On the smallest point on a diagonal quartic threefold

Andreas-Stephan Elsenhans^{∗} and Jörg Jahnel^{∗}

Abstract For the family a_{0}x^{4} = a_{1}y^{4}+a_{2}z^{4}+a_{3}v^{4}+a_{4}w^{4}, a_{0}, …, a_{4} > 0, of diagonal quartic threefolds, we study the behaviour of the height of the smallest rational point versus the Tamagawa type number introduced by E. Peyre.

1 Introduction – A computer experiment

1.1. –––– Let V ⊆ P_{ℚ}^{n} be a Fano variety defined over ℚ. If V (ℚ_{ν}) ≠ ∅ for every ν ∈ Val(ℚ) then it is natural to ask whether V (ℚ) ≠ ∅ (Hasse’s principle). Further, it would be desirable to have an a-priori upper bound for the height of the smallest ℚ-rational point on V as this would allow to effectively decide whether V (ℚ) ≠ ∅ or not.

When V is a conic, Legendre’s theorem on zeroes of ternary quadratic forms proves the Hasse principle and, moreover, yields an effective bound for the smallest point. For quadrics of arbitrary dimension, the same is true by an observation due to J. W. S. Cassels [Ca]. Further, there is a theorem of C. L. Siegel [Si, Satz 1] which provides a generalization to hypersurfaces defined by norm equations. For more general Fano varieties, no theoretical upper bound is known for the height of the smallest ℚ-rational point. Some of these varieties fail the Hasse principle.

In this note, we present some theoretical and experimental results concerning the height of the smallest ℚ-rational point on quartic hypersurfaces in P_{ℚ}^{4} .

1.2. –––– There is a conjecture, due to Yu. I. Manin, that the number of ℚ-ra tional points of anticanonical height < B on a Fano variety V is asymptotically equal to τB log^{rk Pic(V)−1} B, for B → ∞.

Key words and phrases. Diagonal quartic threefold, Diophantine equation, smallest solution, naive height, E. Peyre’s Tamagawa-type number

∗The 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.

In the particular case of a quartic threefold, the anticanonical height is the same as the naive height. Further, rk Pic(V) = 1 and, finally, the coefficient τ ∈ ℝ equals the Tamagawa-type number τ(V) introduced by E. Peyre in [Pe]. Thus, one expects ∼τ(V)B points of height <B. Assuming equidistribution, the height of the smallest point should be ∼ \frac{1}{τ(V)}. Being a bit optimistic, this might lead to the expectation that m(V), the height of the smallest ℚ-rational point on V, is always less than \frac{C}{τ(V)} for a certain absolute constant C.

1.3. –––– To test this expectation, we computed the Tamagawa number and ascertained the smallest ℚ-rational point for each of the quartic threefolds V_{4}^{(−a,b)} ⊂ P_{ℚ}^{4} given by ax^{4} = by^{4} + z^{4} + v^{4} + w^{4} for a, b = 1, …, 1000.

On 516 820 of these varieties, there are no ℚ-rational points as the equation is unsolvable in ℚ_{p} for p = 2, 5, or 29. (Note that, for each prime p different from 2, 5, or 29, there are p-adic points already on the Fermat quartic given by z^{4}+v^{4}+w^{4} = 0.) On each of the remaining varieties, ℚ-rational points were found. In other words, there are no counterexamples to the Hasse principle in this family.

The methods to systematically search for solutions of Diophantine equations and to compute Tamagawa numbers we applied here are described in our earlier papers [EJ1, EJ2].

The results are summarized by the diagrams below.

a, b = 1 … 100 a, b = 1 … 1000 Figure 1: Height of smallest point versus Tamagawa number

It is apparent from the diagrams that the experiment agrees with the expectation above. The slope of a line tangent to the top right of each of the scatter plots is indeed near (−1). However, we show in Section 2 that, in general, the inequality m(V) < \frac{C}{τ(V)} does not hold. The following remains a logical possibility.

1.4. Question. –––– For every ε > 0, does there exist a constant C(ε) such that, for each quartic threefold,

m(V) < \frac{C(ε)}{τ(V)^{1+ε}} ?

1.5. Peyre’s constant. –––– In our situation, rk Pic(V\overline{ℚ} ) = 1. In particular, there is no Brauer-Manin obstruction on V. Recall that, in this case, E. Peyre’s Tamagawa-type number is defined [PT, Definition 2.4] as an infinite product τ(V) := ∏_{ν∈Val(ℚ)} τ_{ν}(V). For a prime number p, the definition of the local factor may be simplified to

τ_{p}(V) := (1 − \frac{1}{p}) · lim_{n→∞} \frac{#V (ℤ/p^{n}ℤ)}{p^{3n}} .

τ_{∞}(V) is described in [Pe, Lemme 5.4.7]. In the case of the diagonal quartic threefold V^{(a_{0},...,a_{4})} given by a_{0}x_{0}^{4} + … + a_{4}x_{4}^{4} = 0 (a_{0} < 0, a_{1}, …, a_{4} > 0) in P_{ℚ}^{4} , this yields

(∗) τ_{∞}(V^{(a_{0},...,a_{4})}) = \frac{1}{4\sqrt[4]{|a_{0}|}} ∫∫∫∫_{R} \frac{1}{(a_{1}y^{4} + a_{2}z^{4} + a_{3}v^{4} + a_{4}w^{4})^{3/4}} dy dz dv dw

where the domain of integration is

R := {(y, z, v, w) ∈ [−1, 1]^{4} | |a_{1}y^{4} + a_{2}z^{4} + a_{3}v^{4} + a_{4}w^{4}| ≤ |a_{0}|} .

1.6. –––– In the case of diagonal quartic threefolds, there is an estimate for m(V) in terms of τ(V). Namely, \frac{1}{τ(V)} admits a fundamental finiteness property. More precisely, in Section 3, we will show the following results.

Theorem. Let 𝔞 = (a_{0}, …, a_{4}) be a vector such that a_{0}, …, a_{4} ∈ ℤ, a_{0} < 0, and a_{1}, …, a_{4} > 0. Denote by V^{𝔞} the quartic in P_{ℚ}^{4} given by a_{0}x_{0}^{4} + … + a_{4}x_{4}^{4} = 0. Then, for each ε > 0 there exists a constant C(ε) > 0 such that

\frac{1}{τ(V^{𝔞})} ≥ C(ε) · H_{naive}(\frac{1}{a_{0}} : …: \frac{1}{a_{4}} )^{\frac{1}{4} −ε}.

Corollary (Fundamental Finiteness). For each B > 0, there are only finitely many quartics V^{𝔞}: a_{0}x_{0}^{4} + … + a_{4}x_{4}^{4} = 0 in P_{ℚ}^{4} such that a_{0} < 0, a_{1}, …, a_{4} > 0, and τ(V^{𝔞}) > B.

Corollary (An inefficient search bound). There exists a monotonically decreasing function S: (0, ∞) → [0, ∞), the search bound, satisfying the following condition. Every quartic V^{𝔞}: a_{0}x_{0}^{4} + … +a_{4}x_{4}^{4} = 0 such that a_{0} < 0 and a_{1}, …, a_{4} > 0 admits, if V^{𝔞}(ℚ) ≠ ∅, already a ℚ-rational point of height ≤ S(τ(V^{𝔞})). Proof. One may simply put S(t) := max_{{τ(V^{𝔞})≥t \atop V^{𝔞}(ℚ)≠∅}} min_{P ∈V^{𝔞}(ℚ)} h(P). □

In other words, we have m(V^{𝔞}) ≤ S(τ(V^{𝔞})) as soon as V^{𝔞}(ℚ) ≠ ∅.

2 A negative result

For a ∈ ℕ, let V^{(−a)} ⊂ P_{ℚ}^{4} be given by ax^{4} = y^{4} + z^{4} + v^{4} + w^{4} and let

m(V^{(−a)}) := min { H_{naive}(x : y : z : v : w) | (x : y : z : v : w) ∈ V^{(−a)}(ℚ)}

be the smallest height of a ℚ-rational point on V^{(−a)}. We compare m(V^{(−a)}) with the Tamagawa type number τ^{(−a)} := τ(V^{(−a)}).

2.1. Notation. –––– For a prime number p and integers y, z, …, not all of which are equal to zero, we write gcd_{p}(y, z, …) for the largest power of p dividing all of the y, z, ….

2.2. Theorem. –––– There is no constant C such that

m(V^{(−a)}) < \frac{C}{τ^{(−a)}}

for all a ∈ ℕ. Proof. The proof consists of several steps.

First step. For a ≥ 4, one has τ_{∞}^{(−a)} = \frac{1}{\sqrt[4]{a}}I where I is an integral independent of a. This follows immediately from formula (∗) above.

Second step. For the height of the smallest point, we have m(V^{(−a)}) ≥ \sqrt[4]{\frac{a}{4}}. |x| ≥ 1 yields y^{4} + z^{4} + v^{4} + w^{4} ≥ a and max\{|y|, |z|, |v|, |w|\} ≥ \sqrt[4]{\frac{a}{4}}.

Third step. There are two positive constants C_{1} and C_{2} such that, for all a ∈ ℕ,

C_{1} < ∏_{{p prime \atop p>13,p∤a}}τ_{p}^{(−a)} < C_{2} .

For a prime p of good reduction, Hensel’s lemma shows τ_{p}(V^{(−a)}) = (1−\frac{1}{p})·\frac{#V^{(−a)}(𝔽_{p})}{p^{3}} . Further, for the number of points on a non-singular variety over a finite field, there are excellent estimates provided by the Weil conjectures, proven by P. Deligne. In our situation, [De, Théorème (8.1)] may be directly applied. It shows #V^{(−a)}(𝔽_{p}) = p^{3} + p^{2} + p + 1 + E^{(−a)} with an error-term |E^{(−a)}| ≤ 60p^{3/2}. Note that dim H^{3}(V, ℝ) = 60 for every smooth quartic threefold V in P_{ℂ}^{4} .

Consequently, 1 − \frac{60(1−1/p)}{p^{3/2}} − \frac{1}{p^{4}} ≤ τ_{p}^{(−a)} ≤ 1 + \frac{60(1−1/p)}{p^{3/2}} − \frac{1}{p^{4}} . Here, the left hand side is positive for p > 13. The infinite product over all 1 − \frac{60(1−1/p)}{p^{3/2}} − \frac{1}{p^{4}} (respectively 1 + \frac{60(1−1/p)}{p^{3/2}} − \frac{1}{p^{4}} ) is convergent.

Fourth step. There is a sequence {a_{i}}_{i∈ℕ} of natural numbers such that

\{ ∏_{{p prime \atop p≤13 or p|a_{i}}}τ_{p}^{(−a_{i})}\}_{i∈ℕ}

is unbounded. Let C ∈ ℝ be given. We will show that

∏_{{p prime \atop p≤13 or p|a}}τ_{p}^{(−a)} > C

when a := p_{1}·. . .·p_{r} is a product of sufficiently many different primes p_{i} ≡ 3 (mod 4) fulfilling a ≡ 1 (mod M) for M := 16 · 3 · 5 · 7 ·11·13.

Let p be a prime such that p|a. We made sure that p^{4} ∤ a. Then, for any point (x : y : z : v : w) ∈ V^{(−a)}(ℤ/p^{n}ℤ), the assumption p | y, z, v, w would imply p^{4}|ax^{4} and p|x. Therefore, gcd_{p}(y, z, v, w) = 1. Further, y^{4} + z^{4} + v^{4} + w^{4} ≡ 0 (mod p).

As p ≡ 3 (mod 4), the number of solutions of that congruence is the same as that of y^{2} + z^{2} + v^{2} + w^{2} ≡ 0 (mod p). Since p remains prime in ℤ[i], this quadratic form is a direct sum of two norm forms. Its number of zeroes in 𝔽_{p}^{4} is therefore equal to 1 + (p − 1)(p + 1)^{2}.

gcd_{p}(y, z, v, w) = 1 implies that Hensel’s lemma is applicable. It shows

#V^{(−a)}(ℤ/p^{n}ℤ) = \frac{p^{n} · (p − 1)(p + 1)^{2}p^{3(n−1)}}{(p − 1)p^{n−1}} = p^{3n−2}(p + 1)^{2}.

Hence, τ_{p}^{(−a)} = (1 − \frac{1}{p})\frac{(p+1)^{2}}{p^{2}} = 1 + \frac{1}{p} − \frac{1}{p^{2}} − \frac{1}{p^{3}} . We may consequently write

∏_{{p prime \atop p≤13 or p|a}}τ_{p}^{(−a)} = ∏_{i=1}^{r} (1 + \frac{1}{p_{i}} − \frac{1}{p_{i}^{2}} − \frac{1}{p_{i}^{3}} ) · ∏_{{p prime \atop p≤13}}τ_{p}^{(−a)}.

The second product is over p = 2 and a finite number of primes of good reduction. The value of the product depends only on the residue of a modulo M = 16 · 3 · 5 · 7 ·11·13 by virtue of [EJ2, Lemma 13.a) and b.ii)]. In particular, our assumption a ≡ 1 (mod M) implies

T := ∏_{{p prime \atop p≤13}}τ_{p}^{(−a)}

is a constant. It is clear that T > 0 as the equation x^{4} = y^{4} + z^{4} + v^{4} + w^{4} admits a non-zero solution in ℤ/mℤ for any m > 1.

It remains to show that there exists a set {p_{1}, …, p_{r}} of primes p_{i} ≡ 3 (mod 4) such that p_{i} > 13,

(+)  ∏_{i=1}^{r} (1 + \frac{1}{p_{i}} − \frac{1}{p_{i}^{2}} − \frac{1}{p_{i}^{3}} ) ≥ \frac{C}{T} ,

and p_{1} · . . . · p_{r} ≡ 1 (mod M).

Condition (+) is easily satisfied as the series ∑_{p≡3 (mod 4)} \frac{1}{p} diverges. We find a set {p_{1}, …, p_{s}} of prime numbers p_{i} > 13 of the form p_{i} ≡ 3 (mod 4) such that ∏_{i=1}^{s} (1 + \frac{1}{p_{i}} − \frac{1}{p_{i}^{2}} − \frac{1}{p_{i}^{3}} ) ≥ \frac{C}{T}. Enlarging \{p_{1}, …, p_{s}\} makes that product even bigger. We may therefore assume p_{1} · . . . · p_{s} ≡ 3 (mod 4).

The numbers M and p_{1} · . . . · p_{s} are relatively prime. By Dirichlet’s prime number theorem, there exists a prime p_{s+1}, larger than each of the p_{1}, …, p_{s}, such that p_{1} · . . . · p_{s+1} ≡ 1 (mod M). This shows, in particular, p_{s+1} ≡ 3 (mod 4). The assertion follows.

Conclusion. The four steps together show that m(V^{(−a)})·τ^{(−a)} is unbounded. □

3 The fundamental finiteness property

3.1 An estimate for the factors at the finite places

3.1.1. 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) 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.

3.1.2. Definition. –––– For (a_{0}, …, a_{4}) ∈ ℤ^{5}, r ∈ ℕ, and ν_{0}, …, ν_{4} ≤ r, put

\begin{matrix}S_{ν_{0},...,ν_{4};a_{0},...,a_{4}}^{(r)} := \{(x_{0}, …, x_{4}) ∈ (ℤ & /p^{r}ℤ)^{5} | \\ ν(x_{0}) = ν_{0}, …, & ν(x_{4}) = ν_{4}; a_{0}x_{0}^{4} + … + a_{4}x_{4}^{4} = 0 ∈ ℤ/p^{r}ℤ\} . \end{matrix}

For the particular case ν_{0} = … = ν_{4} = 0, we will write Z_{a_{0},...,a_{4}}^{(r)} := S_{0,...,0;a_{0},...,a_{4}}^{(r)}. I.e.,

Z_{a_{0},...,a_{4}}^{(r)} = {(x_{0}, …, x_{4}) ∈ [(ℤ/p^{r}ℤ)^{∗}]^{5} | a_{0}x_{0}^{4} + … + a_{4}x_{4}^{4} = 0 ∈ ℤ/p^{r}ℤ} .

We will use the notation z_{a_{0},...,a_{4}}^{(r)} := #Z_{a_{0},...,a_{4}}^{(r)}.

3.1.3. Sublemma. –––– If p^{k}|a_{0}, …, a_{4} and r > k then we have

z_{a_{0},...,a_{4}}^{(r)} = p^{5k} · z_{a_{0}/p^{k},...,a_{4}/p^{k}}^{(r−k)} .

Proof. Since a_{0}x_{0}^{4} +…+a_{4}x_{4}^{4} = p^{k}(a_{0}/p^{k} ·x_{0}^{4} +…+a_{4}/p^{k} ·x_{4}^{4}), there is a surjection

ι: Z_{a_{0},...,a_{4}}^{(r)} ⟶ Z_{a_{0}/p^{k},...,a_{4}/p^{k}}^{(r−k)} ,

given by (x_{0}, …, x_{4}) ↦ ((x_{0} mod p^{r−k}), …, (x_{4} mod p^{r−k})). The kernel of the homomorphism of modules underlying ι is (p^{r−k}ℤ/p^{r}ℤ)^{5}. □

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

z_{a_{0},...,a_{4}}^{(r)} ≤ 8p^{4r+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_{4}) ∈ (ℤ/p^{r}ℤ)^{4}, there appears an equation of the form a_{0}x_{0}^{4} = c. For p odd, it cannot have more than four solutions in (ℤ/p^{r}ℤ)^{∗} as this group is cyclic. On the other hand, in the case p = 2, we have (ℤ/2^{r}ℤ)^{∗} ≅ ℤ/2^{r−2}ℤ × ℤ/2ℤ and up to eight solutions are possible.

The general case follows directly from Sublemma 3.1.3. Indeed, if k < r then

z_{a_{0},...,a_{4}}^{(r)} = p^{5k} · z_{a_{0}/p^{k},...,a_{4}/p^{k}}^{(r−k)} ≤ p^{5k} · 8p^{4(r−k)} = 8p^{4r+k}.

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

z_{a_{0},...,a_{4}}^{(r)} = #Z_{a_{0},...,a_{4}}^{(r)} < p^{5r} ≤ p^{4r+k} < 8p^{4r+k}.  □

3.1.5. Remark. –––– The proof shows that in the case p ≠ 2 the same inequality is true with coefficient 4 instead of 8. If p ≡ 3 (mod 4) then one could even reduce the coefficient to 2. Unfortunately, these observations do not lead to a substantial improvement of our final result.

3.1.6. Lemma. –––– Let r ∈ ℕ and ν_{0}, …, ν_{4} ≤ r. Then,

#S_{ν_{0},...,ν_{4};a_{0},...,a_{4}}^{(r)} = \frac{z_{p^{4ν_{0}}a_{0}, ... ,p^{4ν_{4}}a_{4}}^{(r)} · φ(p^{r−ν_{0}}) · . . . · φ(p^{r−ν_{4}})}{φ(p^{r})^{5}} .

Proof. As p^{4ν_{0}}a_{0}x_{0}^{4} + … + p^{4ν_{4}}a_{4}x_{4}^{4} = a_{0}(p^{ν_{0}}x_{0})^{4} + … + a_{4}(p^{ν_{4}}x_{4})^{4}, we have a surjection

π: Z_{p^{4ν_{0}}a_{0},...,p^{4ν_{4}}a_{4}}^{(r)} ⟶ S_{ν_{0},...,ν_{4};a_{0},...,a_{4}}^{(r)} ,

given by (x_{0}, …, x_{4}) ↦ (p^{ν_{0}}x_{0}, …, p^{ν_{4}}x_{4}).

For i = 0, …, 4, consider the mapping ι: ℤ/p^{r}ℤ → ℤ/p^{r}ℤ, x_{0} ↦ p^{ν_{i}}x_{0}. 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 on 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_{4})}) : 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. □

3.1.7. Corollary. –––– Let (a_{0}, …, a_{4}) ∈ (ℤ\{0})^{5}. Then, for the local factor τ_{p}^{(a_{0},...,a_{4})} := τ_{p}(V^{(a_{0},...,a_{4})}), one has

τ_{p}^{(a_{0},...,a_{4})} = lim_{r→∞} ∑_{ν_{0},...,ν_{4}=0}^{r} \frac{z_{p^{4ν_{0}}a_{0}, ... ,p^{4ν_{4}}a_{4}}^{(r)} · φ(p^{r−ν_{0}}) · . . . · φ(p^{r−ν_{4}})}{p^{4r} · φ(p^{r})^{5}} .

Proof. [PT, Corollary 3.5] implies that

τ_{p}^{(a_{0},...,a_{4})} = lim_{r→∞} ∑_{ν_{0},...,ν_{4}=0}^{r} \frac{#S_{ν_{0},...,ν_{4};a_{0},...,a_{4}}^{(r)}}{p^{4r}} .

Lemma 3.1.6 yields the assertion. □

3.1.8. Proposition. –––– Let (a_{0}, …, a_{4}) ∈ (ℤ\{0})^{5}. Then, for each ε such that 0 < ε < \frac{1}{4}, one has

τ_{p}^{(a_{0},...,a_{4})} ≤ 8(\frac{1}{1 − \frac{1}{p^{1−4ε}}} )(\frac{1}{1 − \frac{1}{p^{ε}}} )^{4} · (a_{0}^{(p)} · . . . · a_{3}^{(p)})^{\frac{1−ε}{4}} (a_{4}^{(p)})^{ε}.

Proof. By Lemma 3.1.4,

z_{p^{4ν_{0}}a_{0}, ... ,p^{4ν_{4}}a_{4}}^{(r)}/p^{4r} ≤ 8 gcd_{p}(p^{4ν_{0}}a_{0}, …, p^{4ν_{4}}a_{4})
= 8 gcd(p^{4ν_{0}}a_{0}^{(p)}, …, p^{4ν_{4}}a_{4}^{(p)}).

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

z_{p^{4ν_{0}}a_{0}, ... ,p^{4ν_{4}}a_{4}}^{(r)}/p^{4r} ≤ 8 gcd(p^{4ν_{0}+k_{0}}, …, p^{4ν_{4}+k_{4}})
= 8p^{min{4ν_{0}+k_{0}, ... ,4ν_{4}+k_{4}}}.

We estimate the minimum by a weighted arithmetic mean with weights \frac{1−ε}{4}, \frac{1−ε}{4}, \frac{1−ε}{4}, \frac{1−ε}{4}, and ε,

min\{4ν_{0} + k_{0}, …, 4ν_{4} + k_{4}\} ≤ \frac{1 − ε}{4} · (4ν_{0} + k_{0}) + … + \frac{1 − ε}{4} · (4ν_{3} + k_{3})
+ ε(4ν_{4} + k_{4})
= (1 − ε)(ν_{0} + … + ν_{3}) + 4εν_{4}

+ \frac{1 − ε}{4}(k_{0} + … + k_{3}) + εk_{4} .

This shows

z_{p^{4ν_{0}}a_{0}, ... ,p^{4ν_{4}}a_{4}}^{(r)}/p^{4r} ≤ 8p^{(1−ε)(ν_{0}+...+ν_{3})+4εν_{4}+ \frac{1−ε}{4} (k_{0}+...+k_{3})+εk_{4}}
= 8p^{(1−ε)(ν_{0}+...+ν_{3})+4εν_{4}} · (a_{0}^{(p)} … a_{3}^{(p)})^{\frac{1−ε}{4}} (a_{4}^{(p)})^{ε}.

We may therefore write

τ_{p}^{(a_{0},...,a_{4})} ≤ 8(a_{0}^{(p)} … a_{3}^{(p)})^{\frac{1−ε}{4}} (a_{4}^{(p)})^{ε}

· lim_{r→∞} ∑_{ν_{0},...,ν_{4}=0}^{r}\frac{p^{(1−ε)(ν_{0}+...+ν_{3})+4εν_{4}} · φ(p^{r−ν_{0}}) · . . . · φ(p^{r−ν_{4}})}{φ(p^{r})^{5}} .

Here, the term under the limit is precisely the product of four 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^{4εν} · φ(p^{r−ν})}{φ(p^{r})} = ∑_{ν=0}^{r−1} \frac{1}{(p^{1−4ε})^{ν}} + \frac{p}{p − 1} \frac{1}{(p^{1−4ε})^{r}} .

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

□

3.1.9. Remark. –––– Unfortunately, the constants

C_{p}^{(ε)} := 8(\frac{1}{1 − \frac{1}{p^{1−4ε}}} )(\frac{1}{1 − \frac{1}{p^{ε}}} )^{4}

have the property that the product ∏_{p} C_{p}^{(ε)} diverges. On the other hand, we have at least that C_{p}^{(ε)} is bounded for p → ∞, say C_{p}^{(ε)} ≤ C^{(ε)}.

3.1.10. Lemma. –––– Let C > 1 be any constant. Then, for each ε > 0, one has

∏_{{p prime \atop p|x}}C ≤ c · x^{ε}

for a suitable constant c (depending on ε). Proof. This follows directly from [Na, Theorem 7.2] together with [Na, Section 7.1, Exercise 7]. □

3.1.11. Proposition. –––– For each ε > 0, there exists a constant c such that

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

for all (a_{0}, …, a_{4}) ∈ (ℤ\{0})^{5}. Proof. As already noticed in the third step of the proof of Theorem 2.2, the product over all primes of good reduction is bounded by consequence of the Weil conjectures. It, therefore, remains to show that

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

For this, we may assume that ε is small, say ε < \frac{1}{4}.

Then, by Proposition 3.1.8, we have at first

τ_{p}^{(a_{0},...,a_{4})} ≤ C_{p}^{(ε)} · (a_{0}^{(p)} · . . . · a_{3}^{(p)})^{\frac{1}{4} − \frac{ε}{5}} · (a_{4}^{(p)})^{\frac{4}{5} ε}
= C_{p}^{(ε)} · (a_{0}^{(p)} · . . . · a_{3}^{(p)}a_{4}^{(p)})^{\frac{1}{4} − \frac{ε}{5}} · (a_{4}^{(p)})^{− \frac{1}{4} +ε}.

Here, the indices 0, …, 4 are interchangeable. Hence, it is even allowed to write

τ_{p}^{(a_{0},...,a_{4})} ≤ C_{p}^{(ε)} · (a_{0}^{(p)} · . . . · a_{4}^{(p)})^{\frac{1}{4} − \frac{ε}{5}} · (max_{i} a_{i}^{(p)})^{− \frac{1}{4} +ε}
= C_{p}^{(ε)} · (a_{0}^{(p)} · . . . · a_{4}^{(p)})^{\frac{1}{4} − \frac{ε}{5}} · min_{i} ∥a_{i}∥_{p}^{\frac{1}{4} −ε}.

Now, we multiply over all prime divisors of 2a_{0} · . . . · a_{4}. 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 ∤ a_{0} · . . . · a_{4}, anyway.

∏_{{p prime \atop p|2a_{0}...a_{4}}}τ_{p}^{(a_{0},...,a_{4})} ≤ ∏_{{p prime \atop p|2a_{0}...a_{4}}}C_{p}^{(ε)} · ∏_{p prime}(a_{0}^{(p)} · . . . · a_{4}^{(p)})^{\frac{1}{4} − \frac{ε}{5}} · ∏_{p prime} min_{i=0,...,4} ∥a_{i}∥_{p}^{\frac{1}{4} −ε}

= ∏_{{p prime \atop p|2a_{0}...a_{4}}}C_{p}^{(ε)} · |a_{0} · . . . · a_{4}|^{\frac{1}{4} − \frac{ε}{5}} · ∏_{p prime} min_{i=0,...,4} ∥a_{i}∥_{p}^{\frac{1}{4} −ε}

when we observe that ∏_{p} a^{(p)} = |a|. Further, we have C_{p}^{(ε)} ≤ C^{(ε)} and, by Lemma 3.1.10,

∏_{{p prime \atop p|2a_{0}...a_{4}}}C^{(ε)} ≤ c · |2a_{0} · . . . · a_{4}|^{\frac{ε}{5}} .

We finally estimate 2^{\frac{ε}{5}} by a constant. The assertion follows. □

3.2 A bound for the factor at the infinite place

We want to estimate the integral τ_{∞}^{(a_{0},...,a_{4})} := τ_{∞}(V^{(a_{0},...,a_{4})}) described in Paragraph 1.5.

3.2.1. Lemma. –––– There exist two constants C_{1} and C_{2} such that

τ_{∞}^{(a_{0},...,a_{4})} ≤ \{ \begin{matrix}\frac{1}{\sqrt[4]{|a_{0}|a_{1}·...·a_{4}}} (C_{1} min(|a_{0}|, 2a_{4})^{1/4}), & if |a_{0}| ≤ a_{4}, \\ \frac{1}{\sqrt[4]{|a_{0}|a_{1}·...·a_{4}}} (C_{1} min(|a_{0}|, 2a_{4})^{1/4} + C_{2} a_{4}^{1/4} log \frac{min(|a_{0}|,3a_{1})}{a_{4}}), & otherwise, \end{matrix}

for all (a_{0}, …, a_{4}) ∈ ℝ^{5} satisfying a_{1} ≥ a_{2} ≥ a_{3} ≥ a_{4} ≥ 1 and a_{0} ≤ −1. Proof. A linear substitution shows

τ_{∞}^{(a_{0},...,a_{4})} = \frac{1}{4} · \frac{1}{\sqrt[4]{|a_{0}|a_{1} · . . . · a_{4}}} ∫∫∫∫_{R^{(0)}} \frac{1}{(y^{4} + z^{4} + v^{4} + w^{4})^{3/4}} dy dz dv dw

where R^{(0)} := {(y, z, v, w) ∈ [−a_{1}^{1/4}, a_{1}^{1/4}] × ⋯ × [−a_{4}^{1/4}, a_{4}^{1/4}] | |y^{4} + z^{4} + v^{4} + w^{4}| ≤ |a_{0}|} .

The integrand is non-negative. We cover R^{(0)} ⊆ R_{1} ∪ R_{2} by two sets as follows,

R_{1} := {(y, z, v, w) ∈ ℝ^{4} | y^{4} + z^{4} + v^{4} + w^{4} ≤ min(|a_{0}|, 2a_{4})} ,
R_{2} := {(y, z, v, w) ∈ ℝ^{4} | a_{4} ≤ y^{4} + z^{4} + v^{4} ≤ min(|a_{0}|, 3a_{1}) and w ∈ [−a_{4}^{1/4}, a_{4}^{1/4}]} ,

and estimate. In the case |a_{0}| ≤ a_{4}, the domain of integration is covered by R_{1} alone and we may omit R_{2}, completely.

By homogeneity, we have

∫∫∫∫_{R_{1}} \frac{1}{(y^{4} + z^{4} + v^{4} + w^{4})^{3/4}} dy dz dv dw = ω_{1} · ∫_{0}^{min(|a_{0}|,2a_{4})^{1/4}} \frac{1}{r^{3}} · r^{3} dr
= ω_{1} · min(|a_{0}|, 2a_{4})^{1/4}

where ω_{1} is the three-dimensional hypersurface measure of the l_{4}-unit hypersphere

S_{1} := {(y, z, v, w) ∈ ℝ^{4} | y^{4} + z^{4} + v^{4} + w^{4} = 1} .

Further,

∫∫∫∫_{R_{2}} \frac{1}{(y^{4} + z^{4} + v^{4} + w^{4})^{3/4}} dy dz dv dw ≤ 2a_{4}^{1/4}∫∫∫_{R_{3}} \frac{1}{(y^{4} + z^{4} + v^{4})^{3/4}} dy dz dv

where

R_{3} := {(y, z, v) ∈ ℝ^{3} | a_{4} ≤ y^{4} + z^{4} + v^{4} ≤ min(|a_{0}|, 3a_{1})} .

The latter integral may be treated in much the same way as the one above. We see

∫∫∫_{R_{3}} \frac{1}{(y^{4} + z^{4} + v^{4})^{3/4}} dy dz dv = ω_{2} · ∫_{a_{4}^{1/4}}^{min(|a_{0}|,3a_{1})^{1/4}} \frac{1}{r^{3}} · r^{2} dr

where ω_{2} is the usual two-dimensional hypersurface measure of the l_{4}-unit sphere

S_{2} := {(y, z, v) ∈ ℝ^{3} | y^{4} + z^{4} + v^{4} = 1} .

Finally,

∫_{a_{4}^{1/4}}^{min(|a_{0}|,3a_{1})^{1/4}} \frac{1}{r^{3}} · r^{2} dr = log \frac{min(|a_{0}|, 3a_{1})^{1/4}}{a_{4}^{1/4}} = \frac{1}{4} log \frac{min(|a_{0}|, 3a_{1})}{a_{4}} .  □

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

τ_{∞}^{(a_{0},...,a_{4})} ≤ C · |a_{0} · . . . · a_{4}|^{− \frac{1}{4} +ε} · min_{i=0,...,4}∥a_{i}∥∞^{\frac{1}{4}}

for each (a_{0}, …, a_{4}) ∈ ℤ^{5} satisfying a_{0} < 0 and a_{1}, …, a_{4} > 0. Proof. We assume without restriction that a_{1} ≥ … ≥ a_{4}. There are two cases to be distinguished. First case. |a_{0}| ≤ a_{4}. Then, by Lemma 3.2.1, we have

τ_{∞}^{(a_{0},...,a_{4})} ≤ |a_{0} · . . . · a_{4}|^{− \frac{1}{4}} · C_{1} min\{|a_{0}|, 2a_{4}\}^{\frac{1}{4}}
= C_{1} · |a_{0} · . . . · a_{4}|^{− \frac{1}{4}} · |a_{0}|^{\frac{1}{4}}
= C_{1} · |a_{0} · . . . · a_{4}|^{− \frac{1}{4}} · min_{i=0,...,4} ∥a_{i}∥∞^{\frac{1}{4}} .

Second case. |a_{0}| > a_{4}. Here, Lemma 3.2.1 shows

τ_{∞}^{(a_{0},...,a_{4})} ≤ |a_{0} · . . . · a_{4}|^{− \frac{1}{4}} (C_{1} min\{|a_{0}|, 2a_{4}\}^{\frac{1}{4}} + C_{2} a_{4}^{1/4} log \frac{min\{|a_{0}|, 3a_{1}\}}{a_{4}})

≤ |a_{0} · . . . · a_{4}|^{− \frac{1}{4}} (C_{1}(2a_{4})^{\frac{1}{4}} + C_{2} a_{4}^{1/4} log \frac{|a_{0}|}{a_{4}})

= |a_{0} · . . . · a_{4}|^{− \frac{1}{4}} · |a_{4}|^{\frac{1}{4}} (C_{1}2^{\frac{1}{4}} + C_{2} log \frac{|a_{0}|}{a_{4}})

= |a_{0} · . . . · a_{4}|^{− \frac{1}{4}} · min_{i=0,...,4} ∥a_{i}∥∞^{\frac{1}{4}} · (C_{1}2^{\frac{1}{4}} + C_{2} log \frac{|a_{0}|}{a_{4}})

≤ |a_{0} · . . . · a_{4}|^{− \frac{1}{4}} · min_{i=0,...,4} ∥a_{i}∥∞^{\frac{1}{4}} · (C_{1}2^{\frac{1}{4}} + C_{2} log |a_{0} · . . . · a_{4}|)

≤ |a_{0} · . . . · a_{4}|^{− \frac{1}{4}} · min_{i=0,...,4} ∥a_{i}∥∞^{\frac{1}{4}} · (C_{3} · |a_{0} · . . . · a_{4}|)^{ε}.  □

3.3 The Tamagawa number

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

\frac{1}{τ^{(a_{0},...,a_{4})}} ≥ C · \frac{H_{naive} (\frac{1}{a_{0}} : …: \frac{1}{a_{4}} )^{\frac{1}{4}}}{|a_{0} · . . . · a_{4}|^{ε}}

for each (a_{0}, …, a_{4}) ∈ ℤ^{5} satisfying a_{0} < 0 and a_{1}, …, a_{4} > 0. Proof. By Proposition 3.2.2, we have

τ_{∞}^{(a_{0},...,a_{4})} ≤ C_{1} · |a_{0} · . . . · a_{4}|^{− \frac{1}{4} + \frac{ε}{2}} · min_{i=0,...,4} ∥a_{i}∥∞^{\frac{1}{4}} .

On the other hand, by Theorem 3.1.11,

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

It follows that

τ^{(a_{0},...,a_{4})} ≤ C_{3} · |a_{0} · . . . · a_{4}|^{\frac{ε}{2}} · ∏_{p prime} min_{i=0,...,4} ∥a_{i}∥p^{\frac{1}{4}} · min_{i=0,...,4} ∥a_{i}∥∞^{\frac{1}{4}} · ∏_{p prime}[ min_{i=0,...,4} ∥a_{i}∥_{p}]^{− \frac{ε}{2}}

and

\frac{1}{τ^{(a_{0},...,a_{4})}} ≥ \frac{1}{C_{3}} · \frac{∏_{p prime}[ min_{i=0,...,4} ∥a_{i}∥_{p}]^{− \frac{1}{4}} · [ min_{i=0,...,4} ∥a_{i}∥_{∞}]^{− \frac{1}{4}}}{|a_{0} · . . . · a_{4}|^{\frac{ε}{2}} · ∏_{p prime}[ min_{i=0,...,4} ∥a_{i}∥_{p}]^{− \frac{ε}{2}}}

= \frac{1}{C_{3}} · \frac{∏_{p prime} max_{i=0,...,4} ∥\frac{1}{a_{i}} ∥p^{\frac{1}{4}} · max_{i=0,...,4} ∥\frac{1}{a_{i}} ∥∞^{\frac{1}{4}}}{|a_{0} · . . . · a_{4}|^{\frac{ε}{2}} · ∏_{p prime}[ max_{i=0,...,4} a_{i}^{(p)}]^{\frac{ε}{2}}}

= \frac{1}{C_{3}} · \frac{H_{naive}(\frac{1}{a_{0}} : …: \frac{1}{a_{4}} )^{\frac{1}{4}}}{|a_{0} · . . . · a_{4}|^{\frac{ε}{2}} · ∏_{p prime}[ max_{i=0,...,4} a_{i}^{(p)}]^{\frac{ε}{2}}} .

It is obvious that max_{i=0,...,4}a_{i}^{(p)} ≤ |a_{0}^{(p)} · . . . · a_{4}^{(p)}| and ∏_{p prime}|a_{0}^{(p)} · . . . · a_{4}^{(p)}| = |a_{0} · . . . · a_{4}|.

This shows

\frac{1}{τ^{(a_{0},...,a_{4})}} ≥ \frac{1}{C_{3}} · \frac{H_{naive}(\frac{1}{a_{0}} : …: \frac{1}{a_{4}} )^{\frac{1}{4}}}{|a_{0} · . . . · a_{4}|^{\frac{ε}{2}} · |a_{0} · . . . · a_{4}|^{\frac{ε}{2}}}

= \frac{1}{C_{3}} · \frac{H_{naive}(\frac{1}{a_{0}} : …: \frac{1}{a_{4}} )^{\frac{1}{4}}}{|a_{0} · . . . · a_{4}|^{ε}} .  □

3.3.2. Lemma. –––– Let (a_{0} : …: a_{4}) ∈ P^{4}(ℚ) be any point such that a_{0} ≠ 0, …, a_{4} ≠ 0. Then,

H_{naive}(a_{0} : …: a_{4}) ≤ H_{naive}(\frac{1}{a_{0}} : …: \frac{1}{a_{4}} )^{4} .

Proof. First, observe that (a_{0} : …: a_{4}) ↦ (\frac{1}{a_{0}} : …: \frac{1}{a_{4}} ) is a welldefined map. Hence, we may assume without restriction that a_{0}, …, a_{4} ∈ ℤ and gcd(a_{0}, …, a_{4}) = 1. This yields H_{naive}(a_{0} : …: a_{4}) = max_{i=0,...,4}|a_{i}|.

On the other hand, (\frac{1}{a_{0}} : …: \frac{1}{a_{4}} ) = (a_{1}a_{2}a_{3}a_{4} : …: a_{0}a_{1}a_{2}a_{3}). Consequently,

H_{naive} (\frac{1}{a_{0}} : …: \frac{1}{a_{4}} ) ≤ [ max_{i=0,...,4}|a_{i}|]^{4} = H_{naive}(a_{0} : …: a_{4})^{4} .

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

3.3.3. Corollary. –––– Let a_{0}, …, a_{4} ∈ ℤ such that gcd(a_{0}, …, a_{4}) = 1. Then,

|a_{0} · . . . · a_{4}| ≤ H_{naive} (\frac{1}{a_{0}} : …: \frac{1}{a_{4}} )^{20}.

Proof. Observe that |a_{0} · . . . · a_{4}| ≤ max_{i=0,...,4}|a_{i}|^{5} = H_{naive}(a_{0} : …: a_{4})^{5} and apply Lemma 3.3.2. □

3.3.4. Theorem. –––– For each ε > 0, there exists a constant C(ε) > 0 such that, for all (a_{0}, …, a_{4}) ∈ ℤ^{5} satisfying a_{0} < 0 and a_{1}, …, a_{4} > 0,

\frac{1}{τ^{(a_{0},...,a_{4})}} ≥ C(ε) · H_{naive}(\frac{1}{a_{0}} : …: \frac{1}{a_{4}} )^{\frac{1}{4} −ε}.

Proof. We may assume that gcd(a_{0}, …, a_{4}) = 1. Then, by Proposition 3.3.1,

\frac{1}{τ^{(a_{0},...,a_{4})}} ≥ C · \frac{H_{naive}(\frac{1}{a_{0}} : …: \frac{1}{a_{4}} )^{\frac{1}{4}}}{|a_{0} · . . . · a_{4}|^{\frac{ε}{20}}} .

Corollary 3.3.3 yields |a_{0} · . . . · a_{4}|^{\frac{ε}{20}} ≤ H_{naive}(\frac{1}{a_{0}} : …: \frac{1}{a_{4}} )^{ε}. □

3.3.5. Corollary (Fundamental finiteness). —– For each B > 0, there are only finitely many quartics V^{(a_{0},...,a_{4})} : a_{0}x_{0}^{4} + … + a_{4}x_{4}^{4} = 0 in P_{ℚ}^{4} such that a_{0} < 0, a_{1}, …, a_{4} > 0, and τ^{(a_{0},...,a_{4})} > B. Proof. This is an immediate consequence of the comparison to the naive height established in Theorem 3.3.4. □

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