The Asymptotics of Points of Bounded Height on Diagonal Cubic and Quartic Threefolds

Andreas-Stephan Elsenhans^{1} and Jörg Jahnel^{2}

Universität Göttingen, Mathematisches Institut, Bunsenstraße 3–5, D-37073 Göttingen, Germany^{⋆}

           1elsenhan@wwwuser.gwdg.de, ^{2}~jjahnel@wwwuser.gwdg.de

Abstract. For the families ax^{3} = by^{3} + z^{3} + v^{3} + w^{3}, a, b = 1, …, 100, and ax^{4} = by^{4} + z^{4} + v^{4} + w^{4}, a, b = 1, …, 100, of projective algebraic threefolds, we test numerically the conjecture of Manin (in the refined form due to Peyre) about the asymptotics of points of bounded height on Fano varieties.

1 Introduction — Manin’s Conjecture

Let V be a projective algebraic variety over ℚ. We fix an embedding ι: V → P_{ℚ}^{n}. In this situation, there is the well-known naive height H_{naive} : V (ℚ) → ℝ which is given by H_{naive}(P) := max_{i=0,...,n} |x_{i}|. Here, (x_{0} : …: x_{n}) := ι(P) ∈ P^{n}(ℚ) where the projective coordinates are integers satisfying gcd(x_{0}, …, x_{n}) = 1.

It is of interest to ask for the asymptotics of the number of ℚ-rational points on V of bounded naive height. This applies particularly to the case V is a Fano variety as those are expected to have many rational points (at least after a finite extension of the ground-field). Simplest examples of Fano varieties are complete intersections in P_{ℚ}^{n} of a multidegree (d_{1}, …, d_{r}) such that d_{1} + … + d_{r} ≤ n. In this case, Manin’s conjecture reads as follows.

Conjecture 1. Let V ⊆ P_{ℚ}^{n} be a non-singular complete intersection of multidegree (d_{1}, …, d_{r}). Assume dim V ≥ 3 and k := n + 1 − d_{1} − … − d_{r} > 0. Then, there exists a Zariski open subset V^{∘} ⊆ V such that

#{x ∈ V^{∘}(ℚ) | H_{naive}(x)^{k} < B} ∼ CB  (1)

for a constant C.

Example 2. Let V ⊂ P_{ℚ}^{4} be a smooth hypersurface of degree 4. Conjecture 1 predicts ∼ CB rational points of height < B. However, the hypersurface

⋆ The computer part of this work was executed on the Linux PCs of the Gauß Laboratory for Scientific Computing at the Göttingen Mathematisches Institut. 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. Special thanks go to two anonymous referees whose valuable comments and suggestions helped to substancially improve this article.

x^{4} + y^{4} = z^{4} + v^{4} + w^{4} contains the line given by x = z, y = v, and w = 0 on which there is quadratic growth, already. This explains the necessity of the restriction to a Zariski open subset V^{∘} ⊆ V.

Remark 3. Conjecture 1 is proven for P^{n}, linear subspaces, and quadrics. Further, it is established [Bi] in the case that the dimension of V is very large compared to d_{1}, …, d_{r}. Generalizations are known to be true in a number of further particular cases. A complete list may be found in the survey article [Pe2, sec. 4].

In this note, we report numerical evidence for Conjecture 1 in the case of the varieties V_{a,b}^{e} given by ax^{e} = by^{e} + z^{e} + v^{e} + w^{e} in P_{ℚ}^{4} for e = 3 and 4.

Remark 4. By the Noether-Lefschetz Theorem, the assumptions made on V imply that Pic(V\overline{ℚ} ) ≅ ℤ [Ha1, Corollary IV.3.2]. This is no longer true in dimension two. See Remark 6.ii) for more details.

The constant. Conjecture 1 is compatible with results obtained by the classical circle method (e.g. [Bi]). Motivated by this, E. Peyre [Pe1] provided a description of the constant C expected in (1). In the situation considered here, Peyre’s constant is equal to the Tamagawa-type number

τ_{H}(V) := ∏_{p∈ℙ∪\{∞\}}(1 − \frac{1}{p}) ω_{H,p}(V (ℚ_{p})).

In this formula, the Tamagawa measure ω_{H,p} is given in local p-adic analytic coordinates x_{1}, …, x_{d} by ∥\frac{∂}{∂x_{1}} ∧ … ∧ \frac{∂}{∂x_{d}} ∥_{p} dx_{1} … dx_{d}. Here, each dx_{i} denotes a Haar measure on ℚ_{p} which is normalized in the usual manner. \frac{∂}{∂x_{1}} ∧ … ∧ \frac{∂}{∂x_{d}} is a section of 𝒪(−K) ≅ 𝒪(−k).

For p finite, one has a canonical model 𝒱 ⊆ P_{ℤ_{p}}^{n} of V. This defines the norm ∥.∥_{p} on 𝒪(−k). It is almost immediate from the definition that

ω_{H,p}(V (ℚ_{p})) = lim_{n→∞} \frac{#V (ℤ/p^{n}ℤ)}{p^{dn}}.

The place at infinity. Here, ∥.∥_{sup} on 𝒪(1) is the hermitian metric, in the sense of complex geometry, defined by ∥x_{i}∥_{sup} := inf_{j=0,...,n}|x_{i}/x_{j}|. This induces the hermitian metric ∥.∥_{∞} := ∥.∥_{sup}^{−k} on 𝒪(−k).

Lemma 5. If V ⊂ P^{n} is a hypersurface defined by the equation f = 0 then

ω_{∞}(V (ℝ)) = \frac{1}{2} ∫_{\begin{matrix}f(x_{0}, ... ,x_{n})=0 \\ |x_{0}|, ... ,|x_{n}|≤1 \end{matrix}} ω_{Leray}.

The Leray measure ω_{Leray} on {(x_{0}, …, x_{n}) ∈ ℝ^{n+1} | f(x_{0}, …, x_{n}) = 0} is related to the usual hypersurface measure by the formula ω_{Leray} = \frac{1}{grad f} ω_{hyp}. On the other hand, one may also write

ω_{Leray} = \frac{1}{|\frac{∂f}{∂x_{i}} (x_{0}, …, x_{n})|} dx_{0} … d̂x_{i} … dx_{n}.

Proof. The equivalence of the two descriptions of the Leray measure is a standard calculation. The main assertion is a particular case of [Pe1, Lemma 5.4.7]. 2 is the number of roots of unity in ℚ. □

Remark 6. There are several ways to generalize Conjecture 1.

i) One may consider more general heights corresponding to the tautological line bundle 𝒪(1). This includes to

a) replace the maximum norm by an arbitrary continuous hermitian metric on 𝒪(1). This would affect the domain of integration for the factor at infinity.

b) multiply H_{naive}(x) with a function that depends on the reduction of x modulo some N ∈ ℕ. This augments Conjecture 1 by an equidistribution statement. ii) Instead of complete intersections, one may consider arbitrary projective Fano varieties V. Then, H_{naive}^{k} needs to be replaced by a height defined by the anticanonical bundle 𝒪(−K_{V}). If Pic(V\overline{ℚ} ) ≅ ℤ̸ then the description of the constant C gets more complicated in several ways. First, there is an additional factor β := #H^{1}(Gal(\overline{ℚ}/ℚ), Pic(V\overline{ℚ} )). Further, instead of the factors (1 − \frac{1}{p}) one has to write 1/L_{p}(1, Pic(V\overline{ℚ} )), L_{p} being the local L-function corresponding to the Picard group. Finally, the Tamagawa measure has to be taken not of the full variety V (𝔸_{ℚ} ) but of the subset which is not affected by the Brauer-Manin obstruction. If Pic(V) ≅ ℤ̸ already over ℚ then the right hand side of (1) has to be replaced by CB log^{t} B. For the exponent of the log-term, there is the expectation that t = rk Pic(V)−1. There are, however, examples [BT] in dimension three in which the exponent is larger. The constant C gets equipped with yet another additional factor α which depends on the structure of the effective cone in Pic(V) and on the position of −K in it [Pe1, Définition 2.4]. iii) Finally, there is a generalization to arbitrary number fields [Pe1].

2 Computation of the Tamagawa number

Counting points over finite fields. We consider the projective varieties V_{a,b}^{e} given by ax^{e} = by^{e} + z^{e} + v^{e} + w^{e} in P_{𝔽_{p}}^{4} . We assume a, b ≠ 0 (and p ∤ e) in order to avoid singularities. Observe that, even for large p, these are at most e^{2} varieties up to obvious 𝔽_{p}-isomorphism as 𝔽_{p}^{∗} consists of no more than e cosets modulo (𝔽_{p}^{∗})^{e}.

It follows from the Weil conjectures, proven by P. Deligne [De, Théorème (8.1)], that

#V_{a,b}^{e}(𝔽_{p}) = p^{3} + p^{2} + p + 1 + E_{a,b}^{e}

where the error-term E_{a,b}^{e} may be estimated by |E_{a,b}^{e}| ≤ C_{e}p^{3/2}.

Here, C_{3} = 10 and C_{4} = 60 as dim H^{3}(𝒱 ^{3}, ℝ) = 10 for every smooth cubic threefold and dim H^{3}(𝒱 ^{4}, ℝ) = 60 for every smooth quartic threefold in P^{4}(ℂ). These dimensions result from the Weak Lefschetz Theorem together with F. Hirzebruch’s formula [Hi, Satz 2.4] for the Euler characteristic which actually works in much more generality.

Remark 7. Suppose e = 3 and p ≡ 2 (mod 3). Then, #V_{a,b}^{3}(𝔽_{p}) = #V_{a,b}^{1}(𝔽_{p}) as gcd(p − 1, 3) = 1. Similarly, for e = 4 and p ≡ 3 (mod 4), one has gcd(p − 1, 4) = 2 and #V_{a,b}^{4}(𝔽_{p}) = #V_{a,b}^{2}(𝔽_{p}). In these cases, the error term vanishes and #V_{a,b}^{e}(𝔽_{p}) = p^{3} + p^{2} + p + 1.

In the remaining cases p ≡ 1 (mod 3) for e = 3 and p ≡ 1 (mod 4) for e = 4, our goal is to compute the number of 𝔽_{p}-rational points on V_{a,b}^{e}. As V_{a,b}^{e} ⊆ P^{4}, there would be an obvious O(p^{4})-algorithm. We can do significantly better than that.

Definition 8. Let K be a field and let x ∈ K^{n} and y ∈ K^{m} be two vectors. Then, their convolution z := x ∗ y ∈ K^{n+m−1} is defined to be z_{k} := ∑_{i+j=k}x_{i}y_{j}.

Theorem 9 (FFT convolution). Let n = 2^{l} and K be a field which contains the 2n-th roots of unity. Then, the convolution x∗y of two vectors x, y of length ≤ n can be computed in O(n log n) steps. Proof. The idea is to apply the Fast Fourier Transform (FFT) [Fo, Satz 20.3]. The connection to the convolution is shown in [Fo, Satz 20.2, or CLR, Theorem 32.8]. □

Theorem 9 is the basis for the following algorithm.

Algorithm 10 (FFT point counting on V_{a,b}^{e}).

i) Initialize a vector x[0 … p] with zeroes. ii) Let r run from 0 to p − 1 and increase x[r^{e} mod p] by 1. iii) Calculate ỹ:= x ∗ x ∗ x by FFT convolution. iv) Normalize by putting y[i] := ỹ[i] + ỹ[i + p] + ỹ[i + 2p] for i = 0, …, p − 1.

v) Initialize N as zero. vi) (Counting points with first coordinate ≠ 0) Let j run from 0 to p − 1 and increase N by y[(a − bj^{4}) mod p]. vii) (Counting points with first coordinate 0 and second coordinate ≠ 0) Increase N by y[(−b) mod p]. viii) (Counting points with first and second coordinate 0) Increase N by (y[0] − 1)/(p − 1). ix) Return N as the number of all 𝔽_{p}-valued points on V_{a,b}^{e}.

Remark 11. For the running-time, step iii) is dominant. Therefore, the runningtime of Algorithm 10 is O(p log p).

To count, for fixed e and p, 𝔽_{p}-rational points on V_{a,b}^{e} with varying a and b, one needs to execute the first four steps only once. Afterwards, one may perform steps v) through ix) for all pairs (a, b) of elements from a system of representatives for 𝔽_{p}^{∗}/(𝔽_{p}^{∗})^{e}. Note that steps v) through ix) alone are of complexity O(p).

We ran this algorithm for all primes p ≤ 10^{6} and stored the cardinalities in a file. This took several days of CPU time.

Remark 12. There is a formula for #V_{a,b}^{e}(𝔽_{p}) in terms of Jacobi sums. A skilful manipulation of these sums should lead to another efficient algorithm which serves the same purpose as Algorithm 10.

The local factors at finite places. We are interested in the Euler product

τ_{a,b,fin}^{e} := ∏_{p∈ℙ} (1 − \frac{1}{p}) lim_{n→∞} \frac{#V_{a,b}^{e}(ℤ/p^{n}ℤ)}{p^{3n}}.

Lemma 13. a) (Good reduction) If p ∤ abe then the sequence (#V_{a,b}^{e}(ℤ/p^{n}ℤ)/p^{3n})_{n∈ℕ} is constant.

b) (Bad reduction)

i) If p divides ab but not e then the sequence (#V_{a,b}^{e}(ℤ/p^{n}ℤ)/p^{3n})_{n∈ℕ} becomes stationary as soon as p^{n} divides neither a nor b. ii) If p = 2 and e = 4 then the sequence (#V_{a,b}^{e}(ℤ/p^{n}ℤ)/p^{3n})_{n∈ℕ} becomes stationary as soon as 2^{n} does not divide 8a or 8b. iii) If p = 3 and e = 3 then the sequence (#V_{a,b}^{e}(ℤ/p^{n}ℤ)/p^{3n})_{n∈ℕ} becomes stationary as soon as 3^{n} divides neither 3a nor 3b.

Theorem 14. For every pair (a, b) of integers such that a, b ≠ 0, the Euler product τ_{a,b,fin}^{e} is convergent. Proof. Let p be a prime bigger than |a|, |b|, and e. Then, the factor at p is τ_{p} := (1 − \frac{1}{p})(1 + p + p^{2} + p^{3} + D_{p}p^{3/2})/p^{3} where |D_{p}| ≤ C_{e} for C_{3} = 10 and C_{4} = 60, respectively.

Taking the logarithm, we consider ∑_{p} log τ_{p}. In the case e = 3, the sum is effectively over the primes p = 1 (mod 3). If e = 4 then summation extends over all primes p = 1 (mod 4). In either case, we take a sum over one-half of all primes. This leads to the following estimate,

∑_{p≥N} | log τ_{p}| ≤ ∑_{p≥N} [\frac{C_{e}}{p^{3/2}} + O(p^{−5/2})] ∼ \frac{C_{e}}{2} ∫_{N}^{∞}\frac{1}{t^{3/2} log t} dt

≤ \frac{C_{e}}{2 log N} ∫_{N}^{∞}t^{−3/2}dt = \frac{C_{e}}{\sqrt{N} log N} . □

Remark 15. We are interested in an explicit upper bound for | ∑_{p≥10^{6}} log τ_{p}|. Using Taylor’s formula, one gets

| ∑_{p≥10^{6}} log τ_{p} − ∑_{p≥10^{6}} \frac{D_{p}}{p^{3/2}} | ≤ 10^{−8}.

Since \frac{D_{p}}{p^{3/2}} is zero for p ≡ 3 (mod 4) (or p ≡ 2 (mod 3)), the sum should be compared with log(ζ_{K}(3/2)). Here, ζ_{K} is the Dedekind Zeta function of K = ℚ(i) or ℚ(ζ_{3}), respectively. This yields

∑_{\begin{matrix}p≥10^{6} \\ p≡1 (mod 4) \end{matrix}} \frac{1}{p^{3/2}} ≤ log \frac{\sqrt{ζ_{ℚ(i)} (3/2)}}{(1 − 2^{−3/2})^{−1/2} · ∏_{p≡3 (mod 4)} (1 − p^{−3})^{−1/2} · ∏_{\begin{matrix}p<10^{6} \\ p≡1 (mod 4) \end{matrix}} (1 − p^{−3/2})^{−1}}

and, for the other case, a similar estimate containing ζ_{ℚ(ζ_{3})} (3/2). Note that the infinite product in the denominator converges a lot faster than the left hand side.

Using Pari, we evaluated the right hand side numerically. We found 0.39% for the quartic and 0.065% for the cubic as upper bounds for the error of approximation.

Remark 16. In practice, the error of the approximation is much smaller. The main reason is that the error-term D_{p} may have a positive or a negative sign. Some cancellations happen during summation. The assumption of a random distribution would result in a higher order of convergence. In fact, we observed this effect numerically.

Approximation of the Euler product. Lemma 13 allows us to determine each factor of the Euler product exactly. As we need to know the numerical value of τ_{a,b,fin}^{e}, we approximate it by a finite product.

Observe that the factor at a good prime p is simply (1 − 1/p) · #V_{a,b}^{e}(𝔽_{p})/p^{3}. In particular, for this factor there are only e^{2} values possible. Even more, these numbers had been precomputed using FFT point counting (Algorithm 10). The algorithm below is based on the fact that the vast majority of the factors actually do not need to be computed. They are available from a list.

Algorithm 17 (Compute an approximate value for τ_{a,b,fin}^{3} (τ_{a,b,fin}^{4})).

i) Let p run over all prime numbers such that p ≡ 2 (mod 3) (p ≡ 3 (mod 4)) and p ≤ N and calculate the product of all values of (1 − 1/p^{4}).

ii) Compute the factor corresponding to p = 3 (p = 2) by Lemma 13.b).

iii) Let p run over all prime numbers such that p ≡ 1 (mod 3) (p ≡ 1 (mod 4)) and p ≤ N. Calculate the product of the factors described below.

If p|ab then the corresponding factor is given by Lemma 13.b). Otherwise, compute the e-th power residue-symbols of a and b and look up the precomputed factor for this 𝔽_{p}-isomorphism class of varieties in the list.

iv) Multiply the two products from steps i) and iii) and the factor from step ii) with each other. Correct the product by taking the bad primes p ≡ 2 (mod 3) (p ≡ 3 (mod 4)) into consideration.

Remark 18. When we meet a bad prime p, we have to count ℤ/p^{n}ℤ-valued points on V_{a,b}^{e}. This is done by an algorithm which is very similar to Algorithm 10.

We used Algorithm 17 to compute the Euler products τ_{a,b,fin}^{3} and τ_{a,b,fin}^{4} for a, b = 1, …, 100. We did all calculations for N = 10^{6}. Note that step i) had to be done only once for e = 3 and once for e = 4. The running-time was a quarter of an hour for either exponent.

The factor at the infinite place. For the quartic V_{a,b}^{4}, we have the integral

ω_{H,∞}(V_{a,b}^{4}(ℝ)) = \frac{1}{4\sqrt[4]{a}}∫∫∫∫_{R} \frac{1}{(by^{4} + z^{4} + v^{4} + w^{4})^{3/4}} dy dz dv dw

over R := {(y, z, v, w) ∈ ℝ^{4} | |y|, |z|, |v|, |w| ≤ 1 and |by^{4} + z^{4} + v^{4} + w^{4}| ≤ a} . The integrand is singular in one point. We used a simple substitution to make it sufficiently smooth for numerical integration.

On the other hand, for the cubic V_{a,b}^{3}, we have to consider

ω_{H,∞}(V_{a,b}^{3}(ℝ)) = \frac{1}{6\sqrt[3]{a}}∫∫∫∫_{R} \frac{1}{(by^{3} + z^{3} + v^{3} + w^{3})^{2/3}} dy dz dv dw

for R := {(y, z, v, w) ∈ ℝ^{4} | |y|, |z|, |v|, |w| ≤ 1 and |by^{3} + z^{3} + v^{3} + w^{3}| ≤ a} . The difficulty here is the handling of the singularity of the integrand. It is located in the zero set of by^{3} + z^{3} + v^{3} + w^{3} in R.

Since (by^{3} + z^{3} + v^{3} + w^{3})^{−2/3} is a homogeneous function, it is enough to integrate over the boundary of R. This reduces the problem to several three-dimensional integrals of functions having a two-dimensional singular locus. If a ≥ b + 3 then R is a cube and the boundary of R is easy to describe. We restricted our attention to this case. We smoothed the singularities by separation of Puiseux expansions and substitutions. The resulting integrals were treated by the Gauß-Legendre formula [Kr].

3 On the geometry of diagonal cubic threefolds

Lemma 19. Let V ⊂ P^{4} be any smooth hypersurface. Then, every (reduced but possibly singular) surface S ⊂ V is a complete intersection V ∩ H_{d} with a hypersurface H_{d} ⊂ P^{4}. Proof. By the Noether-Lefschetz Theorem, we have Pic(V) ≅ ℤ. The surface S is a Weil divisor on V. Hence, 𝒪(S) = 𝒪(d) ∈ Pic(V) for a certain d > 0. The restriction Γ (P^{4}, 𝒪(d)) → Γ (V, 𝒪(d)) is surjective as H^{1}(P^{4}, 𝒪_{V} (d − deg V)) = 0 [Ha2, Theorem III.5.1.b)]. □

Elliptic Cones. Let V ⊂ P^{4}(ℂ) be the diagonal cubic threefold given by the equation x^{3} + y^{3} + z^{3} + v^{3} + w^{3} = 0. Fix ζ ∈ ℂ such that ζ^{3} = 1. Then, for every point (x_{0} : y_{0} : z_{0}) on the elliptic curve F: x^{3} + y^{3} + z^{3} = 0, the line given by (x : y : z) = (x_{0} : y_{0} : z_{0}) and v = −ζw is contained in V. All these lines together form a cone C_{F} over F the cusp of which is (0 : 0 : 0 : −ζ : 1). C_{F} is a singular model of a ruled surface over an elliptic curve. This shows, there are no other rational curves contained in C_{F}.

By permuting coordinates, one finds a total of thirty elliptic cones of that type within V. The cusps of these cones are usually named Eckardt points [Mu,CG]. We call the lines contained in one of these cones the obvious lines lying on V. It is clear that there are an infinite number of lines on V running through each of the thirty Eckardt points.

Proposition 20 (cf. [Mu, Lemma 1.18]). Let V ⊂ P^{4} be the diagonal cubic threefold given by the equation x^{3}+y^{3}+z^{3}+v^{3}+w^{3} = 0. Then, through each point p ∈ V different from the thirty Eckardt points there are precisely six lines on V. Proof. Let P = (x_{0} : y_{0} : z_{0} : v_{0} : w_{0}). A line l through P and another point Q = (x : y : z : v : w) is parametrized by (s : t) ↦ ((sx_{0} +tx) : …: (sw_{0} +tw)). Comparing coefficients at s^{2}t, st^{2}, and t^{3}, we see that the condition that l lies on V may be expressed by the three equations below.

x_{0}^{2}x + y_{0}^{2}y +z_{0}^{2}z + v_{0}^{2}v + w_{0}^{2}w = 0  (2)
x_{0}x^{2} + y_{0}y^{2} + z_{0}z^{2} + v_{0}v^{2} + w_{0}w^{2} = 0  (3)
x^{3} + y^{3} + z^{3} + v^{3} + w^{3} = 0  (4)

The first equation means that Q lies on the tangent hyperplane H_{P} at P while equation (4) just encodes that Q ∈ V. By [Za, Corollary 1.15.b)], H_{P} ∩ V is an irreducible cubic surface.

On the other hand, the quadratic form q on the left hand side of equation (3) is of rank at least 3 as P is not an Eckardt point. Therefore, q is not just the product of two linear forms. In particular, q|_{H_{P}} ≢ 0.

As H_{P} ∩ V is irreducible, Z(q|_{H_{P}}) and H_{P} ∩ V do not have a component in common. By Bezout’s theorem, their intersection in H_{P} is a curve of degree 6. □

Remark 21. It may happen that some of the six lines coincide. Actually, it turns out that a line appears with multiplicity > 1 if and only if it is obvious [Mu, Lemma 1.19]. In particular, for a general point P the six lines through it are different from each other.

Under certain exceptional circumstances it is possible to write down all six lines explicitly. For example, if P = (\sqrt[3]{−4} : 1 : 1 : 1 : 1) then the line (\sqrt[3]{−4}t : (t + s) : (t + is) : (t − s) : (t − is)) through P lies on V. Permuting the three rightmost coordinates yields all six lines.

4 Detection of accumulating subvarieties

The detection of �-rational lines on the cubics. On a cubic threefold V_{a,b}^{3}, quadratic growth is predicted for the number of ℚ-rational points of bounded height. Lines are the only curves with such a growth rate.

The moduli space of the lines on a cubic threefold is well-understood. It is a surface of general type [CG, Lemma 10.13]. Nevertheless, we do not know of a method to find all ℚ-rational lines on a given cubic threefold, explicitly. For that reason, we use the algorithm below which is an irrationality test for the six lines through a given point (x_{0} : y_{0} : z_{0} : v_{0} : w_{0}) ∈ V_{a,b}^{3}(ℚ).

Algorithm 22 (Test the six lines through a given point for irrationality).

i) Let p run through the primes from 3 to N. For each p, solve the system of equations (2), (3), (4) (adapted to V_{a,b}^{3}) in 𝔽_{p}^{5}. If the multiples of (x_{0}, y_{0}, z_{0}, v_{0}, w_{0}) are the only solutions then output that there is no ℚ-rational line through (x_{0} : y_{0} : z_{0} : v_{0} : w_{0}) and terminate prematurely. ii) If the loop comes to its regular end then output that the point is suspicious. It could possibly lie on a ℚ-rational line.

Remark 23. We use a very naive O(p)-algorithm to solve the system of equations over 𝔽_{p}. If, say, x_{0} ≠ 0 then it is sufficient to consider quintuples such that x = 0. We parametrize the projective plane given by (2). Then, we compute all points on the conic given by (2) and (3). For each such point, we compute the cubic form on the left hand side of (4).

We carried out the irrationality test on every ℚ-rational point found on any of the cubics except for the points lying on an obvious line. We worked with N = 600. It turned out that suspicious points are rare and that, at least in our sample, each of them actually lies on a ℚ-rational line.

We found only 42 non-obvious ℚ-rational lines on all of the cubics V_{a,b}^{3} for 100 ≥ a ≥ b ≥ 1 together. Among them, there are only five essentially different ones. We present them in the table below. The list might be enlarged by two, as V_{21,6}^{3} and V_{22,5}^{3} may be transformed into V_{48,21}^{3} and V_{40,22}^{3}, respectively, by an automorphism of P^{4}. Further, each line has six pairwise different images under the obvious operation of the group S_{3}.

Table 1. Sporadic lines on the cubic threefolds

a b Smallest point  Point s.t. x = 0
19 18 (1 : 2 : 3 : -3 : -5) (0 : 7 : 1 : -7 : -18)
21 6 (1 : 2 : 3 : -3 : -3) (0 : 9 : 1 : -10 : -15)
22 5 (1 : -1 : 3 : 3 : -3) (0 : 27 : -4 : -60 : 49)
45 18 (1 : 1 : 3 : 3 : -3) (0 : 3 : -1 : 3 : -8)
73 17 (1 : 5 : -2 : 11 : -15) (0 : 27 : -40 : 85 : -96)

Remark 24. It is a priori unnecessary to search for accumulating surfaces, at least if we assume some conjectures.

First of all, only rational surfaces are supposed to accumulate that many rational points that it could be seen through our asymptotics. Indeed, a surface which is abelian or bielliptic may not have more than O(log^{t} B) points of height < B. Non-rational ruled surfaces accumulate points in curves, anyway. Further, it is expected [Pe2, Conjecture 3.6] that K3 surfaces, Enriques surfaces, and surfaces of Kodaira dimension one may have no more than O(B^{ε}) points of height < B outside a finite union of rational curves. For surfaces of general type, finally, expectations are even stronger (Lang’s conjecture).

A rational surface S is, up to exceptional curves, the image of a rational map φ: P^{2} [figure] V ⊂ P^{4}. There is a birational morphism ε: P → P^{2} such that \overline{φ} := φ ∘ ε is a morphism of schemes. ε is given by a sequence of blowingups [Bv, Theorem II.11]. \overline{φ} is defined by the linear system |dH − E| where d := deg φ, H is a hyperplane section, and E is the exceptional divisor. On the other hand, K := K_{P} = −3H + E. Therefore, if d ≥ 3 then

H_{naive}(\overline{φ}(p)) = H_{dH−E}(p) = H_{3H− \frac{3}{d} E}(p)^{d/3} ≥ c · H_{−K}(p)^{d/3}

for p ∉ supp(E). Manin’s conjecture implies there are O((B log^{t} B)^{\frac{3}{d}}) = o(B^{2}) points of height < B on the Zariski-dense subset \overline{φ}(P \ supp(E)) ⊆ S.

It remains to show that there are no rational maps φ: P^{2} [figure] V of degree d ≤ 2. Indeed, under this assumption, deg \overline{φ(P^{2})} ≤ 4. This implies, by virtue of Lemma 19, that \overline{φ(P^{2})} is necessarily a hyperplane section V ∩ H. Zak’s theorem [Za, Corollary 1.8] shows that V ∩ H contains only finitely many singular points. It is, however, well known that cubic surfaces in 3-space which are the image of P^{2} under a quadratic map have a singular line [Bv, Corollary IV.8].

The detection of �-rational conics on the quartics. On a quartic threefold, linear growth is predicted for the number of ℚ-rational points of bounded height. The assumption b > 0 ensures that there are no ℝ-rational lines contained in V_{a,b}^{4}. The only other curves with at least linear growth one could think about are conics.

We were not able to create an efficient routine to test whether there is a ℚ-rational conic through a given point. The resulting system of equations seems to be too complicated to handle.

Conics through two points. A conic Q through (x_{0} : y_{0} : z_{0} : v_{0} : w_{0}) and (x_{1} : y_{1} : z_{1} : v_{1} : w_{1}) may be parametrized in the form

(s : t) ↦ ((λx_{0}s^{2} + μx_{1}t^{2} + xst) : …: (λw_{0}s^{2} + μw_{1}t^{2} + wst))

for some x, y, z, v, w, λ, μ ∈ ℤ. The condition that Q is contained in V_{a,b}^{4} leads to a system G of seven equations in x, y, z, v, w, and λμ. The phenomenon that λ and μ do not occur individually is explained by the fact that they are not invariant under the automorphisms of P^{1} which fix 0 and ∞.

Algorithm 25 (Test for conic through two points).

i) Let p run through the primes from 3 to N. In the exceptional case that G could allow a solution such that p|x, y, z, v, w but p^{2} ∤ λμ, do nothing. Otherwise, solve G in 𝔽_{p}^{6}. If (0, 0, 0, 0, 0, 0) is the only solution then output that there is no ℚ-rational conic through (x_{0} : y_{0} : z_{0} : v_{0} : w_{0}) and (x_{1} : y_{1} : z_{1} : v_{1} : w_{1}) and terminate prematurely. ii) If the loop comes to its regular end then output that the pair is suspicious. It could possibly lie on a ℚ-rational conic.

To solve the system G in 𝔽_{p}^{6}, we use an O(p)-algorithm. Actually, comparison of coefficients at s^{7}t and st^{7} yields two linear equations in x, y, z, v, and w. We parametrize the projective plane I given by them. Comparison of coefficients at s^{6}t^{2} and s^{2}t^{6} leads to a quadric O and an equation λμ = q(x, y, z, v, w)/M with a quadratic form q over ℤ and an integer M ≠ 0. The case p|M sends us to the next prime immediately. Otherwise, we compute all points on the conic I ∩O. For each of them, we test the three remaining equations.

Conics through three points. Three points P_{1}, P_{2}, and P_{3} on V_{a,b}^{4} define a projective plane P. The points together with the two tangent lines P ∩ T_{P_{1}} and P ∩ T_{P_{2}} determine a conic Q, uniquely. It is easy to transform this geometric insight into a formula for a parametrization of Q. We then need a test whether a conic given in parametrized form is contained in V_{a,b}^{4}. This part is algorithmically simple but requires the use of multiprecision integers.

Detecting conics. For each quartic V_{a,b}^{4}, we tested every pair of ℚ-rational points of height < 10 000 for a conic through them. The existence of a conic through (P, Q) is equivalent to the existence of a conic through (gP, gQ) for g ∈ (ℤ/2ℤ)^{4} ⋊ S_{3} ⊆ Aut(V_{a,b}^{4}). This reduces the running time by a factor of about 96. Further, pairs already known to lie on the same conic were excluded from the test.

For each pair (P, Q) found suspicious, we tested the triples (P, Q, R) for R running through the ℚ-rational points of height < 10 000, until a conic was found. Due to the symmetries, one finds several conics at once. For each conic detected, all points on it were marked as lying on this conic.

Actually, there were a few pairs found suspicious through which no conic could be found. In any of these cases, it was easy to prove by hand that there is actually no ℝ-rational conic passing through the two points. This means, we detected every conic which meets at least two of the rational points of height < 10 000.

The conics found. Up to symmetry, we found a total of 1 533 ℚ-rational conics on all of the quartics V_{a,b}^{4} for 1 ≤ a, b ≤ 100 together.

Among them, 1 410 are contained in a plane of type z = v + w and Y x−Xy = 0 for (X, Y, t) a rational point on the genus one curve aX^{4}−bY^{4} = 2t^{2}. Further, there are 90 conics which are slight modifications of the above with y interchanged with z, v, or w. This is possible if b is a fourth power.

There is a geometric explanation for the occurrence of these conics. The hyperplane given by z = v + w intersects V_{a,b}^{4} in a surface S with the two singular points (0 : 0 : −1 : e^{±2πi/3} : e^{∓2πi/3}). The linear projection π: S [figure] P^{1} to the first two coordinates is undefined only in these two points. Its fibers are plane quartics which split into two conics as (v +w)^{4} +v^{4} +w^{4} = 2(v^{2} +vw +w^{2})^{2}. After resolution of singularities, the two conics become disjoint. S̃ is a ruled surface over a twofold cover of P^{1} ramified in the four points such that ax^{4} − by^{4} = 0, i.e. over a curve of genus one.

In the case a is twice a square, a different sort of conics comes from the equations v = z + Dy and w = Ly when (L, D) is a point on the affine genus three curve C_{b}: L^{4}+b = D^{4}. We found 28 conics of this type. C_{b} has a ℚ-rational point for b = 5, 15, 34, 39, 65, 80, and 84. The conics actually admit a ℚ-rational point for a = 2, 18, 32, and 98.

The remaining five conics are given as follows. For a = 3, 12, 27, or 48 and b = 10, intersect with the plane given by v = y + z and w = 2y + z. For a = 17 and b = 30, put v = 2x + y and w = x + 3y + z.

Remark 26. Again, it is not necessary to search for accumulating surfaces. Here, rational maps φ: P^{2} [figure] V ⊂ P^{4} such that deg φ ≤ 3 need to be taken into consideration. We claim, such a map is impossible.

If deg φ = 3 then we had φ: (λ : μ : ν) ↦ (K_{0}(λ, μ, ν) : …: K_{4}(λ, μ, ν)) where K_{0}, …, K_{4} are cubic forms defined over ℚ. K_{0} = 0 defines a plane cubic which has infinitely many real points, automatically. As the image of φ is assumed to be contained in V_{a,b}^{4}, we have that K_{0}(λ, μ, ν) = 0 implies K_{1}(λ, μ, ν) = … = K_{4}(λ, μ, ν) = 0 for λ, μ, ν ∈ ℝ. By consequence, K_{1}, …, K_{4} are divisible by K_{0} (or by a linear factor of K_{0} in the case it is reducible) and φ is not of degree three.

For deg φ ≤ 2, we had deg \overline{φ(P^{2})} ≤ 4 such that \overline{φ(P^{2})} = V ∩ H is a hyperplane section. Zak’s theorem [Za, Corollary 1.8] shows it has at most finitely many singular points. On the other hand, a quartic in P^{3} which is the image of a quadratic map from P^{2} is a Steiner surface. It is known [Ap, p. 40] to have one, two, or (in generic case) three singular lines.

5 The final results

A technology to find solutions of Diophantine equations. In [EJ1] and [EJ2], we described a modification of D. Bernstein’s [Be] method to search efficiently for all solutions of naive height < B of a Diophantine equation of the particular form f(x_{1}, …, x_{n}) = g(y_{1}, …, y_{m}). The expected running-time of our algorithm is O(B^{max{n,m}}). Its basic idea is as follows.

Algorithm 27 (Search for solutions of a Diophantine equation).

i) (Writing) Evaluate f on all points of the cube {(x_{1}, …, x_{n}) ∈ ℤ^{n} | |x_{i}| < B} of dimension n. Store the values within a hash table H. ii) (Reading) Evaluate g on all points of the cube {(y_{1}, …, y_{m}) ∈ ℤ^{m} | |y_{i}| < B}. For each value, start a search in order to find out whether it occurs in H. When a coincidence is detected, reconstruct the corresponding values of x_{1}, …, x_{n} and output the solution.

Remark 28. In the case of a variety V_{a,b}^{e}, the running-time is obviously O(B^{3}). We decided to store the values of z^{e} +v^{e} +w^{e} into the hash table. Afterwards, we have to look up the values of ax^{e} − by^{e}.

In this form, the algorithm would lead to a program in which almost the entire running-time is consumed by the writing part. Observe, however, the following particularity of our method. When we search on up to O(B) threefolds, differing only by the values of a and b, simultaneously, then the running-time is still O(B^{3}).

We worked with B = 5 000 for the cubics and B = 10 000 for the quartics. In either case, we dealt with all threefolds arising for a, b = 1, …, 100, simultaneously. For the quartics, the running-time was around four days of CPU time.

This is approximately only three times longer than searching on a single threefold had lasted. For the cubics, a program with integrated line detection took us approximately ten days.

The result for the cubics. We counted all ℚ-rational points of height less than 5 000 on the threefolds V_{a,b}^{3} where a, b = 1, …, 100 and b ≤ a. Note that V_{a,b}^{3} ≅ V_{b,a}^{3}. Points lying on one of the elliptic cones or on a sporadic ℚ-rational line in V_{a,b} were excluded from the count. The smallest number of points found is 3 930 278 for (a, b) = (98, 95). The largest numbers of points are 332 137 752 for (a, b) = (7, 1) and 355 689 300 in the case that a = 1 and b = 1.

On the other hand, for each threefold V_{a,b}^{3} whereas a, b = 1, …, 100 and b + 3 ≤ a, we calculated the expected number of points and the quotients

# { points of height < B found } / # { points of height < B expected }.

Let us visualize the quotients by two histograms.

Fig. 1. Distribution of the quotients for B = 1 000 and B = 5 000.

The statistical parameters are listed in the table below.

Table 2. Parameters of the distribution in the cubic case

B = 1 000 B = 2 000 B = 5 000

mean value  0.981 79 0.988 54 0.993 83 standard deviation 0.012 74 0.008 23 0.004 55

The results for the quartics. We counted all ℚ-rational points of height less than 10 000 on the threefolds V_{a,b}^{4} where a, b = 1, …, 100. It turns out that on 5 015 of these varieties, there are no ℚ-rational points occurring at all as the equation is unsolvable in ℚ_{p} for some small p. In this situation, Manin’s conjecture is true, trivially.

Further, there is the case (a, b) = (58, 87) in which the smallest 96 solutions are the images of (6 465 : 637 : 4 321 : 6 989 : 17 719) under the obvious operation of the group (ℤ/2ℤ)^{4} ⋊ S_{3}. Here, τ_{H}(V_{58,87}^{4}) ≈ 0.002 722.

For the remaining varieties, the points lying on a known ℚ-rational conic in V_{a,b} were excluded from the count. Table 3 shows the quartics sorted by the numbers of points remaining.

Table 3. Numbers of points of height < 10 000 on the quartics.

a b #points # not on conic # expected
29 29  2  2  13.5
58 58  2  2  38.8
51 71  96  96  319.8
87 87  98  98  35.7
⋮ ⋮  ⋮  ⋮  ⋮
34 1 995 808  569 088  567 300
17 64 581 640  581 640  564 300
1 14 682 830  598 038  648 300
3 1 1 262 048  739 008  752 600

We see that the variation of the quotients is higher than in the cubic case.

Fig. 2. Distribution of the quotients for B = 1 000 and B = 10 000.

The statistical parameters are listed in the table below.

Table 4. Parameters of the distribution in the quartic case

B = 1 000 B = 10 000
mean value 0.9853 0.9957
standard deviation 0.3159  0.1130

Interpretation of the result. The results suggest that Manin’s conjecture should be true for the two families of threefolds considered. In the cubic case, the standard deviation is by far smaller than in the case of the quartics. This, however, is not very surprising as on a cubic there tend to be much more rational points than on a quartic. This makes the sample more reliable.

Remark 29. The data we collected might be used to test the sharpening of the asymptotic formula (1) suggested by Sir P. Swinnerton-Dyer [S-D].

Question 30. Our calculations seem to indicate that the number of rational points often approaches its expected value from below. Is that more than an accidental effect?

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