Kummer surfaces and the computation of the Picard group

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

Abstract

We test R. van Luijk’s method for computing the Picard group of a K3 surface. The examples considered are the resolutions of Kummer quartics in P^{3}. Using the theory of abelian varieties, in this case, the Picard group may be computed directly. Our experiments show that the upper bounds provided by R. van Luijk’s method are sharp when sufficiently many primes are used. In fact, there are a lot of primes that yield a value close to the exact one. However, for many but not all Kummer surfaces V of Picard rank 18, we have rk Pic(V_{\overline{𝔽}_{p}}) ≥ 20 for a set of primes of density ≥ \frac{1}{2}.

1 Introduction

1.1. –––– For a general K3 surface V, the methods to compute the geometric Picard group are limited up to now. As shown, for example in [vL2], [Kl], [EJ1], [EJ3], or [EJ5], it is possible to construct K3 surfaces of rank two or four with a prescribed Picard group. But when a K3 surface is given, say, by an equation with rational coefficients, it is not entirely clear whether its geometric Picard rank may be determined using the methods presently known.

1.2. –––– To be concrete, one may always establish a lower bound by specifying divisors explicitly and verifying that their intersection matrix is nondegenerate.

∗Mathematisches Institut, Universität Bayreuth, Univ’straße 30, D-95440 Bayreuth, Germany,

Stephan.Elsenhans@uni-bayreuth.de, Website: http://www.staff.uni-bayreuth.de/∼btm216

‡Département Mathematik, Universität Siegen, Walter-Flex-Str. 3, D-57068 Siegen, Germany,

~jjahnel@mathematik.uni-siegen.de, Website: http://wwwuser.gwdg.de/~jjahnel

∗The first author was supported in part by the Deutsche Forschungsgemeinschaft (DFG) through a funded research project.

∗^{‡}The computer part of this work was executed on the servers of the chair for Computer Algebra at the University of Bayreuth. Both authors are grateful to Prof. M. Stoll for the permission to use these machines as well as to the system administrators for their support.

On the other hand, for upper bounds, the method of R. van Luijk is available, which is based on reduction modulo p. It is not at all clear whether the upper bounds provided by van Luijk’s method are always sharp.

1.3. Remark. –––– Conjecturally, the Picard rank of a K3 surface over \overline{𝔽}_{p} is always even. In particular, if, for V a K3 surface over ℚ, rk Pic(V\overline{ℚ} ) is odd then there is no prime p of good reduction such that rk Pic(V_{\overline{𝔽}_{p}}) = rk Pic(V\overline{ℚ} ). Even more, the rank over \overline{ℚ} being even or odd, there is no obvious reason why there should exist a prime number p such that rk Pic(V_{\overline{𝔽}_{p}}) is at least close to rk Pic(V\overline{ℚ} ).

1.4. Definition. –––– Let V be a K3 surface over ℚ and p be a prime of good reduction. Then, we will call p good if the geometric Picard rank of the reduction modulo p does not exceed the Picard rank over \overline{ℚ} by more than one.

1.5. –––– In this article, we will report on our experiments concerning van Luijk’s method on a sample of Kummer surfaces. Kummer surfaces are particular K3 surfaces allowing a two-to-one covering by an abelian surface. The geometric Picard group of a Kummer surface is closely related to the Néron-Severi group of the abelian surface. In practice, it may be computed this way.

Nevertheless, for testing van Luijk’s method, Kummer surfaces have big advantages. Knowing the Picard ranks anyway, the usual question whether the lower bound or the upper bound needs to be improved, does not appear. Further, using the particularities of a Kummer surface, one may massively optimize the point counting step. In fact, it is very well possible to compute rk Pic(V_{\overline{𝔽}_{p}}) for primes p up to 10 000.

1.6. –––– Our sample consists of the resolutions of 9452 Kummer quartics with small coefficients. For each of these surfaces, we computed the upper bounds that were found using the primes p ≤ 997. It turned out that good primes existed in every example. The upper bounds found turned out to be equal to the geometric Picard ranks in all cases.

1.7. Question. –––– Do there exist good primes for every K3 surface over ℚ?

1.8. The method of van Luijk in detail. –––– The geometric Picard group of a K3 surface over ℚ is isomorphic to ℤ^{n} where n may range from 1 to 20. An upper bound for the geometric Picard rank may be computed as follows. One has the inequality

rk Pic(V\overline{ℚ} ) ≤ rk Pic(V_{\overline{𝔽} _{p}})

that is true for every smooth variety V over ℚ and every prime p of good reduction. This is worked out in detail in [vL1, Remark 2.6.3], the main input being [Fu, Example 20.3.6].

Further, for a K3 surface 𝒱 over the finite field 𝔽_{p}, one has the first Chern class homomorphism

c_{1} : Pic(𝒱_{\overline{𝔽}_{p}}) ⟶ H_{ét}^{2}(𝒱_{\overline{𝔽}_{p}}, ℚ_{l}(1))

into l-adic cohomology. There is a natural operation of Frobenius on H_{ét}^{2}(𝒱_{\overline{𝔽}_{p}}, ℚ_{l}(1)). All eigenvalues are of absolute value 1. The Frobenius operation on the Picard group is compatible with the operation on cohomology.

Every divisor is defined over a finite extension of the ground field. Consequently, on the subspace Pic(𝒱_{\overline{𝔽}_{p}})⊗_{ℤ} \overline{ℚ} _{l} ↪ H_{ét}^{2}(𝒱_{\overline{𝔽}_{p}}, ℚ_{l}(1)) , all eigenvalues are roots of unity. These correspond to eigenvalues of the Frobenius on H_{ét}^{2}(𝒱_{\overline{𝔽}_{p}}, ℚ_{l}) being of the form pζ for ζ a root of unity. One may therefore estimate the rank of the Picard group Pic(𝒱_{\overline{𝔽}_{p}}) from above by counting how many eigenvalues are of this particular form.

Doing this for one prime, one obtains an upper bound for rk Pic(V_{\overline{𝔽}_{p}}) that is always even. The Tate conjecture asserts that this bound is actually sharp. For this reason, one tries to combine information from two primes. The assumption that the surface would have Picard rank 2r over \overline{ℚ} and \overline{𝔽}_{p} implied that the discriminants of both Picard groups, Pic(V\overline{ℚ} ) and Pic(V_{\overline{𝔽}_{p}}), were in the same square class. Note here that reduction modulo p respects the intersection product. When combining information from two primes, it may happen that one finds the rank bound 2r twice, but the square classes of the discriminants are not the same. Then, these data are incompatible with Picard rank 2r over \overline{ℚ} . There is a rank bound of (2r − 1).

1.9. Remark. –––– There are refinements of the method of van Luijk described in [EJ3] and [EJ5]. We will not test these refinements here.

1.10. Example. –––– Let V be a K3 surface of geometric Picard rank 1. We denote by

V^{n} := _{i=1}^{n} V

the n-fold cartesian product. Then, the Picard rank of V^{n} is equal to n.

Indeed, as we have H^{1}(V (ℂ), ℤ) = 0, the Künneth formula shows that H^{2}(V (ℂ)^{n}, ℤ) ≅ H^{2}(V (ℂ), ℤ)^{n}. There is an analogous isomorphism for cohomology with complex coefficients, which is compatible with Hodge structures. I.e., H^{1,1}(V (ℂ)^{n}) ≅ H^{1,1}(V (ℂ))^{n}. The assertion now follows from the Lefschetz theorem on (1, 1)-classes [GH, p. 163].

Assuming the Tate conjecture, one sees that the Picard rank of the reduction of V^{n} at an arbitrary prime is at least 2n. This shows that there is no good prime. Not knowing the decomposition of V^{n} into a direct product, we could not determine its Picard rank.

1.11. Convention. –––– Let V be a projective variety over a field k. In this article, unless stated otherwise, the Picard rank of V shall always mean the geometric Picard rank, i.e., the rank of Pic(V\overline{k}).

The analytic discriminant – The Artin-Tate formula. For the final step in 1.8, one needs to know the discriminant of the Picard lattice. One possibility to compute this is to use the Artin-Tate formula.

1.12. Conjecture (Artin-Tate). —– Let V be a K3 surface over a finite field 𝔽_{q}. Denote by ρ the rank and by Δ the discriminant of the Picard group of V, defined over 𝔽_{q}. Then,

|Δ| = \frac{lim_{T →q} \frac{Φ(T)}{(T −q)^{ρ}}}{q^{21−ρ}#Br(V)} .

Here, Φ denotes the characteristic polynomial of Frob on H_{ét}^{2}(V_{\overline{𝔽}_{q}}, ℚ_{l}) . Finally, Br(V) is the Brauer group of V.

1.13. Remarks. –––– i) The Artin-Tate conjecture was first formulated, more generally than just for K3 surfaces, as Conjecture (C) in [Ta, p. 426]. ii) Conjecture 1.12 is proven for most K3 surfaces. Most notably, the Tate conjecture implies the Artin-Tate conjecture [Mi1, Theorem 6.1]. In these cases, #Br(V) is a perfect square. On its part, the Tate conjecture is proven for K3 surfaces under various additional assumptions. For example, it is true for elliptic K3 surfaces [ASD]. For ordinary K3 surfaces, it is known, too [NO], but we will not need this fact. iii) The Artin-Tate formula allows to compute the square class of the discriminant of the Picard group over a finite field. No knowledge of explicit generators is necessary.

2 Singular quartics

Singular quartic surfaces were extensively studied by the geometers of the 19th century, particularly by A. Cayley and E. E. Kummer. For example, the concept of a trope is due to this period [Je].

2.1. Definition. –––– Let Q ⊂ P^{3} be any quartic surface. Then, by a trope on Q, we mean a plane E such that Q ∩ E is a double conic. This is equivalent to the condition that the equation defining Q becomes a perfect square on E.

2.2. Remark. –––– A trope yields a singular point on the surface Q^{∨} ⊂ (P^{3})^{∨} dual to Q.

2.3. Lemma (Kummer). —– A quartic surface without singular curves may have at most 16 singular points. □

A classical family. A classification of the singular quartic surfaces with at least eight singularities of type A_{1} was given by K. Rohn [Ro], cf. [Je, Chapter I]. In this article, we will deal with one of the most important classical families.

2.4. Lemma (Kummer). —– A three-dimensional family of quartics in P^{3} such that the generic member has exactly 16 singularities of type A_{1} and no others is given by the equation

16kxyzw − ϕ^{2} = 0 .

Here,

k := a^{2} + b^{2} + c^{2} − 1 − 2abc ,
ϕ := x^{2} + y^{2} + z^{2} + w^{2} + 2a(yz + xw) + 2b(xz + yw) + 2c(xy + zw)

for parameters a, b, and c.

2.5. Remarks. –––– i) E. E. Kummer introduces this family in section 10 of his report [Ku]. ii) We will write Q_{[a,b,c]} for the quartic corresponding to the triple [a, b, c]. Up to isomorphism, this surface is independent of the order of a, b, c. Further, there is the isomorphism Q_{[a,b,c]} ⟶^{≅} Q_{[−a,−b,c]} given by (x : y : z : w) ↦ ((−x) : (−y) : z : w). iii) When one of the coefficients is equal to ±1, Q_{[a,b,c]} contains a singular line. For example, the surfaces for a = ±1 contain the singular line, given by x+aw = y+az = 0. iv) On the generic fiber, there are twelve obvious singularities defined over quadratic extensions of ℚ(a, b, c). These are given by x = y = 0, z^{2} + w^{2} + 2czw = 0 and the analogous conditions with the roles of the variables interchanged. Further, there are four singular points forming a Galois orbit.

v) On a Kummer quartic, there are 16 tropes. Four of them are obvious. They are explicitly given by the coordinate planes. Each trope passes through six of the 16 singular points and each singular point is contained in six tropes [Hu, Chapter I]. On an obvious trope, the conic is of discriminant 2abc + 1 − a^{2} − b^{2} − c^{2} = −k. Thus, these conics are nondegenerate except for the case that Q is non-reduced itself. vi) For a generic Kummer quartic, every singular point on Q^{∨} comes from a trope.

3 The desingularization

3.1. Lemma. –––– Let π: Q̃ → Q be the desingularization of a normal quartic surface Q such that all singularities are of type A_{1}. Then, Q̃ is a K3 surface. Proof. On the smooth part of Q, the adjunction formula [GH, Sec. 1.1, Example 3] may be applied as usual. As, for the canonical sheaf, one has K_{P^{3}} = 𝒪(−4), this shows that the invertible sheaf Ω_{Q^{reg}}^{2} is trivial. Consequently, KQ̃ is given by a linear combination of the exceptional curves.

However, for an exceptional curve E, we have E^{2} = −2. Hence, according to the adjunction formula, KQ̃E = 0, which shows that KQ̃ is trivial. The classification of algebraic surfaces [Be] assures that Q̃ is either a K3 surface or an abelian surface.

Further, a standard application of the theorem on formal functions implies R^{1}π_{∗}𝒪Q̃ = 0. Hence, χ_{alg}(Q̃) = χ_{alg}(Q) = 2. This shows that Q̃ is actually a K3 surface. □

3.2. Remarks. –––– i) For the assertion of the lemma, it is actually sufficient to assume that the singularities of Q are of types A, D, or E [Li]. ii) In general, the desingularization of a normal quartic surface is a K3 surface, a rational surface, a ruled surface over an elliptic curve, or a ruled surface over a curve of genus three [IN]. The last possibility is caused by a quadruple point. The existence of a triple point implies that surface is rational. It is, however, also possible that there is a double point, not of type A, D, or E. Then, Q̃ is rational or a ruled surface over an elliptic curve.

3.3. Lemma. –––– Let π: Q̃ → Q be the desingularization of a proper surface Q having only A_{1}-singularities.

a) Then, the exceptional curves define a nondegenerate orthogonal system in Pic(Q̃).

b) In particular, the Picard rank of Q̃ is strictly bigger than the number of singularities of Q.

Proof. a) The exceptional curves have self-intersection number (−2) and do not meet each other.

b) For H the hyperplane section, π^{∗}𝒪_{Q}(H) is orthogonal to the exceptional curves.

□

4 Abelian surfaces and Kummer quartics

Let A be an abelian surface. Denote by ϕ: A → A the involution given by p ↦ (−p). Then, the quotient A/∼ for ∼ := {(p, ϕ(p)) | p ∈ A} has precisely 16 singular points. We call such a quotient an abstract Kummer surface.

4.1. Fact. –––– Let A be an abelian surface over a field k of characteristic zero and V be the resolution of the corresponding Kummer surface. Then, rk Pic(V\overline{k}) = rk NS(A\overline{k}) + 16. Proof. A standard argument [EGA4, Proposition (8.9.1)] allows us to assume that k is finitely generated over ℚ. Then, in particular, k allows an embedding into ℂ. The canonical injection ι: H^{2}(V (ℂ), ℤ) → H^{2}(A(ℂ), ℤ) yields a bijection of H^{2}(V (ℂ), ℤ) with ⟨E_{1}, …, E_{16}⟩^{⊥}. As ι respects the (1, 1)-classes, the assertion follows. Observe that base change to ℂ does not change the Picard and Néron-Severi ranks. □

4.2. Lemma (Nikulin). —– Let Q be a quartic surface over an algebraically closed field k of characteristic zero with precisely 16 singular points of type A_{1} and no others. Then, Q is isomorphic to an abstract Kummer surface. Proof. This result is shown in [Ni]. We include a sketch of the proof for the convenience of the reader.

Again, we may assume that k is a subfield of ℂ. As shown in Lemma 3.1, the desingularization Q̃ is a K3 surface. We have to prove that Q̃ admits a double cover ramified exactly at the 16 exceptional curves E_{1}, …, E_{16}. This is equivalent to the assertion that 𝒪(E_{1} + ⋯ + E_{16}) ∈ Pic(Q̃) is divisible by two.

Consider, more generally, the set C of all ℚ-divisors D = c_{1}E_{1} +⋯+c_{16}E_{16} that define an element of Pic(Q̃). Clearly, c_{1}, …, c_{16} ∈ \frac{1}{2}ℤ as, otherwise, the intersection numbers with E_{1}, …, E_{16} would not be integers. Thus, C defines a sub-vector space \overline{C} of

⨁_{i=1}^{16} \frac{1}{2}ℤE_{i} / ⨁_{i=1}^{16} ℤE_{i} ≅ 𝔽_{2}^{16} .

We claim that dim \overline{C} ≥ 5. Indeed, otherwise, the lattice C ⊂ Pic(Q̃) would have a basis containing twelve of the standard elements E_{1}, …, E_{16}. As the quotients H^{2}(Q̃(ℂ), ℤ)/ Pic(Q̃) and Pic(Q̃)/C have no torsion, H^{2}(Q̃(ℂ), ℤ) still has a basis containing twelve of the E_{i}. But then, the 22 × 22-matrix of the cup product form contains a symmetric 12 × 12-block consisting entirely of even entries. This ensures that the determinant is even and, hence, is a contradiction to the unimodularity of H^{2}(Q̃(ℂ), ℤ) .

Further, every vector in \overline{C} is a sum of precisely eight or 16 standard basis vectors. In fact, if it is a sum of l basis vectors then it defines a double cover P′ of Q̃ ramified at exactly l of the 16 exceptional curves E_{1}, …, E_{16}. Its minimal model P, obtained by blowing down the l exceptional curves, clearly has trivial canonical class. It is therefore either an abelian surface, χ_{top}(P) = 0, or a K3 surface, χ_{top}(P) = 24. But a direct calculation shows χ_{top}(P) = 48 − 3l.

Finally, it is a well-known result from coding theory [HP, Theorem 2.7.4] that there is no five-dimensional subspace of 𝔽_{2}^{16} such that every non-zero vector has exactly eight components equal to 1. Indeed, adding the vector (1, 1, …, 1) would yield a code contradicting the optimality of the [16, 5, 8]-Hadamard code. □

4.3. –––– Consider the particular case that A = J(C) is the Jacobian of a curve C of genus two. Then, a projective model of the corresponding Kummer surface may be obtained as follows.

For r a Weierstraß point of C, put θ := {[x] − [r] | x ∈ C} ⊂ J(C). This is an ample divisor on the Jacobian J(C) such that θ^{2} = 2. The Riemann-Roch theorem shows dim Γ(J(C), 2θ) = 4. Hence, 2θ defines a morphism ι: J(C) → P^{3} of degree eight. Actually, ι is a two-to-one map inducing an embedding of J(C)/∼ [Be, Chapter VIII, Exercise 4]. The image of ι is a quartic surface.

4.4. –––– It is a classical result that every Kummer quartic Q may be constructed from a genus-2 curve C in this way. We may therefore ask for an explicit construction of such a curve from a given Kummer quartic. This may indeed be done as follows.

Construction. i) There are 16 tropes. We choose one of them, which we call D. ii) The intersection Q ∩ D is a double conic. Let I be the underlying reduced curve. Six of the singular points on Q are contained in I. iii) Take the double cover C of I ramified at these six points. This is a genus-2 curve.

4.5. Remarks. –––– i) This construction clearly yields a genus-2 curve C on the abelian surface A. The Albanese property of the Jacobian guarantees that A is at least isogenous to J(C). They are actually isomorphic to each other. ii) If Q is defined over a base field k and D over an extension k′ ⊇ k then C is defined over k′. Indeed, the six ramification points form a Gal(\overline{k′}/k′)-invariant set. We will apply the construction only to the obvious tropes of the Kummer family, which are defined over the base field.

4.6. Fact. –––– Let V′ be an abstract Kummer surface over a finite field 𝔽_{q} and V its resolution of singularities. Then, the Gal(\overline{𝔽}_{q}/𝔽_{q})-module H_{ét}^{2}(V_{\overline{𝔽}_{q}}, ℚ_{l}) is reducible. A direct summand is isomorphic to H_{ét}^{2}(A_{\overline{𝔽}_{q}}, ℚ_{l}) for A the abelian surface covering V′. Its complement is described by the Galois operation on the 16 singular points. □

4.7. Remark (Frobenius eigenvalues for Kummer surfaces). —– In order to determine the eigenvalues of the Frobenius on H_{ét}^{2}(V_{\overline{𝔽}_{q}}, ℚ_{l}) , the usual method is to count the points on V defined over 𝔽_{q} and some of its extensions and to apply the Lefschetz trace formula [Mi2, Chapter VI, Theorem 12.3].

For Kummer surfaces, there is, however, a far better method. In fact, 16 eigenvalues are determined by the operation of Frobenius on the 16 singular points. Further, for A isogenous to the Jacobian J(C), we have H_{ét}^{2}(A_{\overline{𝔽}_{q}}, ℚ_{l}) ≅ Λ^{2}H_{ét}^{1}(C_{\overline{𝔽}_{q}}, ℚ_{l}). Thus, in order to determine the remaining six eigenvalues, it suffices to count the points on C. This is faster as the problem is reduced to dimension one.

4.8. Proposition. –––– Let A be an abelian surface over an algebraically closed field. Suppose that End(A) is an order of a real quadratic number field. Then, rk NS(A) = 2.

Proof. According to [Mu, sec. 21, Appl. III], one has NS(A)⊗ℚ ≅ (End(A)⊗ℚ)^{†} where † denotes the Rosati involution. As that is positive [Mu, sec. 21, Theorem 1], it cannot be the conjugation on a real quadratic number field. Hence, † = id which implies the assertion. □

4.9. Remark. –––– If the real multiplication is by an order in ℚ(\sqrt{d}) then the discriminant of the Néron-Severi lattice is of square class (−d). Indeed, [Mu, sec. 21, Thm. 1] shows that, for H ample, Φ^{∗}(H)·H is a scalar multiple of Tr(Φ^{2}). Working with Φ = 1 and Φ = 1 + \sqrt{d}, we find the intersection matrix \binom{2 2(1+d)}{2(1+d) 2(1−d)^{2}} of determinant (−16d).

4.10. Proposition. –––– Let A be an abelian surface over ℚ. Suppose that A has an endomorphism N defined only over a quadratic extension F = ℚ(\sqrt{D}). Then, for every prime p, inert in F and of good reduction, the following is true. If λ is an eigenvalue of Frob_{p} on H_{ét}^{1}(A_{\overline{𝔽}_{p}}, ℚ_{l}) then (−λ) is an eigenvalue, too.

Proof. N induces an endomorphism of A_{𝔽_{p}} , which we denote by \underline{N}. Clearly, \underline{N} is defined over 𝔽_{p^{2}} but not over 𝔽_{p}. This means, in the endomorphism ring R_{p} of A_{\overline{𝔽}_{p}}, we have Frob_{p^{2}}^{−1} \underline{N} Frob_{p^{2}} = \underline{N} but the analogous statement is not true for Frob_{p}.

Thus, under the operation of Frob_{p^{2}} on R_{p} by conjugation, \underline{N} lies in the (+1)-eigenspace. For the corresponding operation of Frob_{p}, this space decomposes into a (+1)-eigenspace and a (−1)-eigenspace. The latter is nonzero as \underline{N} is not fixed under conjugation by Frob_{p}. Hence, there is some J ∈ R_{p} anticommuting with Frob_{p}. This implies the assertion. □

4.11. Corollary. –––– Let V be a Kummer surface over ℚ covered by the abelian surface A. Suppose that A has an endomorphism N defined only over a quadratic extension F = ℚ(\sqrt{D}). Then, for every prime p, inert in F and of good reduction, rk Pic(V_{\overline{𝔽}_{p}}) ≥ 20.

Proof. Recall that Kummer surfaces are elliptic [Be, Chapter IX, Exercise 6]. Hence, the Tate conjecture is true for V_{\overline{𝔽}_{p}}.

Under the assumptions made, it is possible that the Frobenius eigenvalues on H_{ét}^{1}(A_{\overline{𝔽}_{p}}, ℚ_{l}) are ±\sqrt{p} and ±i\sqrt{p}. This yields Picard rank 22 over \overline{𝔽}_{p}. Except for this case, the Frobenius eigenvalues must be ±λ and ±\overline{λ} for a suitable λ ∈ ℂ. On H_{ét}^{2}(A_{\overline{𝔽}_{p}}, ℚ_{l}) , this leads to the eigenvalues p and (−p), both with multiplicity two, as well as (−λ^{2}) and (−\overline{λ}^{2}). The Picard rank is at least 20. □

5 The tetrahedroid

The tetrahedroid is another family of quartic surfaces studied in the 19th century. It was first considered by A. Cayley in [Ca1].

5.1. Lemma. –––– A family of quartics in P^{3} such that every member has exactly 16 A_{1}-singularities and no others is given by the equation

det ( \begin{matrix}0 & x_{0}^{2} & x_{1}^{2} & x_{2}^{2} & x_{3}^{2} \\ x_{0}^{2} & 0 & a_{01}^{2} & a_{02}^{2} & a_{03}^{2} \\ x_{1}^{2} & a_{01}^{2} & 0 & a_{12}^{2} & a_{13}^{2} \\ x_{2}^{2} & a_{02}^{2} & a_{12}^{2} & 0 & a_{23}^{2} \\ x_{3}^{2} & a_{03}^{2} & a_{13}^{2} & a_{23}^{2} & 0 \end{matrix} ) = 0 .

for parameters a_{01}, a_{02}, a_{03}, a_{12}, a_{13}, a_{23} ≠ 0.

5.2. Remarks. –––– i) In this form, the equation of the tetrahedroid appears in [Ca2, p. 286]. ii) We will write T_{[a_{01},a_{02},a_{03},a_{12},a_{13},a_{23}]} for the quartic corresponding to the particular coefficient vector (a_{01}, a_{02}, a_{03}, a_{12}, a_{13}, a_{23}). iii) Let the group 𝔾_{m}^{4} operate on the parameters according to the rule

(i, j, k, l)[a_{01}, a_{02}, a_{03}, a_{12}, a_{13}, a_{23}] := [ija_{01}, ika_{02}, ila_{03}, jka_{12}, jla_{13}, kla_{23}] .

Then, the quartics defined by a whole orbit are all isomorphic to each other, as one can see by left and right multiplying the matrix above by diag(1, i^{2}, j^{2}, k^{2}, l^{2}). Consequently, the tetrahedroid defines only a two-dimensional family in the moduli stack of all K3 surfaces. Actually, it is a subfamily of the Kummer quartics [Ca2], [Hu, §56].

5.3. Remarks. –––– a) The sixteen singularities are (0 : ±a_{01} : ±a_{02} : ±a_{03}), (±a_{01} : 0 : ±a_{12} : ±a_{13}), (±a_{02} : ±a_{12} : 0 : ±a_{23}), and (±a_{03} : ±a_{13} : ±a_{23} : 0).

b) The four planes, given by ±a_{23}x_{1} ± a_{13}x_{2} ± a_{12}x_{3} = 0, clearly contain six singular points each. For example, ((±a_{01}) : 0 : a_{12} : (−a_{13})), ((±a_{02}) : a_{12} : 0 : (−a_{23})), and ((±a_{03}) : a_{13} : (−a_{23}) : 0) satisfy the equation a_{23}x_{1} + a_{13}x_{2} + a_{12}x_{3} = 0. There are twelve more tropes obtained in an analogous manner by distinguishing the first, second, or third coordinate instead of the zeroth one.

c) Besides the tropes, there are four other particular planes related to this family of quartics. Actually, the coordinate planes contain exactly four singularities each. As these form a tetrahedron, they gave this family its name. There are no planes containing exactly four singular points on a general Kummer quartic.

5.4. Proposition. –––– Let E_{1} and E_{2} be two elliptic curves. Fix an isomorphism of groups ϕ: E_{1}[2] → E_{2}[2] and let

A := (E_{1} × E_{2})/⟨(x, ϕ(x)) | x ∈ E_{1}[2]⟩

be the corresponding abelian surface, covered four-to-one by E_{1} × E_{2}.

Then, the Kummer surface corresponding to A is given by a tetrahedroid.

Proof. We describe the elliptic curves as intersections of two quadrics in P^{3},

E_{1} : x_{1}^{2} = x_{0}^{2} − x_{2}^{2}, x_{3}^{2} = x_{0}^{2} − κ_{1}x_{2}^{2} ,
E_{2} : y_{1}^{2} = y_{0}^{2} − y_{2}^{2}, y_{3}^{2} = y_{0}^{2} − κ_{2}y_{2}^{2} .

We have j(E_{1}) = 256(κ_{1}^{2} − κ_{1} + 1)^{3}/κ_{1}^{2}(κ_{1} − 1)^{2} and the analogous formula for E_{2}. Thus, these equations define general families of elliptic curves. The morphism

E_{1} × E_{2} ⟶ P^{3}, ((x_{0} : x_{1} : x_{2} : x_{3}), (y_{0} : y_{1} : y_{2} : y_{3})) ↦ (x_{2}y_{3} : x_{1}y_{1} : x_{3}y_{2} : x_{0}y_{0})

is generically eight-to-one onto the tetrahedroid T_{[\sqrt{κ_{2}−1},1,\sqrt{κ_{2}},\sqrt{κ_{1}−1},i,\sqrt{κ_{1}}]}. It factors through A and even through the Kummer surface associated with it. □

5.5. Remarks. –––– i) It is not hard to see that every tetrahedroid is obtained from two elliptic curves in this way. ii) From the point of view of the present article, Proposition 5.4 is a purely algebraic statement. We even checked the assertions on the morphism using magma. It was, however, originally discovered by H. Weber [We, p. 353] in the guise of a parametrization of the tetrahedroid by elliptic functions. Cf. [Hu, Chapter XVIII].

5.6. Proposition (Kummer quartics with two coefficients equal). —– Let V := V_{[a,a,c]} be the Kummer quartic for the coefficients [a, a, c]. Then, V is linearly isomorphic to the tetrahedroid

T_{[\sqrt{c+1},\sqrt{c−1},X\sqrt{c−1},X\sqrt{c−1},\sqrt{c−1},2X(X+a)\sqrt{c−1}]} .

Here, X is a solution of the equation X^{2} + 2aX + 1 = 0. Proof. The isomorphism from the tetrahedroid to V is given explicitly by the linear map P^{3} → P^{3},

(t_{1} : t_{2} : t_{3} : t_{4}) ↦ ((−t_{2} − \frac{t_{3}−Xt_{4}}{1−X^{2}} ) : (−t_{1} + \frac{t_{4}−Xt_{3}}{1−X^{2}} ) : (t_{2} − \frac{t_{3}−Xt_{4}}{1−X^{2}} ) : (t_{1} + \frac{t_{4}−Xt_{3}}{1−X^{2}} )) . □

5.7. Remark. –––– One might ask to determine the two elliptic curves E_{1}, E_{2} which correspond to V_{[a,a,c]}, i.e., those satisfying (E_{1} × E_{2})/⟨(x, ϕ(x))⟩ ≅ V_{[a,a,c]}. This leads to a simple calculation but the explicit formulas become rather lengthy. Interestingly, the two j-invariants are defined in the quadratic field extension ℚ(a, c)(\sqrt{4a^{2} − 2c − 2}) and conjugate to each other. Their trace is

(1024a^{10}c^{2} + 2048a^{10}c + 1024a^{10} − 512a^{8}c^{3} − 4608a^{8}c^{2} − 7680a^{8}c − 3584a^{8}
+ 32a^{6}c^{4} + 1568a^{6}c^{3} + 7776a^{6}c^{2} + 10976a^{6}c + 4736a^{6} − 72a^{4}c^{4} − 1680a^{4}c^{3}
− 6016a^{4}c^{2} − 7280a^{4}c − 2872a^{4} + 54a^{2}c^{4} + 702a^{2}c^{3} + 2010a^{2}c^{2} + 2106a^{2}c

+ 760a^{2} − \frac{27}{2}c^{4} − 81c^{3} − 180c^{2} − 175c − \frac{125}{2})/(a − 1)(a + 1)(b − 1)^{2}(b + 1)^{2}, while their norm turns out to be

\frac{(16a^{4}b^{2} + 48a^{4} − 24a^{2}b^{2} − 32a^{2}b − 72a^{2} + 9b^{2} + 30b + 25)^{3}}{16(a − 1)^{2}(a + 1)^{2}(b − 1)^{4}(b + 1)^{2}} .

5.8. Remarks. –––– i) The case of three equal coefficients is even more special. In some sense, the quartics V_{[a,a,a]} are tetrahedroids in three distinct ways. It turns out that, in this situation, the resulting elliptic curves are related by an isogeny of order 3. In fact, it is easy to check that the resulting pair of j-invariants is a zero of the third classical modular polynomial. Consequently, the Picard rank of a Kummer surface with three equal coefficients is at least 19. The additional divisor leading to a Picard rank higher than 18 is the image of the graph of the 3-isogeny under the two-to-one covering described in Proposition 5.4. ii) There is another case, which is particular. Consider the quartics V_{[0,0,c]}. Then, the j-invariants of the corresponding elliptic curves are defined in ℚ(c) and equal to each other. We have j(E_{1}) = j(E_{2}) = \frac{1728c^{3}+8640c^{2}+14400c+8000}{c^{3}−c^{2}−c+1}. Consequently, the Picard rank of a Kummer surface with two coefficients zero is at least 19. The additional divisor leading to a Picard rank higher than 18 is the image of the diagonal.

6 Experiments – The Picard ranks over \overline{�}

A sample of Kummer surfaces. We inspected the Kummer surfaces Q_{[a,b,c]} given by the Kummer coefficients a, b, c = −30, …, 30. Because of symmetry, the considerations were restricted to the case |a| ≤ b ≤ c. Recall that one may always change the signs of two coefficients simultaneously. Hence, b, c ≥ 0 was assumed. The coefficient vectors [3, 3, 17], [2, 2, 7], and [2, 7, 26] as well as those containing ±1 were excluded from the sample as the corresponding surfaces have singularities of types worse than A_{1}.

6.1. –––– For each surface Q in the sample, first, using Construction 4.4, we determined the genus-2 curve C such that V is the Kummer surface corresponding to J(C). Then, for every prime number below 1000, we counted the numbers of points on C over 𝔽_{p} and 𝔽_{p^{2}}. From these data, we computed the characteristic polynomial of the Frobenius on the l-adic cohomology of the resolution V.

From the characteristic polynomial, we read off the rank of Pic(V_{\overline{𝔽}_{p}}) and, using the Artin-Tate formula 1.12, computed the square class of the discriminant. Note that the Artin-Tate formula is applicable, since every Kummer surface is elliptic.

The Picard ranks over \overline{�} . A generic Kummer surface is of geometric Picard rank 17. In the case that two Kummer coefficients are of the same absolute value, Proposition 5.6 shows together with Remark 2.5.ii) that the surface is a tetrahedroid. Then, the Picard rank is at least 18. Thus, we distinguished between these two cases. The possibilities that all three coefficients coincide, at least up to symmetry, or that two coefficients vanish were treated as being somehow exceptional.

Being a bit sloppy at first, in the first case, we tested whether an upper bound of 17 is provable by van Luijk’s method while, in the second case, we awaited an upper bound of 18. The table below shows the distribution of the biggest prime that had to be considered in order to prove the expectation.

prime #cases finished #cases left
7  57  7656
11  287  7369
13  713  6656
17  1229  5427
19  1308  4119
23  1215  2904
29  1004  1900
31  759  1141
37  551  590
41  320  270
43  143  127
47  59  68
53  28  40
59  17  23
61  6  17
67  3  14
73  1  13
83  1  12

prime #cases finished #cases left
5  156  1495
7  66  1429
11  193  1236
13  253  983
17  288  695
19  132  563
23  117  446
29  116  330
31  82  248
37  81  167
41  73  94
43  24  70
47  18  52
53  15  37
59  13  24
61  6  18
67  3  15
71  2  13
73  4  9
79  2  7
101  1  6

Table 1: Distribution of the biggest prime used for rank 17 (left) and rank 18 (right)

The 18 examples left. Let us take a closer look at the Kummer quartics left.

6.2. Examples. –––– Among the Kummer quartics, the coefficients of which had three distinct absolute values, twelve examples remained. For these, only a rank bound of 18 could be established. Using magma, we calculated the corresponding genus-2 curves C_{i} and determined their periods at high precision.

i) Consider the Kummer quartics for the coefficient vectors [2, 3, 13], [−3, 4, 19], [−3, 5, 11], [−2, 7, 23], [−2, 8, 17], [−2, 9, 14], and [0, 4, 7]. In these cases, it turned out that the Jacobians J(C_{i}) are isogenous to products of two elliptic curves. Hence, the geometric Picard ranks are indeed equal to 18. The isogenies are all of degree 16. Their kernels are groups of type ℤ/4ℤ × ℤ/4ℤ. The j-invariants of the elliptic curves are conjugate to each other in quadratic number fields. We summarize them in the table below.

vector  j_{1}, j_{2}
[ 2, 3, 13]  \frac{8000}{21609} [38155±16152\sqrt{2}]
[−3, 4, 19]  \frac{64}{164025} [1082783±399784\sqrt{−2}]
[−3, 5, 11]  \frac{16}{5625} [17903±64596\sqrt{−1}]
[−2, 7, 23] \frac{2000}{10673289} [−614135±4744012\sqrt{−2}]
[−2, 8, 17]  \frac{250}{21609} [−50045±45683\sqrt{−3}]
[−2, 9, 14] \frac{16}{6426225} [327552721±229629540\sqrt{−1}]
[ 0, 4, 7]  \frac{16}{5625} [17903±64596\sqrt{−1}]

Table 2: j-invariants of the corresponding elliptic curves

ii) Consider the Kummer quartics given by the coefficient vectors [2, 7, 17], [2, 9, 26], [2, 17, 26], [3, 9, 19], and [0, 8, 15]. Here, our calculations showed that the corresponding abelian surfaces have real multiplication by orders in ℚ(\sqrt{2}), ℚ(\sqrt{3}), ℚ(\sqrt{5}), ℚ(\sqrt{5}), and ℚ(\sqrt{5}), respectively. This implies that the Picard ranks are equal to 18. The non-trivial endomorphisms are expected to be defined over the quadratic number fields ℚ(\sqrt{30}), ℚ(\sqrt{11}), ℚ(\sqrt{−2}), ℚ(\sqrt{−1}), and ℚ(\sqrt{2}). In fact, the primes leading to Picard rank 18 are all split for the corresponding field. Compare Corollary 4.11.

6.3. –––– Consider the Kummer quartics for the coefficient vectors [5, 5, 17], [2, 2, 17], [−4, 4, 9], [−3, 7, 7], [−2, 11, 11], and [0, 5, 5]. For these, the situation is as follows. One finds rank 20 at several primes. Discriminants of various square classes appear such that a rank bound of 19 is established. As two of the Kummer coefficients are equal, the corresponding abelian surfaces are isogenous to products of two elliptic curves. Specializing the calculation discussed in Remark 5.7, one may determine the corresponding j-invariants. It turns out in every case that the corresponding elliptic curves are isogenous to each other. Thus, we have Picard rank 19. The isogenies are of degrees 5, 5, 4, 4, 4, and 4, respectively.

6.4. Examples. –––– Let us present two of these examples in detail.

i) Let T_{1} be the Kummer quartic for the coefficient vector [5, 5, 17]. We find rank 20 at p = 5, 7, 13, 17, 19, 23, 29 and several other primes. The rank bound 19 is proven as many distinct square classes of discriminants occur. Further, as two of the Kummer coefficients are equal, the corresponding abelian surface is isogenous to a product of two elliptic curves. Specializing the calculation discussed in Remark 5.7, one finds the j-invariants j_{1} = \frac{85184}{3} and j_{2} = \frac{58591911104}{243}. The pair (j_{1}, j_{2}) is a zero of the fifth modular polynomial. Hence, between the two elliptic curves, there is an isogeny of order five. We have Picard rank 19. ii) Let T_{2} be the Kummer quartic for the coefficient vector [2, 2, 17]. For this surface, we find rank 20 at p = 7, 11, 13, 17, 23, 29 and several other primes. Many distinct square classes of discriminants appear. Hence, the rank bound 19 is proven. Here, the two j-invariants are defined in ℚ(\sqrt{−5}). They are the roots of the polynomial X^{2} + \frac{21180800}{243}X + \frac{1693669888000}{729}. Again, the corresponding elliptic curves turn out to be 5-isogenous. This confirms Picard rank 19.

Expected rank 19. In the case of three equal coefficients or two coefficients equal to zero, we know that the Picard rank is at least equal to 19. In 84 of the 88 surfaces, the reductions modulo p provided an upper bound of 19. The biggest prime that had to be used was 37.

The cases [0, 0, 0], [−5, 5, 5], [−2, 2, 2], and [7, 7, 7] remained. Here, the corresponding elliptic curves have complex multiplication. This shows that the corresponding Kummer surfaces indeed have geometric Picard rank 20.

6.5. Example. –––– Consider, for instance, the case [7, 7, 7]. Then, the two j-invariants are the roots of the polynomial X^{2} −37018076625X +153173312762625. The corresponding elliptic curves have complex multiplication by an order in ℚ(\sqrt{−15}).

Testing isomorphy. As a byproduct of the computations, we tried to prove that the surfaces in our sample are pairwise non-isomorphic. For this, it would suffice to show that, for each pair of surfaces, there exists a prime where both have good reduction, but the geometric Picard groups differ in rank or discriminant. Actually, the data for p ≤ 59 contained enough information for this but there were 41 pairs of surfaces that could not be separated.

The point here is that the test actually tries to prove that the corresponding abelian surfaces are non-isogenous. But in these 41 cases, the surfaces are isogenous to each other. To be more precise, we found 17 pairs, four triples, and two quadruples of mutually isogenous abelian surfaces.

6.6. Example. –––– The abelian surfaces corresponding to V_{[2,2,9]} and V_{[3,3,19]} are isogenous. Hence, the test described above has no chance to work.

In fact, V_{[2,2,9]} is covered eight-to-one by E_{1} × E_{2} while V_{[3,3,19]} is covered eightto-one by E_{3} × E_{4} for j(E_{1}), j(E_{2}) the zeroes of X^{2} + \frac{1114112}{25}X + \frac{589752696832}{225} and j(E_{3}), j(E_{4}) the zeroes of X^{2} − \frac{281615072}{2025}X + \frac{15000601854041872}{164025}. It is easy to check that E_{1} and E_{3}, as well as E_{2} and E_{4}, are connected by isogenies of order four. Hence, E_{1} × E_{2} and E_{3} × E_{4} are 16-isogenous.

An isomorphism between the quotients as described in Proposition 5.4 would yield a 16-isogeny

E_{1} × E_{2} ⟶ E_{1} × E_{2}/⟨(x, ϕ(x)) | x ∈ E_{1}[2]⟩ ≅ E_{3} × E_{4}/⟨(x, ϕ′(x)) | x ∈ E_{3}[2]⟩ ⟶ E_{3} × E_{4} ,

too. But, in its kernel, there are the 2-torsion points (x, ϕ(x)) for x ∈ E_{1}[2] that are clearly not in the kernel of the direct product of two 4-isogenies. This shows that V_{[2,2,9]} and V_{[3,3,19]} are not isomorphic, either.

Testing isomorphy II. For each of the 41 pairs, we numerically calculated the periods of the corresponding abelian surfaces. From these, we determined a minimal isogeny. It turned out that the surfaces corresponding to the coefficient vectors [−3, 7, 7] and [0, 5, 5] were actually isomorphic to each other. This was, however, the only such case among the critical pairs.

Summary. We considered the resolutions of 9452 Kummer quartics with exactly 16 singularities of type A_{1}. It turned out that the upper bounds for the Picard ranks provided by the reductions modulo p were sharp in every case. However, at several examples, rather large primes up to p = 101 had to be considered. We had Picard rank 17, 7701 times, Picard rank 18, 1657 times, and Picard rank 19, 90 times. Further, there were four surfaces of Picard rank 20 in the sample.

7 Some more statistics

7.1. Example (All primes less than 10 000 for a typical surface). —– Let us take a closer look at a particular example. We selected the surface with Kummer coefficients [3, 11, 21], but many others would be representative, as well.

There are only five primes p ≤ 10 000 such that the reduction modulo p of V_{[3,11,21]} is not a quartic having 16 singular points of type A_{1}. These are 2, 3, 5, 11, and 17. In the range considered, 1224 primes lead to a reduction of Picard rank 18. Further, there are 69 primes leading to a reduction of rank 20. These seem to be rather equidistributed within the range, the smallest one being 7, the largest one being 9677. Finally, there is the prime 4583 that leads to a reduction of Picard rank 22.

In the cases of reduction to Picard rank 18, we found 586 different square classes for the discriminant. As for many of the surfaces in our sample, the most frequent square class was (−1). In the example selected, it appeared 376 times.

Discriminants – The special case of rank 17. In the special case of a rank-17 surface, we counted how many square classes of discriminants occurred when reducing to surfaces of Picard rank 18 modulo various primes. There are 168 prime numbers in our computational range, e.g. less than 1000. For a fixed surface, between 44 and 89 distinct square classes were found.

Figure 1: Number of distinct square classes of discriminants at primes with reduction to rank 18

In total, we found 541 distinct square classes of discriminants. Some of them occurred only for one surface and one prime. On the other hand, the class of (−1) appeared 134 553 times. The surfaces with Kummer coefficients [−3, 9, 17] and [−3, 10, 29] both had the most repetitions for one square class. This was the class of (−1) occurring 43 times.

The average value for a prime. For simplicity, let us restrict our considerations to surfaces of Picard rank 17. For every prime number p, we counted how many of the surfaces in our sample had good reduction modulo p. We determined the proportion of those having reduction to rank >18. The results are visualized by the graph below. According to this graph, the proportion is close to \frac{C}{\sqrt{p}} for C ≈ 2.

Figure 2: Distribution of the proportion of the surfaces with reduction to rank >18

The average value for a surface. On the other hand, for every surface of Picard rank ≤18 in the sample, we counted how many primes below 1000 lead to a reduction of geometric Picard rank 18 over \overline{𝔽}_{p}. Let us visualize the result in a histogram.

Figure 3: Distribution of the number of the primes with reduction to rank 18

The histogram clearly suggests that there are two kinds of examples. For the first kind, the probability that the reduction has rank 18 is between 1/4 and 1/2. For the second kind, this probability is between 3/4 and 1.

It turns out that most of the examples with two Kummer coefficients equal (up to sign) belong to the first kind. The only examples in the first group not being of this form are given by the coefficient vectors [3, 9, 19], [2, 3, 13], [2, 7, 17], [2, 9, 26], [2, 17, 26], [−3, 4, 19], [−3, 5, 11], [−2, 7, 23], [−2, 8, 17], [−2, 9, 14], [0, 4, 7], and [0, 8, 15]. Further, there are some examples with two coefficients equal belonging to the second group. These are [3, 3, 9], [3, 3, 15], [4, 4, 13], [4, 4, 23], [4, 4, 29], [6, 6, 21], [7, 7, 25], [8, 8, 29], [2, 2, 5], [−6, 6, 27], [−5, 5, 23], [−4, 4, 19], [−3, 3, 15], [−2, 2, 11], and [−2, 2, 25].

An explanation. For the tetrahedroid case, an explanation is given by the following fact.

7.2. Fact. –––– Let V_{[a,a,c]} be a Kummer surface with two coefficients equal. Suppose that 4a^{2} − 2c − 2 is not a perfect square. Then, for every prime p, inert in F = ℚ(\sqrt{4a^{2} − 2c − 2}) and of good reduction, rk Pic(V_{\overline{𝔽}_{p}}) ≥ 20.

Proof. The corresponding abelian surface is isogenous to the product of two elliptic curves. As noticed in Remark 5.7, the j-invariants are two elements conjugate in F. Reducing the surface modulo a prime inert in F leads to two elliptic curves isogenous via the Frobenius endomorphism. This shows that all inert primes yield an upper bound of at least 20 for the geometric Picard rank. □

7.3. Questions. –––– i) For a surface V, put N_{V} (B) := #{p ∈ P_{V} | p ≤ B}, where

P_{V} := #\{ p prime | rk Pic(V_{\overline{𝔽}_{p}}) > 18 or V has bad reduction at p \} .

Is there a monotonically decreasing function h_{V} such that

N_{V} (B) ∼ ∫_{2}^{B} \frac{h_{V} (t)}{log t} dt?

Can h_{V} be given explicitly? ii) Suppose that rk Pic(V\overline{ℚ} ) = 17. Does then h_{V} converge to 0 for t → ∞? The graph in Figure 2 might suggest that h_{V} (t) = \frac{C_{V}}{\sqrt{t}} for a constant C_{V}. Is h_{V} perhaps independent of V? iii) For a fixed Kummer surface of geometric Picard rank 17, are there infinitely many primes with reduction to rank 18? Are there infinitely many primes with reduction to rank >18?

7.4. Remark. –––– In relation with these questions, the reader might want to consult [MP], for example Conjecture 5.1 formulated there.

7.5. Remark. –––– When rk Pic(V\overline{ℚ} ) = 18, the situation is typically different. For example, when two Kummer coefficients are equal, we saw in Fact 7.2 that P_{V_{[a,a,c]}} has density at least \frac{1}{2} unless 4a^{2} − 2c − 2 is a perfect square. According to Proposition 4.10, the same is true when the abelian surface corresponding to V has real multiplication by an endomorphism defined over a proper field extension of ℚ. Note that the latter case actually subsumes the former as the abelian variety corresponding to V_{[a,a,c]} is isogenous to the product of two elliptic curves and, therefore, has real multiplication.

8 Our data

8.1. –––– The raw data of the experiments are available on both authors’ web pages.

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