it is finitely generated. Further, it is annihilated by #G.

H^{1}(G, F/F_{0}) depends only on the subgroup G ⊂ W(E_{6}) occurring. For that, there are finitely many possibilities. This implies the claim. □

2.1.4. Remarks. –––– i) A more precise consideration [Ma, Proposition 31.3] yields a canonical isomorphism

H^{1}(Gal(\overline{ℚ}/ℚ), Pic(S\overline{ℚ} )) Hom((NF ∩ F_{0})/NF_{0}, ℚ/ℤ) .

Here, N is the norm map under the operation of G. As an application of this, one may inspect the 350 conjugacy classes of subgroups of W(E_{6}) using GAP. The calculations show that the lemma is actually true for c = 9. ii) Diagonal cubic surfaces actually provide only 16 of the 350 conjugacy classes. Eight of them may be realized over ℚ, the others over ℚ(ζ_{3}) [CTKS].

2.1.5. Lemma. –––– There are positive constants c_{1} and c_{2} such that, for all smooth cubic surfaces S over ℚ satisfying S(𝔸_{ℚ} ) ∅,

c_{1} ≤ α(S) ≤ c_{2}.

Proof. Again, we claim that α(S) is completely determined by the group G ⊂ W(E_{6}). Thus, suppose that we do not have the full information available about what surface S is but are given the group G only.

The assumption S(𝔸_{ℚ} ) ∅ makes sure that Pic(S) Pic(S\overline{ℚ} )^{G} [KT, Remark 3.2.ii)]. We may therefore write Pic(S) (F/F_{0})^{G}. The effective cone

Λ_{eff}(S) ⊂ Pic(S)⊗_{ℤ} ℂ (F/F_{0})^{G}⊗_{ℤ} ℂ

is generated by the symmetrizations of the classes ℓ_{1}, …, ℓ_{27} of the 27 lines in F. In particular, it is determined by G, completely. Further, we have K = −\frac{1}{9}(ℓ_{1}+…+ℓ_{27}). These data are sufficient to compute α(S) according to its very definition. □

2.1.6. Remark. –––– Here, we do not know the optimal values of c_{1} and c_{2} in explicit form. α(S) has not yet been computed in all cases.

2.2. An estimate for the L-factor

2.2.1. –––– In the case of the diagonal cubic surface S^{(a_{0},...,a_{3})} ⊂ P_{ℚ}^{3} , given by a_{0}x_{0}^{3} + … + a_{3}x_{3}^{3} = 0 for a_{0}, …, a_{3} ∈ ℤ \ {0}, the 27 lines on S^{(a_{0},...,a_{3})} may easily be written down explicitly. Indeed, for each pair (i, j) ∈ (ℤ/3ℤ)^{2}, the system

\sqrt[3]{a_{0}} x_{0} + ζ_{3}^{i}\sqrt[3]{a_{1}} x_{1} = 0
\sqrt[3]{a_{2}} x_{2} + ζ_{3}^{j}\sqrt[3]{a_{3}} x_{3} = 0

of equations defines a line on S^{(a_{0},...,a_{3})}. Decomposing the index set {0, …, 3} differently into two subsets of two elements each yields all the lines. In particular, we see that the 27 lines may be defined over K = ℚ(ζ_{3}, \sqrt[3]{a_{1}/a_{0}}, \sqrt[3]{a_{2}/a_{0}}, \sqrt[3]{a_{3}/a_{0}}).

2.2.2. –––– This is an abelian extension of ℚ(ζ_{3}). Therefore, the irreducible representations of Gal(K/ℚ) are at most two-dimensional. Besides the trivial representation, there is the non-trivial Dirichlet character λ of ℚ(ζ_{3})/ℚ. The two-dimensional irreducible representations are actually representations of a factor group of the form Gal(ℚ(ζ_{3}, \sqrt[3]{a_{0}^{e_{0}} · . . .· a_{3}^{e_{3}}})/ℚ) S_{3} for e_{0}, …, e_{3} ∈ {0, 1, 2}.

2.2.3. Lemma. –––– Let a and b be integers different from zero. Then,

|Disc(ℚ(ζ_{3},\sqrt[3]{ab^{2}})/ℚ)| ≤ 3^{9}a^{4}b^{4}.

Proof. We have, at first,

|Disc(ℚ(ζ_{3},\sqrt[3]{ab^{2}})/ℚ)|≤ |Disc(ℚ(ζ_{3})/ℚ)|^{3} · Disc(ℚ(\sqrt[3]{ab^{2}})/ℚ)^{2}
= 27 · Disc(ℚ(\sqrt[3]{ab^{2}})/ℚ)^{2}.

Further, by [De, §4], we know

|Disc(ℚ(\sqrt[3]{ab^{2}})/ℚ)|≤ 3^{3}a^{2}b^{2} .

This shows |Disc(ℚ(ζ_{3},\sqrt[3]{ab^{2}})/ℚ)| ≤ 3^{9}a^{4}b^{4}. □

2.2.4. Proposition. –––– For each ε > 0, there exist positive constants c_{1} and c_{2} such that

c_{1} · |a_{0} · . . .· a_{3}|^{−ε} < lim_{s→1}(s − 1)^{t}L (s, χ_{Pic(S(a_{0},...,a_{3}) ; \overline{ℚ})}) < c_{2} · |a_{0} · . . .· a_{3}|^{ε}

for all (a_{0}, …, a_{3}) ∈ (ℤ\{0})^{4}. Here, t = rkPic(S). Proof. The Galois representation Pic(S(a_{0},...,a_{3}) ; \overline{ℚ}) ⊗_{ℤ} ℂ contains the trivial representation t times as a direct summand. Therefore,

L(s, χ_{Pic(S(a_{0},...,a_{3}) ; \overline{ℚ})}) = ζ(s)^{t}· L (s, χ_{P})

where ζ denotes the Riemann zeta function and P is a representation which does not contain trivial components. All we need to show is

c_{1} · |a_{0} · . . .· a_{3}|^{−ε} < L (1, χ_{P}) < c_{2} · |a_{0} · . . .· a_{3}|^{ε}.

L ( · , χ_{P}) is the product [Ne, Chapter VII, Theorem (10.4).ii)] of not more than six factors of the form L ( · , λ) for λ the non-trivial Dirichlet character of ℚ(ζ_{3})/ℚ and at most three factors which are Artin-L-functions L ( · , ν^{K}) for two-dimensional irreducible representations.

Here, K = ℚ(ζ_{3}, \sqrt[3]{a_{0}^{e_{0}} · . . .· a_{3}^{e_{3}}}) for certain e_{0}, …, e_{3} ∈ \{0, 1, 2\}. As L (1, λ) does not depend on a_{0}, …, a_{3}, at all, it will suffice to show

c_{1}(ε) · |a_{0} · . . .· a_{3}|^{−ε} < L (1, ν^{K}) < c_{2}(ε) · |a_{0} · . . .· a_{3}|^{ε}

for each ε > 0.

ν^{K} is the only irreducible two-dimensional character of Gal(K/ℚ) S_{3}. For that reason, by virtue of [Ne, Chapter VII, Corollary (10.5)], we have

ζ_{K}(s) = ζ_{ℚ} (s) · L (s, λ) · L (s, ν^{K})^{2}
= ζ_{ℚ(ζ_{3})} (s) · L (s, ν^{K})^{2}

for a complex variable s. It, therefore, suffices in our particular situation to estimate the residue res_{s=1} ζ_{K}(s) of the Dedekind zeta function of K.

An estimate from above has been given by C. L. Siegel. In view of the analytic class number formula, his [Si, Satz 1] gives

res_{s=1}ζ_{K}(s) < C [logDisc(K/ℚ)]^{5}
≤ C [log(3^{9}a_{0}^{4}a_{1}^{4}a_{2}^{4}a_{3}^{4})]^{5}
= C [4 log|a_{0} · . . .· a_{3}| + 9 log3]^{5}

for a certain constant C. The final term is less than c_{2}(ε) · |a_{0} · . . .· a_{3}|^{ε} for every ε > 0. On the other hand, H. M. Stark [St, formula (1)] shows

res_{s=1}ζ_{K}(s) > C(ε)·Disc(K/ℚ)^{−ε/4}

for every ε > 0 which implies res_{s=1}ζ_{K}(s) > c_{1}(ε)·|a_{0} · . . .· a_{3}|^{−ε}. □

2.3. An estimate for the factors at the finite places

2.3.1. Lemma. –––– There are two positive constants c_{1} and c_{2} such that, for all a_{0}, …, a_{3} ∈ ℤ\{0},

c_{1} < ∏_{{p prime \atop p∤3a_{0}···· ·a_{3}}} τ_{p}(S^{(a_{0},...,a_{3})}(ℚ_{p})) < c_{2} .

Proof. For a prime p of good reduction, Hensel’s Lemma implies

τ_{p}(S^{(a_{0},...,a_{3})}(ℚ_{p})) = det(1 − p^{−1} Frob_{p} | Pic(S\overline{ℚ} ))· \frac{#S^{(a_{0},...,a_{3})}(𝔽_{p})}{p^{2}} .

Further, for the number of points on a non-singular cubic surface over a finite field, the Lefschetz trace formula can be made completely explicit [Ma, Theorem 27.1]. It shows

#S^{(a_{0},...,a_{3})}(𝔽_{p}) = p^{2} + p · tr(Frob_{p} | Pic(S\overline{ℚ} ))+ 1 .

Denoting the eigenvalues of the Frobenius on Pic(S\overline{ℚ} ) by λ_{1}, …, λ_{7}, we find

τ_{p}(S^{(a_{0},...,a_{3})}(ℚ_{p})) = (1 − λ_{1}p^{−1})(1 − λ_{2}p^{−1}) · . . . · (1 − λ_{7}p^{−1})

· [1 + (λ_{1} + ⋯+ λ_{7})p^{−1} + p^{−2}]

= (1 − σ_{1}p^{−1} + σ_{2}p^{−2} − σ_{3}p^{−3} + …− σ_{7}p^{−7})(1 + σ_{1}p^{−1} + p^{−2})
= 1 + (1 − σ_{1}^{2} + σ_{2})p^{−2} − (σ_{1} − σ_{1}σ_{2} + σ_{3})p^{−3} + …
… − (σ_{5} − σ_{1}σ_{6} + σ_{7})p^{−7} + (σ_{6} − σ_{1}σ_{7})p^{−8} − σ_{7}p^{−9}

where σ_{i} denote the elementary symmetric functions in λ_{1}, …, λ_{7}.

We know |λ_{i}| = 1 for all i. Estimating very roughly, we have |σ_{j}| ≤ ({7 \atop j}) ≤ 7^{j} and see

1 − 99p^{−2} − 7·99p^{−3} − …− 7^{7}·99p^{−9} ≤ τ_{p}(S^{(a_{0},...,a_{3})}(ℚ_{p})) ≤ ≤ 1 + 99p^{−2} + 7·99p^{−3} +…+ 7^{7}·99p^{−9} .

I.e., 1 − 99p^{−2} \frac{1}{1−7/p} < τ_{p}(S^{(a_{0},...,a_{3})}(ℚ_{p})) < 1 + 99p^{−2} \frac{1}{1−7/p} . The infinite product over all 1 − 99p^{−2} \frac{1}{1−7/p} (respectively 1 + 99p^{−2} \frac{1}{1−7/p}) is convergent.

The left hand side is positive for p > 13. For the small primes remaining, we need a better lower bound. For this, note that a cubic surface over a finite field 𝔽_{p} always has at least one 𝔽_{p}-rational point. This yields τ_{p}(S^{(a_{0},...,a_{3})}(ℚ_{p})) ≥ (1 − 1/p)^{7}/p^{2} > 0. □

2.3.2. Remark. –––– It will require by far more labour to estimate the product over the finitely many bad primes, uniformly over all diagonal cubic surfaces.

2.3.3. Notation. –––– i) For a prime number p and an integer x 0, we put x^{(p)} := p^{ν_{p}(x)}. Note x^{(p)} = 1/∥x ∥_{p}for the normalized p-adic valuation. ii) For integers x_{1}, …, x_{n}, not all equal to zero, we write

gcd_{p}(x_{1}, …, x_{n}) := [gcd(x_{1}, …, x_{n})]^{(p)}.

Observe, if x_{1}, …, x_{n} 0 then we have gcd_{p}(x_{1}, …, x_{n}) = gcd(x_{1}^{(p)}, …, x_{n}^{(p)}). iii) By putting ν(x) := min_{{ξ∈ℤp \atop x=(ξ mod p^{r})}}ν(ξ), we carry the p-adic valuation from ℤ_{p} over to ℤ/p^{r}ℤ.

Note that any 0 x ∈ ℤ/p^{r}ℤ has the form x = ε·p^{ν(x)} where ε ∈ (ℤ/p^{r}ℤ)^{∗} is a unit. Clearly, ε is unique only in the case ν(x) = 0.

2.3.4. Definition. –––– For (a_{0}, …, a_{3}) ∈ ℤ^{4}, r ∈ ℕ, and ν_{0}, …, ν_{3} ≤ r, put

\begin{matrix}N_{ν_{0},...,ν_{3};a_{0},...,a_{3}}^{(r)} := \{ (x_{0}, …, x_{3}) ∈ (ℤ \\ /p^{r}ℤ)^{4} | \end{matrix} ; ν(x_{0}) = ν_{0}, …,ν(x_{3}) = ν_{3}; a_{0}x_{0}^{3} + …+ a_{3}x_{3}^{3} = 0 ∈ ℤ/p^{r}ℤ \} .

For the particular case ν_{0} = …= ν_{3} = 0, we will write Z_{a_{0},...,a_{3}}^{(r)} := N_{0,...,0;a_{0},...,a_{3}}^{(r)}, i.e.,

Z_{a_{0},...,a_{3}}^{(r)} = { (x_{0}, …, x_{3}) ∈ [(ℤ/p^{r}ℤ)^{∗}]^{4} | a_{0}x_{0}^{3} + …+ a_{3}x_{3}^{3} = 0 ∈ ℤ/p^{r}ℤ } .

We will use the notation z_{a_{0},...,a_{3}}^{(r)} := #Z_{a_{0},...,a_{3}}^{(r)}.

2.3.5. Sublemma. –––– If p^{k}|a_{0}, …, a_{3} and r > k then we have

z_{a_{0},...,a_{3}}^{(r)} = p^{4k} · z_{a_{0}/p^{k},...,a_{3}/p^{k}}^{(r−k)}.

Proof. Since a_{0}x_{0}^{3} + …+ a_{3}x_{3}^{3} = p^{k}(a_{0}/p^{k} · x_{0}^{3} + …+ a_{3}/p^{k} · x_{3}^{3}), there is a surjection

ι: Z_{a_{0},...,a_{3}}^{(r)} −→ Z_{a_{0}/p^{k},...,a_{3}/p^{k}}^{(r−k)},

given by (x_{0}, …, x_{3}) ↦ ((x_{0} mod p^{r−k}), …, (x_{3} mod p^{r−k})). The kernel of the homomorphism of modules underlying ι is (p^{r−k}ℤ/p^{r}ℤ)^{4}. □

2.3.6. Lemma. –––– Assume gcd_{p}(a_{0}, …, a_{4}) = p^{k}. Then, there is an estimate

z_{a_{0},...,a_{4}}^{(r)} ≤ 3p^{3r+k}.

Proof. Suppose first that k = 0. This means, one of the coefficients is prime to p. Without restriction, assume p ∤ a_{0}.

For any (x_{1}, x_{2}, x_{3}) ∈ (ℤ/p^{r}ℤ)^{3}, there appears an equation of the form a_{0}x_{0}^{3} = c. It cannot have more than three solutions in (ℤ/p^{r}ℤ)^{∗}. Indeed, for p odd, this follows directly from the fact that (ℤ/p^{r}ℤ)^{∗} is a cyclic group. On the other hand, in the case p = 2, we have (ℤ/2^{r}ℤ)^{∗} ℤ/2^{r−2}ℤ × ℤ/2ℤ. Again, there are only up to three solutions possible.

The general case may now easily be deduced from Sublemma 2.3.5. Indeed, if k < r then

z_{a_{0},...,a_{3}}^{(r)} = p^{4k} · z_{a_{0}/p^{k},...,a_{3}/p^{k}}^{(r−k)} ≤ p^{4k} · 3p^{3(r−k)} = 3p^{3r+k}.

On the other hand, if k ≥ r then the assertion is completely trivial since

z_{a_{0},...,a_{3}}^{(r)} = #Z_{a_{0},...,a_{3}}^{(r)} < p^{4r} ≤ p^{3r+k} < 3p^{3r+k}.  □

2.3.7. Remark. –––– The proof shows that in the case p ≡ 2 (mod 3) one could reduce the coefficient to 1. Unfortunately, this observation does not lead to a substantial improvement of our final result.

2.3.8. Lemma. –––– Let r ∈ ℕ and ν_{0}, …, ν_{3} ≤ r. Then,

#N_{ν_{0},...,ν_{3};a_{0},...,a_{3}}^{(r)} = \frac{z_{p^{3ν_{0}}a_{0}, ...,p^{3ν_{3}}a_{3}}^{(r)} · φ(p^{r−ν_{0}}) · . . .· φ(p^{r−ν_{3}})}{φ(p^{r})^{4}}.

Proof. As p^{3ν_{0}}a_{0}x_{0}^{3} + …+ p^{3ν_{3}}a_{3}x_{3}^{3} = a_{0}(p^{ν_{0}}x_{0})^{3} + …+ a_{3}(p^{ν_{3}}x_{3})^{3}, we have a surjection

π: Z_{p^{3ν_{0}}a_{0},...,p^{3ν_{3}}a_{3}}^{(r)} −→ N_{ν_{0},...,ν_{3};a_{0},...,a_{3}}^{(r)},

given by (x_{0}, …, x_{3}) ↦ (p^{ν_{0}}x_{0}, …, p^{ν_{3}}x_{3}).

For i = 0, …, 3, consider the mapping ι: ℤ/p^{r}ℤ → ℤ/p^{r}ℤ, x ↦ p^{ν_{i}}x. If ν_{i} = r then ι is the zero map. All φ(p^{r}) = (p − 1)p^{r−1} units are mapped to zero. Otherwise, observe that ι is p^{ν_{i}} : 1 onto its image. Further, ν(ι(x)) = ν_{i} if and only if x is a unit. By consequence, π is (K^{(ν_{0})} · . . .· K^{(ν_{3})}) : 1 when we put K^{(ν)} := p^{ν} for ν < r and K^{(r)} := (p − 1)p^{r−1}. Summarizing, we could have written K^{(ν)} := φ(p^{r})/φ(p^{r−ν}). The assertion follows. □

2.3.9. Corollary. –––– Let (a_{0}, …, a_{3}) ∈ (ℤ\{0})^{4}. Then, for the local factor τ_{p}(S^{(a_{0},...,a_{3})}(ℚ_{p})), one has

τ_{p}(S^{(a_{0},...,a_{3})}(ℚ_{p})) = det(1 − p^{−1} Frob_{p} | Pic(S\overline{ℚ} )^{I_{p}})

· lim_{r→∞} ∑_{ν_{0},...,ν_{3}=0}^{r} \frac{z_{p^{3ν_{0}}a_{0}, ...,p^{3ν_{3}}a_{3}}^{(r)} · φ(p^{r−ν_{0}}) · . . .· φ(p^{r−ν_{3}})}{p^{3r} · φ(p^{r})^{4}}.

Proof. [PT, Corollary 3.5] implies that

τ_{p}(S^{(a_{0},...,a_{3})}(ℚ_{p})) = det(1 − p^{−1} Frob_{p} | Pic(S\overline{ℚ} )^{I_{p}})· lim_{r→∞} ∑_{ν_{0},...,ν_{3}=0}^{r} \frac{#N_{ν_{0},...,ν_{3};a_{0},...,a_{3}}^{(r)}}{p^{3r}}.

Lemma 2.3.8 yields the assertion. □

2.3.10. Proposition. –––– Let (a_{0}, …, a_{3}) ∈ (ℤ\\{0\})^{4}. Then, for each ε such that 0 < ε < \frac{1}{3}, one has

τ_{p}(S^{(a_{0},...,a_{3})}(ℚ_{p})) ≤ (1 + \frac{1}{p})^{7} · 3(\frac{1}{1 − \frac{1}{p^{1−3ε}}} )(\frac{1}{1 − \frac{1}{p^{ε}}} )^{3} · (a_{0}^{(p)}a_{1}^{(p)}a_{2}^{(p)})^{\frac{1−ε}{3}} (a_{3}^{(p)})^{ε}.

Proof. We use the formula from Corollary 2.3.9. The eigenvalues of the Frobenius on Pic(S\overline{ℚ} )^{I_{p}} are all roots of unity. Therefore, the first factor is at most (1 + 1/p)^{7}. Further, by Lemma 2.3.6,

z_{p^{3ν_{0}}a_{0}, ...,p^{3ν_{3}}a_{3}}^{(r)}/p^{3r} ≤ 3 gcd_{p}(p^{3ν_{0}}a_{0}, …, p^{3ν_{3}}a_{3})
= 3 gcd(p^{3ν_{0}}a_{0}^{(p)}, …, p^{3ν_{3}}a_{3}^{(p)}).

Writing k_{i} := ν_{p}(a_{i}) = ν_{p}(a_{i}^{(p)}), we see

z_{p^{3ν_{0}}a_{0}, ...,p^{3ν_{3}}a_{3}}^{(r)}/p^{3r} ≤ 3 gcd(p^{3ν_{0}+k_{0}}, …, p^{3ν_{3}+k_{3}})
= 3p^{min{3ν_{0}+k_{0}, ...,3ν_{3}+k_{3}}}.

We estimate the minimum by a weighted arithmetic mean with weights \frac{1−ε}{3} , \frac{1−ε}{3} , \frac{1−ε}{3} , and ε,

min\{3ν_{0} + k_{0}, …, 3ν_{3} + k_{3}\}≤ \frac{1 − ε}{3} · (3ν_{0} + k_{0}) + \frac{1 − ε}{3} · (3ν_{1} + k_{1})

+ \frac{1 − ε}{3} · (3ν_{2} + k_{2}) + ε(3ν_{3} + k_{3})

= (1 − ε)(ν_{0} + ν_{1} + ν_{2}) + 3εν_{3}

+ \frac{1 − ε}{3} (k_{0} + k_{1} + k_{2}) + εk_{3} .

This shows

z_{p^{3ν_{0}}a_{0}, ...,p^{3ν_{3}}a_{3}}^{(r)}/p^{3r} ≤ 3p^{(1−ε)(ν_{0}+ν_{1}+ν_{2})+3εν_{3}+ \frac{1−ε}{3} (k_{0}+k_{1}+k_{2})+εk_{3}}
= 3p^{(1−ε)(ν_{0}+ν_{1}+ν_{2})+3εν_{3}} · (a_{0}^{(p)}a_{1}^{(p)}a_{2}^{(p)})^{\frac{1−ε}{3}} (a_{3}^{(p)})^{ε}.

We may therefore write

τ_{p}(S^{(a_{0},...,a_{3})}(ℚ_{p})) ≤ (1 + \frac{1}{p})^{7} · 3(a_{0}^{(p)}a_{1}^{(p)}a_{2}^{(p)})^{\frac{1−ε}{3}} (a_{3}^{(p)})^{ε}

· lim_{r→∞} ∑_{ν_{0},...,ν_{3}=0}^{r}\frac{p^{(1−ε)(ν_{0}+ν_{1}+ν_{2})+3εν_{3}} · φ(p^{r−ν_{0}}) · . . .· φ(p^{r−ν_{3}})}{φ(p^{r})^{4}}.

Here, the term under the limit is precisely the product of three copies of the finite sum

∑_{ν=0}^{r} \frac{p^{(1−ε)ν} · φ(p^{r−ν})}{φ(p^{r})} = ∑_{ν=0}^{r−1} \frac{1}{(p^{ε})^{ν}} + \frac{p}{p − 1} \frac{1}{(p^{ε})^{r}}

and one copy of the finite sum

∑_{ν=0}^{r} \frac{p^{3εν} · φ(p^{r−ν})}{φ(p^{r})} = ∑_{ν=0}^{r−1} \frac{1}{(p^{1−3ε})^{ν}} + \frac{p}{p − 1} \frac{1}{(p^{1−3ε})^{r}} .

For r → ∞, geometric series do appear while the additional summands tend to zero. □

2.3.11. Remark. –––– The constants

C_{p}^{(ε)} := (1 + \frac{1}{p})^{7} · 3(\frac{1}{1 − \frac{1}{p^{1−3ε}}} )(\frac{1}{1 − \frac{1}{p^{ε}}} )^{3}

are clearly not optimal in any sense. Note, in particular, that we did not put much effort into the bound for det(1 − p^{−1} Frob_{p} | Pic(S\overline{ℚ} )^{I_{p}}).

However, and this is what is important for our application, we clearly have that C_{p}^{(ε)} is bounded for p → ∞, say C_{p}^{(ε)} ≤ C^{(ε)}. We do not know of an improvement which would make the product ∏ _{p} C_{p}^{(ε)} converge.

2.3.12. Proposition. –––– For each ε such that 0 < ε < \frac{1}{3}, there exists a constant c such that

∏_{p prime}τ_{p}(S^{(a_{0},...,a_{3})}(ℚ_{p})) ≤ c · |a_{0} · . . .· a_{3}|^{\frac{1}{3}− \frac{ε}{8}} · ∏_{p prime} min_{i=0,...,3}∥a_{i} ∥_{p}^{\frac{1}{3}−ε}

for all (a_{0}, …, a_{3}) ∈ (ℤ\{0})^{4}. Proof. The product over all primes of good reduction is bounded by virtue of Lemma 2.3.1 above. It, therefore, remains to show that

∏_{{p prime \atop p|3a_{0}...a_{3}}}τ_{p}(S^{(a_{0},...,a_{3})}(ℚ_{p})) ≤ c · |a_{0} · . . .· a_{3}|^{\frac{1}{3}− \frac{ε}{8}} · ∏_{p prime} min_{i=0,...,3} ∥a_{i} ∥_{p}^{\frac{1}{3}−ε}.

For this, by Proposition 2.3.10, we have at first

τ_{p}(S^{(a_{0},...,a_{3})}(ℚ_{p})) ≤ C_{p}^{(ε)} · (a_{0}^{(p)}a_{1}^{(p)}a_{2}^{(p)})^{\frac{1}{3}− \frac{ε}{4}} · (a_{3}^{(p)})^{\frac{3}{4}ε}
= C_{p}^{(ε)} · (a_{0}^{(p)}a_{1}^{(p)}a_{2}^{(p)}a_{3}^{(p)})^{\frac{1}{3}− \frac{ε}{4}} · (a_{3}^{(p)})^{− \frac{1}{3}+ε} .

Here, the indices 0, …, 3 are interchangeable. Hence, it is even allowed to write

τ_{p}(S^{(a_{0},...,a_{3})}(ℚ_{p})) ≤ C_{p}^{(ε)} · (a_{0}^{(p)}a_{1}^{(p)}a_{2}^{(p)}a_{3}^{(p)})^{\frac{1}{3}− \frac{ε}{4}} · (max_{i}a_{i}^{(p)})^{− \frac{1}{3}+ε}
= C_{p}^{(ε)} · (a_{0}^{(p)}a_{1}^{(p)}a_{2}^{(p)}a_{3}^{(p)})^{\frac{1}{3}− \frac{ε}{4}} · min_{i}∥a_{i} ∥_{p}^{\frac{1}{3}−ε}.

Now, we multiply over all prime divisors of a_{0} · . . . · a_{3}. Thereby, on the right hand side, we may twice write the product over all primes since the two rightmost factors are equal to one for p ∤ 3a_{0} · . . .· a_{3}, anyway.

∏_{{p prime \atop p|3a_{0}...a_{3}}}τ_{p}(S^{(a_{0},...,a_{3})}(ℚ_{p})) ≤ ∏_{{p prime \atop p|3a_{0}...a_{3}}}C_{p}^{(ε)} · ∏_{p prime}(a_{0}^{(p)}a_{1}^{(p)}a_{2}^{(p)}a_{3}^{(p)})^{\frac{1}{3}− \frac{ε}{4}} · ∏_{p prime} min_{i=0,...,3} ∥a_{i} ∥_{p}^{\frac{1}{3}−ε}

= ∏_{{p prime \atop p|3a_{0}...a_{3}}}C_{p}^{(ε)} · |a_{0} · . . .· a_{3}|^{\frac{1}{3}− \frac{ε}{4}} · ∏_{p prime} min_{i=0,...,3} ∥a_{i} ∥_{p}^{\frac{1}{3}−ε}

when we observe that ∏ _{p} a^{(p)} = |a|. Further, we have C_{p}^{(ε)} ≤ C^{(ε)} and, by [Na, Theorem 7.2] together with [Na, Section 7.1, Exercise 7],

∏_{{p prime \atop p|3a_{0}...a_{3}}}C^{(ε)} ≤ c · |3a_{0} · . . .· a_{3}|^{\frac{ε}{8}}.

We finally estimate 3^{\frac{ε}{8}} by a constant. The assertion follows. □

2.4. An estimate for the factor at the infinite place

2.4.1. Proposition. –––– For real numbers 0 < b_{0} ≤ b_{1} ≤ b_{2} ≤ b_{3}, we have

∫_{\begin{matrix}CS^{(1,...,1)(ℝ)} \\ |x_{0}|≤b_{0}, ...,|x_{3}|≤b_{3} \end{matrix}} ω_{Leray}^{CS^{(1,...,1)}(ℝ)} ≤ (64 + \frac{64}{3} log3 + \frac{1}{3}\sqrt[3]{3}ω_{2})b_{0} + 64b_{0} log\frac{b_{1}}{b_{0}}

where ω_{2} is the two-dimensional hypersurface measure of the l_{3}-unit sphere

S^{2} := { (x_{1}, x_{2}, x_{3}) ∈ ℝ^{3} | |x_{1}|^{3} + |x_{2}|^{3} + |x_{3}|^{3} = 1 } .

Proof. First step. We cover the domain of integration by 25 sets as follows. We put

R_{0} := [−b_{0}, b_{0}]^{4} ∩ CS^{(1,...,1)}(ℝ) .

Further, for each σ ∈ S_{4}, we set

R_{σ} := { (x_{0}, …, x_{3}) ∈ ℝ^{4} | |x_{σ(0)}| ≤ ⋯≤ |x_{σ(3)}|, |x_{i}| ≤ b_{i}, and b_{0}≤ |x_{σ(3)}| }
∩ CS^{(1,...,1)}(ℝ) .

Second step. One has ∫_{R_{σ}} ω_{Leray}^{CS^{(1,...,1)}(ℝ)} ≤ ∫_{R_{id}}ω_{Leray}^{CS^{(1,...,1)}(ℝ)} for every σ ∈ S_{4}.

Consider the map i_{σ} : ℝ^{4} → ℝ^{4} given by (x_{0}, …, x_{3}) ↦ (x_{σ(0)}, …, x_{σ(3)}). Since CS^{(1,...,1)}(ℝ) is defined by a symmetric cubic form, it is invariant under i_{σ}. We claim that

i_{σ}(R_{σ}) ⊆ R_{id} .

Indeed, let (x_{0}, …, x_{3}) ∈ R_{σ}. Then, i_{σ}(x_{0}, …, x_{3}) = (x_{σ(0)}, …, x_{σ(3)}) has the properties |x_{σ(0)}| ≤ … ≤ |x_{σ(3)}| and b_{0} ≤ |x_{σ(3)}|. In order to show i_{σ}(x_{0}, …, x_{3}) ∈ R_{id}, all we need to verify is |x_{σ(i)}| ≤ b_{i} for i = 0, …, 3.

For this, we use that the b_{i} are sorted. We have |x_{σ(3)}| ≤ b_{σ(3)} ≤ b_{3}. Further, |x_{σ(2)}| ≤ b_{σ(2)} and |x_{σ(2)}| ≤ |x_{σ(3)}| ≤ b_{σ(3)} one of which is at most equal to b_{2}. Similarly, |x_{σ(1)}| ≤ b_{σ(1)}, |x_{σ(1)}| ≤ |x_{σ(2)}| ≤ b_{σ(2)}, and |x_{σ(1)}| ≤ |x_{σ(3)}| ≤ b_{σ(3)}, the smallest of which is not larger than b_{1}. Finally, |x_{σ(0)}| ≤ b_{σ(0)}, |x_{σ(0)}| ≤ |x_{σ(1)}| ≤ b_{σ(1)}, |x_{σ(0)}| ≤ |x_{σ(2)}| ≤ b_{σ(2)}, and |x_{σ(0)}| ≤ |x_{σ(3)}| ≤ b_{σ(3)}. This shows |x_{σ(0)}| ≤ b_{0}.

Since x_{0}^{3} + … + x_{3}^{3} is a symmetric form, the Leray measure on CS^{(1,...,1)}(ℝ) is invariant under the canonical operation of S_{4} on CS^{(1,...,1)}(ℝ) ⊂ ℝ^{4}. Therefore, we have (i_{σ})_{∗}ω_{Leray}^{CS^{(1,...,1)}(ℝ)} = ω_{Leray}^{CS^{(1,...,1)}(ℝ)} for each σ ∈ S_{4}.

Altogether,

∫_{R_{σ}} ω_{Leray}^{CS^{(1,...,1)}(ℝ)} ≤ ∫_{i_{σ}^{−1}(R_{id})} ω_{Leray}^{CS^{(1,...,1)}(ℝ)} = ∫_{R_{id}} (i_{σ})_{∗}ω_{Leray}^{CS^{(1,...,1)}(ℝ)} = ∫_{R_{id}} ω_{Leray}^{CS^{(1,...,1)}(ℝ)} .

Third step. We have ∫_{R_{0}}ω_{Leray}^{CS^{(1,...,1)}(ℝ)} ≤ \frac{1}{3} \sqrt[3]{3}ω_{2}b_{0}.

By definition,

∫_{R_{0}} ω_{Leray}^{CS^{(1,...,1)}(ℝ)} = \frac{1}{3} ∫_{R_{0}} \frac{1}{x_{3}^{2}} dx_{0} ∧ dx_{1} ∧ dx_{2}

= \frac{1}{3} ∫ ∫ ∫_{π(R_{0})} \frac{1}{(x_{0}^{3} + x_{1}^{3} + x_{2}^{3})^{2/3}} dx_{0} dx_{1} dx_{2}

where π: CS^{(1,...,1)}(ℝ) → ℝ^{3}, (x_{0}, x_{1}, x_{2}, x_{3}) ↦ (x_{0}, x_{1}, x_{2}), denotes the projection to the first three coordinates.

We enlarge the domain of integration to

R′:= { (x_{1}, x_{2}, x_{3}) ∈ ℝ^{3} | |x_{0}|^{3} + |x_{1}|^{3} + |x_{2}|^{3} ≤ 3b_{0}^{3} } .

Then, by homogeneity, we see

∫ ∫ ∫_{R′} \frac{1}{(x_{0}^{3} + x_{1}^{3} + x_{2}^{3})^{2/3}} dx_{0} dx_{1} dx_{2} = ω_{2} ·∫_{0}^{\sqrt[3]{3}b_{0}} \frac{1}{r^{2}} · r^{2} dr = ω_{2} · \sqrt[3]{3}b_{0} .

Fourth step. We have ∫_{R_{id}}ω_{Leray}^{CS^{(1,...,1)}(ℝ)} ≤ (\frac{8}{3} + \frac{8}{9} log3)b_{0} + \frac{8}{3}b_{0} log \frac{b_{1}}{b_{0}}. Observe |x_{3}| = |\sqrt[3]{x_{0}^{3} + x_{1}^{3} + x_{2}^{3}} | ≤ \sqrt[3]{|x_{0}|^{3} + |x_{1}|^{3} + |x_{2}|^{3}}. For (x_{0}, …, x_{3}) ∈ R_{id}, this implies |x_{3}| ≤ \sqrt[3]{3}|x_{2}| and |x_{2}| ≥ b_{0}/\sqrt[3]{3}. We find

∫_{R_{id}}ω_{Leray}^{CS^{(1,...,1)}(ℝ)} = \frac{1}{3} ∫_{R_{id}} \frac{1}{x_{3}^{2}} dx_{0} ∧ dx_{1} ∧ dx_{2}

≤ \frac{1}{3} ∫_{R_{id}} \frac{1}{x_{2}^{2}} dx_{0} ∧ dx_{1} ∧ dx_{2}

< \frac{1}{3} ∫_{−b_{0}}^{b_{0}} ∫_{|x_{1}|∈[|x_{0}|,b_{1}]} ∫_{{|x_{2}|≥b_{0}/^{3} \sqrt{3} \atop |x_{2}|≥|x_{1}|}}\frac{1}{x_{2}^{2}} dx_{2} dx_{1} dx_{0}

≤ \frac{1}{3}∫_{−b_{0}}^{b_{0}} ∫_{|x_{1}|∈[|x_{0}|,b_{1}]}\frac{2}{max\{b_{0}/\sqrt[3]{3}, |x_{1}|\}}dx_{1} dx_{0}

≤ \frac{2}{3} [∫_{−b_{0}}^{b_{0}} ∫_{|x_{1}|∈[|x_{0}|,b_{0}/\sqrt[3]{3}]} \frac{\sqrt[3]{3}}{b_{0}} dx_{1} dx_{0} + ∫_{−b_{0}}^{b_{0}} ∫_{|x_{1}|∈[b_{0}/\sqrt[3]{3},b_{1}]} \frac{1}{|x_{1}|}dx_{1} dx_{0}]

≤ \frac{2}{3} · \frac{4b_{0}^{2}}{\sqrt[3]{3}} · \frac{\sqrt[3]{3}}{b_{0}} + \frac{2}{3} ∫_{−b_{0}}^{b_{0}}2 log\frac{\sqrt[3]{3}b_{1}}{b_{0}} dx_{0}

= \frac{8}{3}b_{0} + \frac{8}{3}b_{0} log\frac{\sqrt[3]{3}b_{1}}{b_{0}}

= (\frac{8}{3} + \frac{8}{9} log3)b_{0} + \frac{8}{3}b_{0} log\frac{b_{1}}{b_{0}}.  □

2.4.2. Corollary. –––– For every ε > 0, there exists a constant c such that

τ_{∞} (S^{(a_{0},...,a_{3})}(ℝ)) ≤ c · |a_{0} · . . .· a_{3}|^{− \frac{1}{3}+ε} · min_{i=0,...,3}∥a_{i} ∥∞^{\frac{1}{3}}

for each (a_{0}, …, a_{3}) ∈ (ℤ\{0})^{4}. Proof. Our first claim is

τ_{∞} (S^{(a_{0},...,a_{3})}(ℝ)) = \frac{1}{2\sqrt[3]{|a_{0} · . . .· a_{3}|}} ∫_{\begin{matrix}CS^{(1,...,1)}(ℝ) \\ |x_{0}|≤\sqrt[3]{|a_{0}|}, ...,|x_{3}|≤\sqrt[3]{|a_{3}|} \end{matrix}} ω_{Leray}^{CS^{(1,...,1)}(ℝ)} .

Indeed, according to the definition of τ_{∞} (S^{(a_{0},...,a_{3})}(ℝ)), we need to show

\frac{1}{6 |a_{0}|} ∫_{\begin{matrix}CS^{(a_{0},...,a_{3})}(ℝ) \\ |x_{0}|≤1, ...,|x_{3}|≤1 \end{matrix}} \frac{1}{x_{0}^{2}} dx_{1} ∧ dx_{2} ∧ dx_{3} = \frac{1}{6\sqrt[3]{|a_{0} · . . .· a_{3}|}} ∫_{\begin{matrix}CS^{(1,...,1)}(ℝ) \\ |X_{0}|≤\sqrt[3]{|a_{0}|}, ...,|X_{3}|≤\sqrt[3]{|a_{3}|} \end{matrix}} \frac{1}{X_{0}^{2}} dX_{1} ∧ dX_{2} ∧ dX_{3} .

For that, consider the linear mapping l: CS^{(a_{0},...,a_{3})}(ℝ) → CS^{(1,...,1)}(ℝ) given by

(x_{0}, …, x_{3}) ↦ (\sqrt[3]{a_{0}}x_{0}, …, \sqrt[3]{a_{3}}x_{3}) .

Then,

l^{∗}(\frac{1}{X_{0}^{2}} dX_{1} ∧ dX_{2} ∧ dX_{3}) = \frac{\sqrt[3]{a_{1}a_{2}a_{3}}}{a_{0}^{2/3}} \frac{1}{x_{0}^{2}} dx_{1} ∧ dx_{2} ∧ dx_{3}.

When we take into consideration that orientations are chosen in such a way that both integrals are positive, this immediately yields the claim.

To obtain the asserted inequality, we assume without restriction that |a_{0}| ≤ … ≤ |a_{3}|. Then, Proposition 2.4.1 shows that, for certain explicit positive constants c_{1} and c_{2},

τ_{∞} (S^{(a_{0},...,a_{3})}(ℝ)) ≤ |a_{0} · . . .· a_{3}|^{− \frac{1}{3}} · (c_{1}|a_{0}|^{\frac{1}{3}} + c_{2}|a_{0}|^{\frac{1}{3}} log\sqrt[3]{\frac{|a_{1}|}{|a_{0}|}})
= |a_{0} · . . .· a_{3}|^{− \frac{1}{3}} · |a_{0}|^{\frac{1}{3}}(c_{1} + \frac{1}{3} c_{2} log \frac{|a_{1}|}{|a_{0}|})
≤ |a_{0} · . . .· a_{3}|^{− \frac{1}{3}} · min_{i=0,...,3} ∥a_{i} ∥∞^{\frac{1}{3}} · (c_{1} + \frac{1}{3} c_{2} log|a_{0} · . . .· a_{3}|).

There is a constant c such that c_{1} + \frac{1}{3} c_{2} log|a_{0} · . . . · a_{3}| ≤ c|a_{0} · . . . · a_{3}|^{ε} for every (a_{0}, …, a_{3}) ∈ (ℤ\{0})^{4}. □

2.5. The Tamagawa number

2.5.1. Proposition. –––– For every ε > 0, there exists a constant C > 0 such that

\frac{1}{τ^{(a_{0},...,a_{3})}} ≥ C · \frac{H_{naive} (\frac{1}{a_{0}} : …: \frac{1}{a_{3}})^{\frac{1}{3}}}{|a_{0} · . . .· a_{3}|^{ε}}

for each (a_{0}, …, a_{3}) ∈ (ℤ\{0})^{4}. Proof. We may assume that ε is small, say ε < \frac{2}{3}. Then, immediately from the definition of τ^{(a_{0},...,a_{3})}, we have

τ^{(a_{0},...,a_{3})}
= α(S^{(a_{0},...,a_{3})})·β(S^{(a_{0},...,a_{3})}) · lim_{s→1}(s − 1)^{t}L (s, χ_{Pic(S(a_{0},...,a_{3}) ; \overline{ℚ})})· τ_{H} (S^{(a_{0},...,a_{3})}(𝔸_{ℚ} )^{Br})
≤ α(S^{(a_{0},...,a_{3})})·β(S^{(a_{0},...,a_{3})}) · lim_{s→1}(s − 1)^{t}L (s, χ_{Pic(S(a_{0},...,a_{3}) ; \overline{ℚ})})· τ_{H} (S^{(a_{0},...,a_{3})}(𝔸_{ℚ} ))
= α(S^{(a_{0},...,a_{3})})·β(S^{(a_{0},...,a_{3})}) · lim_{s→1}(s − 1)^{t}L (s, χ_{Pic(S(a_{0},...,a_{3}) ; \overline{ℚ})}) · ∏_{ν∈Val(ℚ)}τ_{ν}(S^{(a_{0},...,a_{3})}(ℚ_{ν})).

Let us collect estimates for the factors. First, by Proposition 2.2.4, we have

lim_{s→1}(s − 1)^{t}L(s, χ_{Pic(S(a_{0},...,a_{3}) ; \overline{ℚ})}) < c_{1} · |a_{0} · . . .· a_{3}|^{\frac{ε}{16}}

for a certain constant c_{1}. Further, Proposition 2.3.12 yields

∏_{p prime}τ_{p}(S^{(a_{0},...,a_{3})}(ℚ_{p})) ≤ c_{2} · |a_{0} · . . .· a_{3}|^{\frac{1}{3}− \frac{ε}{16}} · ∏_{p prime} min_{i=0,...,3} ∥a_{i} ∥_{p}^{\frac{1}{3}− \frac{ε}{2}} .

Finally, Corollary 2.4.2 shows

τ_{∞} (S^{(a_{0},...,a_{3})}(ℝ)) ≤ c · |a_{0} · . . .· a_{3}|^{− \frac{1}{3}+ \frac{ε}{2}} · min_{i=0,...,3}∥a_{i} ∥∞^{\frac{1}{3}} .

We assert that the three inequalities together imply the following estimate for Peyre’s constant τ^{(a_{0},...,a_{3})} = τ(S^{(a_{0},...,a_{3})}),

τ^{(a_{0},...,a_{3})} ≤ c_{3} · |a_{0} · . . .· a_{3}|^{\frac{ε}{2}} · ∏_{p prime} min_{i=0,...,3} ∥a_{i} ∥p^{\frac{1}{3}} · min_{i=0,...,3} ∥a_{i} ∥∞^{\frac{1}{3}} · ∏_{p prime}[ min_{i=0,...,3} ∥a_{i} ∥_{p}]^{− \frac{ε}{2}} .

Indeed, this is trivial in the case τ^{(a_{0},...,a_{3})} = 0. Otherwise, S^{(a_{0},...,a_{3})} has an adelic point. Lemmata 2.1.5 and 2.1.3 show that the factors α and β are bounded from above by constants. By consequence,

\frac{1}{τ^{(a_{0},...,a_{3})}} ≥ \frac{1}{c_{3}} · \frac{∏_{p prime}[ min_{i=0,...,3} ∥a_{i} ∥_{p}]^{− \frac{1}{3}} · [ min_{i=0,...,3} ∥a_{i} ∥_{∞} ]^{− \frac{1}{3}}}{|a_{0} · . . .· a_{3}|^{\frac{ε}{2}} · ∏_{p prime}[ min_{i=0,...,3} ∥a_{i} ∥_{p}]^{− \frac{ε}{2}}}

= \frac{1}{c_{3}} · \frac{∏_{p prime} max_{i=0,...,3} ∥\frac{1}{a_{i}} ∥p^{\frac{1}{3}} · max_{i=0,...,3} ∥\frac{1}{a_{i}} ∥∞^{\frac{1}{3}}}{|a_{0} · . . .· a_{3}|^{\frac{ε}{2}} · ∏_{p prime}[ max_{i=0,...,3} a_{i}^{(p)}]^{\frac{ε}{2}}}

= \frac{1}{c_{3}} · \frac{H_{naive}(\frac{1}{a_{0}} : …: \frac{1}{a_{3}})^{\frac{1}{3}}}{|a_{0} · . . .· a_{3}|^{\frac{ε}{2}} · ∏_{p prime}[ max_{i=0,...,3} a_{i}^{(p)}]^{\frac{ε}{2}}}.

It is obvious that max_{i=0,...,3}a_{i}^{(p)} ≤ |a_{0}^{(p)} · . . .· a_{3}^{(p)}| and ∏_{p prime}|a_{0}^{(p)} · . . .· a_{3}^{(p)}| = |a_{0} · . . .· a_{3}|. This shows

\frac{1}{τ^{(a_{0},...,a_{3})}} ≥ \frac{1}{c_{3}} · \frac{H_{naive}(\frac{1}{a_{0}} : …: \frac{1}{a_{3}})^{\frac{1}{3}}}{|a_{0} · . . .· a_{3}|^{\frac{ε}{2}} · |a_{0} · . . .· a_{3}|^{\frac{ε}{2}}}

= \frac{1}{c_{3}} · \frac{H_{naive}(\frac{1}{a_{0}} : …: \frac{1}{a_{3}})^{\frac{1}{3}}}{|a_{0} · . . .· a_{3}|^{ε}} .  □

2.5.2. Lemma. –––– Let (a_{0} : …: a_{3}) ∈ P^{3}(ℚ) be any point such that a_{0} 0, …, a_{3} 0. Then,

H_{naive}(a_{0} : …: a_{3}) ≤ H_{naive}(\frac{1}{a_{0}} : …: \frac{1}{a_{3}})^{3} .

Proof. First, observe that (a_{0} : …: a_{3}) ↦ (\frac{1}{a_{0}} : …: \frac{1}{a_{3}}) is a well-defined map. Hence, we may assume without restriction that a_{0}, …, a_{3} ∈ ℤ and gcd(a_{0}, …, a_{3}) = 1. This yields H_{naive}(a_{0} : …: a_{3}) = max_{i=0,...,3}|a_{i}|.

On the other hand, (\frac{1}{a_{0}} : …: \frac{1}{a_{3}}) = (a_{1}a_{2}a_{3} : …: a_{0}a_{1}a_{2}). Consequently,

H_{naive} (\frac{1}{a_{0}} : …: \frac{1}{a_{3}}) ≤ [ max_{i=0,...,3}|a_{i}|]^{3} = H_{naive}(a_{0} : …: a_{3})^{3} .

From this, the asserted inequality emerges when the roles of a_{i} and \frac{1}{a_{i}} are interchanged. □

2.5.3. Corollary. –––– Let a_{0}, …, a_{3} ∈ ℤ such that gcd(a_{0}, …, a_{3}) = 1. Then,

|a_{0} · . . .· a_{3}| ≤ H_{naive} (\frac{1}{a_{0}} : …: \frac{1}{a_{3}})^{12}.

Proof. Observe that |a_{0} · . . .· a_{3}| ≤ max_{i=0,...,3}|a_{i}|^{4} = H_{naive}(a_{0} : …: a_{3})^{4} and apply Lemma 2.5.2. □

2.5.4. Theorem. –––– For each ε > 0, there exists a constant C(ε) > 0 such that, for all (a_{0}, …, a_{3}) ∈ (ℤ\{0})^{4},

\frac{1}{τ^{(a_{0},...,a_{3})}} ≥ C(ε) · H_{naive}(\frac{1}{a_{0}} : …: \frac{1}{a_{3}})^{\frac{1}{3}−ε}.

Proof. We may assume that gcd(a_{0}, …, a_{3}) = 1. Then, by Proposition 2.5.1,

\frac{1}{τ^{(a_{0},...,a_{3})}} ≥ C(ε) · \frac{H_{naive}(\frac{1}{a_{0}} : …: \frac{1}{a_{3}})^{\frac{1}{3}}}{|a_{0} · . . .· a_{3}|^{\frac{ε}{12}}} .

Corollary 2.5.3 yields |a_{0} · . . .· a_{3}|^{\frac{ε}{12}} ≤ H_{naive}(\frac{1}{a_{0}} : …: \frac{1}{a_{3}})^{ε}. □

2.5.5. Corollary (Fundamental finiteness). —– For each T > 0, there are only finitely many diagonal cubic surfaces S^{(a_{0},...,a_{3})}: a_{0}x_{0}^{3} + … + a_{3}x_{3}^{3} = 0 in P_{ℚ}^{3} such that τ^{(a_{0},...,a_{3})} > T. Proof. This is an immediate consequence of the comparison to the naive height established in Theorem 2.5.4. □

3. The varieties of Batyrev-Tschinkel

3.1. Lemma. –––– Let m, n be positive integers such that m ≤ n + 1 and ι: P^{m}−→ P^{3} a surjective linear map. Then, there exists a constant C such that, for every (a_{0} : …: a_{3}) ∈ P^{3}(ℚ),

∑_{{x∈P^{m}(ℚ) \atop ι(x)=(a_{0}:...:a_{3})}}\frac{1}{H_{naive}^{n}(x)} ≤ C· \frac{1}{H_{naive}^{n−m+3}(a_{0}: …:a_{3})} .

Proof. An automorphism of P^{m} changes the naive height by a factor which is bounded. We may therefore suppose that ι is given by (x_{0} : …: x_{m}) ↦ (x_{0} : …: x_{3}). Further, assume a_{0}, …, a_{3} ∈ ℤ such that gcd(a_{0}, …, a_{3}) = 1. Finally, we write H := H_{naive}(a_{0}: …:a_{3}).

Let N ≥ H be an arbitrary integer. There are two ways a point x = (x_{0} : …: x_{m}) ∈ P^{m}(ℚ) such that ι(x) = (a_{0} : …: a_{3}) may have height exactly equal to N. Either, one of the coordinates x_{4}, …, x_{m} is equal to N. There are at most

2[\frac{N}{H}](m − 3)(2N + 1)^{m−4} ≤ \frac{C_{1}}{H}N^{m−3}

such points. Or, one of the coordinates x_{0}, …, x_{3} is equal to N. This is possible only when N = kH is an exact multiple. Then, there are at most

(2N + 1)^{m−3} ≤ C_{2}N^{m−3}

such points. All in all, we find the estimate

∑_{{x∈P^{m}(ℚ) \atop ι(x)=(a_{0}:...:a_{3})}}\frac{1}{H_{naive}^{n}(x)} ≤ \frac{C_{1}}{H} ∑_{N=H}^{∞} \frac{1}{N^{n−m+3}} + \frac{C_{2}}{H^{n−m+3}} ∑_{k=1}^{∞} \frac{1}{k^{n−m+3}} ≤ \frac{C}{H^{n−m+3}} .

The assumption m ≤ n + 1 assures that all the series occurring are convergent. □

3.2. Proposition. –––– Let ι: P^{n}− → (P^{3})^{∨} be a linear map. Suppose that either dimimι ≥ 2 or n = 1. Then, the series

∑ x∈Pn(ℚ) Sι(x)non−singular rkPic(Sι(x))=4

\frac{1}{H_{naive}^{n}(x)}

is convergent. Proof. Note that Picard rank 4 is the maximal value which is possible for a non-singular diagonal cubic surface. It occurs for S^{(a_{0}:...:a_{3})} if and only if all the quotients a_{i}/a_{0} are perfect cubes in ℚ. We will distinguish three cases. First case. dimimι = 3. There are at most 4(2N + 1)^{3} quadruples (a_{0} : …: a_{3}) of naive height N^{3} such that all the quotients a_{i}/a_{0} are perfect cubes. According to Lemma 3.1, the series to be considered is dominated by ∑_{N} 4(2N + 1)^{3}\frac{C}{(N^{3})^{3}} ≤ 108C ∑_{N} \frac{1}{N^{6}} which converges.

Second case. dimimι = 2. Then, ι is the restriction of a surjective linear map P^{n+1} → P^{3} to a hyperplane. Estimating very roughly, we find the convergent series ∑_{N} 4(2N + 1)^{3}\frac{C}{(N^{3})^{2}} ≤ 108C ∑_{N} \frac{1}{N^{3}} .

Third case. dimimι = 1. Here, by assumption, n = 1. An automorphism of P^{1} changes the naive height by a factor which is bounded. Thus, without restriction, we may suppose that ι is given by

(x_{0} : x_{1}) ↦ (x_{0} : x_{1} : l_{1}(x_{0}, x_{1}) : l_{2}(x_{0}, x_{1}))

for two linear forms l_{1}, l_{2}. As H_{naive}(x_{0} : x_{1} : l_{1}(x_{0}, x_{1}) : l_{2}(x_{0}, x_{1})) ≥ H_{naive}(x_{0} : x_{1}), the contribution of (x_{0} : x_{1}) ∈ P^{1}(ℚ) is estimated by \frac{1}{H_{naive}(x_{0}:x_{1})}. Further, we only consider pairs such that x_{1}/x_{0} a perfect cube. There are ≤ 2(2N + 1) such pairs (x_{0} : x_{1}) of naive height N^{3}. The series ∑_{N} 2(2N + 1)\frac{1}{N^{3}} ≤ 6 ∑_{N} \frac{1}{N^{2}} converges. □

3.3. Corollary (The Batyrev-Tschinkel varieties). —– Let X ⊂ P^{n}× P^{3} be a smooth hypersurface given by a bihomogeneous form of the shape

ι_{0}(x_{0}, …, x_{n}) y_{0}^{3} + ⋯ + ι_{3}(x_{0}, …, x_{n}) y_{3}^{3} .

Suppose that ι_{0}, …, ι_{3} are linear forms, not all proportional to each other. Then, the series

∑_{x∈P^{n}(ℚ) S^{ι(x)}non−singular rkPic(S^{ι(x)})=4} \frac{1}{H_{naive}^{n}(x)} τ(S^{ι(x)})

converges. Here, ι: P^{n}−→ (P^{3})^{∨} is the linear map defined by ι_{0}, …, ι_{3}. Proof. Theorem 2.5.4 immediately implies that the factors τ(S^{ι(x)}) are bounded. Further, as X is smooth [BT, Proposition 1.1], we have dimimι = min(n, 3). Thus, the assertion is a direct consequence of Lemma 3.2. □

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