EXAMPLES OF K3 SURFACES WITH REAL MULTIPLICATION

ANDREAS-STEPHAN ELSENHANS AND JÖRG JAHNEL

Abstract. We construct explicit examples of K3 surfaces over ℚ having real multiplication. Our examples are of geometric Picard rank 16. The standard method for the computation of the Picard rank provably fails for the surfaces constructed.

1. Introduction

It is well known that the endomorphism algebra of a general elliptic curve X over ℂ is equal to ℤ, while for certain exceptional curves the endomorphism algebra is larger. There are only countably many exceptions and these have complex multiplication. I.e., End(X)⊗_{ℤ} ℚ is an imaginary quadratic number field.

There is a rich theory about CM elliptic curves, cf. [38, Chapter II] or [6, Chapter 3]. We will not go into details, but mention only a few facts that are relevant for what follows. First of all, the construction of CM elliptic curves in an analytic setting is very classical [41, 17. bis 23. Abschnitt]. The situation becomes slightly more complicated, however, when explicit equations are asked for.

Concerning their arithmetic, CM elliptic curves are, may be, even more special. There are only nine imaginary quadratic number fields that may occur as the endomorphism field of a CM elliptic curve, defined over ℚ, those being of class number one. Further, on all general elliptic curves, the traces of the Frobenii Frob_{p} ∈ End(H_{ét}^{1}((X_{p})_{\overline{𝔽}_{p}}, ℚ_{l})) have the same statistics [33], while, in the CM case, a different statistics occurs.

The whole theory generalizes to higher dimensions. The most obvious situation is certainly that of an abelian surface. Here, once again, the general case is that the endomorphism algebra is equal to ℤ.

But there is more than one way, in which an abelian surface may be exceptional. A new feature is, for example, that real multiplication occurs [18]. Concerning the possible statistics of the Frobenii on an abelian surface over ℚ, interesting investigations have been undertaken by K. S. Kedlaya and A.V. Sutherland [23].

From the point of view of the classification of algebraic surfaces, there are natural generalizations of elliptic curves to dimension two, other than abelian surfaces. One kind of these is provided by the so-called K3 surfaces. Indeed, elliptic curves may be characterized as being the curves with trivial canonical class.

By definition, a K3 surface is a simply connected, projective algebraic surface with trivial canonical class. The property of being K3 determines the Hodge diamond. The Picard group of a complex K3 surface is isomorphic to ℤ^{n}, where n may range from 1 to 20.

Key words and phrases. K3 surface, real multiplication, Hodge structure, van Luijk’s method

2010 Mathematics Subject Classification. Primary 14J28; Secondary 14C30, 11G15, 14C22

Examples include the classical Kummer surfaces, smooth space quartics and double covers of P^{2}, branched over a smooth sextic curve. As long as the singularities are rational, the minimal resolutions of singular quartics and double covers of P^{2}, branched over a singular sextic, are K3 surfaces, too.

Since K3 surfaces do not carry a natural group structure, the endomorphism algebra may no longer be defined naively. One has to take a look at the cohomology. For 𝔛 a K3 surface, one has H^{1}(𝔛, ℚ) = 0, but H := H^{2}(𝔛, ℚ) is non-trivial. It is a pure weight-2 Hodge structure of dimension 22, consisting of the image P of the Picard group under the Chern class map and its orthogonal complement T := P^{⊥}, which is called the transcendental part of H.

The endomorphism algebra End(T) may only be ℚ, a totally real number field, or a CM field. If E ⫌ ℚ is totally real then T (or the underlying K3 surface 𝔛) is said to have real multiplication. If E is CM then one speaks of complex multiplication. Observe the difference to the situation of an elliptic curve or abelian surface, where End(H^{1}(𝔛, ℚ)) is considered.

The Kummer surface Kum(E_{1}×E_{2}) attached to the product of two elliptic curves E_{1} and E_{2} has complex multiplication if one of the elliptic curves has. On the other hand, a Kummer surface does not inherit the property of having real multiplication from the underlying abelian surface 𝔄. Indeed, in this case, \sqrt{d} ∈ ℚ(\sqrt{d}) operates on H^{1,0}(𝔄, ℂ) with eigenvalues ±\sqrt{d}. Consequently, ℚ(\sqrt{d}) operates on Λ^{2}H^{1,0}(𝔄, ℂ) = H^{2,0}(𝔄, ℂ) ↪ H_{ℂ} := H^{2}(Kum 𝔄, ℂ) via multiplication by the norm, and the same is true for the whole T_{ℂ} ⊂ H_{ℂ} .

Nonetheless, B. van Geemen showed that there exist K3 surfaces of Picard rank 16 that have real multiplication by ℚ(\sqrt{d}), as soon as d is an odd number being the sum of two squares [17, Example 3.4]. Van Geemen’s approach is analytic and does not lead to explicit equations. He poses the problem to construct explicit examples in [17, paragraph 3.1]. We shall give van Geemen’s argument in a slightly more general form in an appendix.

1.1. The results. In this note, we will present algorithms to efficiently test a K3 surface X over ℚ for real multiplication. These do not provide a proof, but only strong evidence. Experiments using the algorithms delivered two families of K3 surfaces of geometric Picard rank 16 and an isolated example.

For infinitely many members X^{(2,t)} of the first family, we will prove in this note (4.13 and Theorem 5.2) that they have real multiplication by ℚ(\sqrt{2}). To our knowledge, these are the first explicit examples of K3 surfaces, for which real multiplication is proven.

The members of the second family are highly suspicious to have real multiplication by ℚ(\sqrt{5}), while the isolated example is highly suspicious to have real multiplication by ℚ(\sqrt{13}).

Unfortunately, no practical algorithm is known that proves real (or complex) multiplication for a given K3 surface. It seems that, in principle, there is such an algorithm under the assumption of the Hodge conjecture, cf. the indications given in the proof of [4, Theorem 6]. But the idea is to inspect the Hilbert scheme of X × X, which is far from realistic to do in practice.

That is why our proof for real multiplication is an indirect one. It is based on showing that #X_{p}(𝔽_{p}) ≡ 1 (mod p) for all primes p ≡ 3, 5 (mod 8). In order to do this, we analyze in detail one of the elliptic fibrations the surfaces X^{(2,t)} have. We do not have a geometric explanation for the occurrence of real multiplication.

1.2. An application. Van Luijk’s method. This is the standard method to determine the geometric Picard rank of a K3 surface over ℚ. Its fundamental idea is that, for every prime p of good reduction, one has rk Pic X\overline{ℚ} ≤ rk Pic X_{\overline{𝔽}_{p}}. Further, the method relies on the hope to find good primes such that

rk Pic X_{\overline{𝔽}_{p}} ≤ rk Pic X\overline{ℚ} + 1 .  (1)

To see the method at work, the reader is advised to consult the original papers of R. van Luijk [26, 27] or some of the authors’ previous articles [10, 12, 13]. Further, there is the remarkable work of N. Elkies and A. Kumar [9], where they compute, among other data, the Néron-Severi ranks of all Hilbert-Blumenthal surfaces corresponding to the real quadratic fields of discriminants up to 100. Several of them are K3.

Quite recently, F. Charles [4] provided a theoretical analysis on the existence of primes fulfilling condition (1). The result is that such primes always exist, unless X has real multiplication by a number field E and (22 − rk Pic X\overline{ℚ} )/[E:ℚ] is odd. In particular, for the explicit examples of K3 surfaces with real multiplication, provided by our experiments, the method is bound to fail in its original form.

For K3 surfaces having real multiplication, there is a modification of van Luijk’s method, cf. [4, Proposition 18]. We will make use of this in the proof of Theorem 5.2. But, still, things will fail whenever a surface is met that has real multiplication, but for which this fact is not known or cannot be assured.

2. Hodge structures

Recall the following definition, cf. [7, Définition 2.1.10 and Proposition 2.1.9].

Definition 2.1. i) A (pure ℚ-) Hodge structure of weight i is a finite dimensional ℚ-vector space V together with a decomposition

V_{ℂ} := V ⊗_{ℚ} ℂ = H^{0,i} ⊕ H^{1,i−1} ⊕ … ⊕ H^{i,0}

having the property that \overline{H^{m,n}} = H^{n,m}, for every m, n ∈ ℕ_{0} such that m + n = i. A morphism f: V → V′ of (pure ℚ-) Hodge structures is a ℚ-linear map such that f_{ℂ} : V_{ℂ} → V′_{ℂ} respects the decompositions. ii) A Hodge structure of weight 2 is said to be of K3 type if dim_{ℂ} H^{2,0} = 1.

Remark 2.2. Hodge structures of weight i form an abelian category [7, 2.1.11]. Further, this category is semisimple. I.e., every Hodge structure is a direct sum of primitive ones [7, Définition 2.1.4 and Proposition 2.1.9].

Examples 2.3. i) Let X be a smooth, projective variety over ℂ. Then H^{i}(X(ℂ), ℚ) is in a natural way a pure ℚ-Hodge structure of weight i. ii) In H^{2}(X(ℂ), ℚ), the image of c_{1} : Pic(X)⊗_{ℤ} ℚ → H^{2}(X(ℂ), ℚ) defines a sub-Hodge structure P such that H_{P}^{0,2} = H_{P}^{2,0} = 0. iii) If X is a surface then H := H^{2}(X(ℂ), ℚ) is actually a polarized pure Hodge structure, the polarization ⟨. , .⟩: H × H → ℚ being given by the cup product, together with Poincaré duality. The sub-Hodge structure P and its orthogonal complement T are polarized Hodge structures, too.

Yu. G. Zarhin [42, Theorem 1.6.a) and Theorem 1.5.1] proved that, for T a polarized weight-2 Hodge structure of K3 type, E := End(T) is either a totally real field or a CM field. Thereby, every φ ∈ E operates as a self-adjoint mapping. I.e.,

⟨φ(x), y⟩ = ⟨x, \overline{φ}(y)⟩ ,

for the identity map in the case that E is totally real and the complex conjugation in the case that it is a CM field. Observe that, in either case, T carries the structure of an E-vector space. Further, in the case of real multiplication, dim_{E} T = 1 is impossible [42, Remark 1.5.3.c)].

Remark 2.4. Motivated by the analysis of F. Charles, we are interested in K3 surfaces having real multiplication and an odd E-dimensional T. The simplest case one can think of is that E = ℚ(\sqrt{d}) is real quadratic and dim_{E} T = 3, i.e. dim_{ℚ} T = 6.

3. Some arithmetic consequences of real multiplication

Let X be a K3 surface over ℚ. We put

P := im(c_{1} : Pic(X_{ℂ} )⊗_{ℤ} ℚ ↪ H^{2}(X(ℂ), ℚ)) ,

T := P^{⊥}, and write E for the endomorphism algebra of the Hodge structure T.

Further, let us choose a prime number l and turn to l-adic cohomology. This essentially means to tensor with ℚ_{l}, as there is the canonical comparison isomorphism [34, Exposé XI, Théorème 4.4.iii)]

H^{2}(X(ℂ), ℚ)⊗_{ℚ} ℚ_{l} ⟵^{≅} H_{ét}^{2}(X\overline{ℚ} , ℚ_{l}) .

An important feature of the l-adic cohomology theory is that it is acted upon by the absolute Galois group of the base field. I.e., there is a continuous representation

ϱ_{l}: Gal(\overline{ℚ}/ℚ) ⟶ GL(H_{ét}^{2}(X\overline{ℚ} , ℚ_{l})) .

The image of ϱ_{l} is an l-adic Lie group. Its Zariski closure is an algebraic group G_{l}, called the algebraic monodromy group associated to ϱ_{l}.

On the other hand, there are, in a way compatible with Betti cohomology, the image P_{l} of Pic(X\overline{ℚ} )⊗_{ℤ}ℚ_{l} under the Chern map and its orthogonal complement T_{l}. The image of ϱ_{l}, and hence the whole of G_{l}, maps P_{l} to P_{l}. Thereby, orthogonality is preserved with respect to the pairing ⟨. , .⟩. Thus, the algebraic monodromy group G_{l} must map T_{l} into itself, as well.

Theorem 3.1 ((Tankeev, Zarhin)). The neutral component G_{l}^{∘} of the algebraic monodromy group with respect to the Zariski topology is equal to the centralizer of E in GO(T_{l}, ⟨. , .⟩). In particular, the operation of E on T_{l} ⊂ H_{ét}^{2}(X\overline{ℚ} , ℚ_{l}) commutes with that of G_{l}^{∘}.

Proof. This follows from the Mumford-Tate conjecture, proven by S. G. Tankeev [39, 40], together with Yu. G. Zarhin’s explicit description of the Mumford-Tate group in the case of a K3 surface [42, Theorem 2.2.1]. We refer the reader to the original articles and to the discussion in [4, Section 2.2]. □

For every prime p, choose an absolute Frobenius element Frob_{p} ∈ Gal(\overline{ℚ}/ℚ). If p ≠ l is a prime, at which X has good reduction then, by virtue of the smooth base change theorem [34, Exp. XVI, Corollaire 2.5], there is a canonical isomorphism

H_{ét}^{2}(X\overline{ℚ} , ℚ_{l}) ≅ H_{ét}^{2}((X_{p})_{\overline{𝔽}_{p}}, ℚ_{l}) .

Here, the vector space on the right hand side is naturally acted upon by Gal(\overline{𝔽}_{p}/𝔽_{p}) and the operation of Frob_{p} ∈ Gal(\overline{ℚ}/ℚ) on the left hand side is compatible with that of Frob ∈ Gal(\overline{𝔽}_{p}/𝔽_{p}) on the right.

Corollary 3.2. There is a positive integer f such that the following statement is true. For every prime number p and every prime number l, the operation of (Frob_{p})^{f} on T_{l} commutes with that of E.

Proof. By definition, ϱ_{l}(Frob_{p}) ∈ G_{l}. Hence, for f := #(G_{l}/G_{l}^{∘}), we have ϱ_{l}((Frob_{p})^{f}) ∈ G_{l}^{∘}. Further, the groups G_{l}/G_{l}^{∘} are canonically isomorphic to each other, for the various values of l, as was proven by M. Larsen and R. Pink [24, Proposition 6.14]. □

Notation 3.3. For every prime p, choose l ≠ p and denote by χ_{p^{n}}^{T} the characteristic polynomial of (Frob_{p})^{n} on T_{l}. This has coefficients in ℚ and is independent of l, whether X has good reduction at p [8, Théorème 1.6] or not [31, Theorem 3.1]. χ_{p^{n}}^{T} is nothing but the characteristic polynomial of (Frob_{p})^{n} on the transcendental part of H_{ét}^{2}(X\overline{ℚ} , ℚ_{l}). It might have zeroes of the form p^{n} times a root of unity.

We write χ_{p^{n}}^{tr} ∈ ℚ[Z] for the corresponding cofactor. If p is a good prime then, according to the Tate conjecture, χ_{p^{n}}^{tr} is the characteristic polynomial of Frob^{n} on the transcendental part of H_{ét}^{2}(X_{\overline{𝔽}_{p}}, ℚ_{l}).

For the remainder of this section, we assume that E = ℚ(\sqrt{d}), for d ≠ 1 a squarefree integer. I.e., that X has real or complex multiplication by a quadratic number field.

Proposition 3.4. Let p be a prime number and l be a prime that is ramified or inert in ℚ(\sqrt{d}). Then the polynomial χ_{p^{f}}^{T} ∈ ℚ[Z] splits as

χ_{p^{f}}^{tr} = g_{l}g_{l}^{σ},

for g_{l} ∈ ℚ_{l}(\sqrt{d})[Z] and σ: ℚ_{l}(\sqrt{d}) → ℚ_{l}(\sqrt{d}) the conjugation.

Proof. The assumption ensures that ℚ_{l}(\sqrt{d}) is a quadratic extension field. Further, T_{l} is a ℚ_{l}(\sqrt{d})-vector space and, by Corollary 3.2, ϱ_{l}((Frob_{p})^{f}) commutes with the operation of \sqrt{d} ∈ E. In other words, ϱ_{l}((Frob_{p})^{f}) is a ℚ_{l}(\sqrt{d})- linear map. For the corresponding characteristic polynomial c_{l} ∈ ℚ_{l}(\sqrt{d})[Z], we have χ_{p^{f}}^{T} = c_{l}c_{l}^{σ}.

c_{l} may be divisible by polynomials of the form Φ(Z/p), for Φ a cyclotomic polynomial. The assertion follows, for g_{l} the cofactor corresponding to the product of all these factors, normalized to a monic polynomial. □

Theorem 3.5. Let p be a prime of good reduction of the K3 surface X over ℚ, having real or complex multiplication by the quadratic number field ℚ(\sqrt{d}). Then at least one of the following two statements is true.

i) The polynomial χ_{p}^{tr} ∈ ℚ[Z] splits in the form

χ_{p}^{tr} = gg^{σ},

for g ∈ ℚ(\sqrt{d})[Z] and σ: ℚ(\sqrt{d}) → ℚ(\sqrt{d}) the conjugation. ii) The polynomial χ_{p^{f}}^{tr} is a square in ℚ[Z].

Proof. According to [43], χ_{p}^{tr} = h^{k}, for an irreducible polynomial h ∈ ℚ[Z] and k ∈ ℕ. The polynomial h^{(f)} having the zeroes x_{i}^{f}, for x_{i} the zeroes of h, might once again be a perfect power. Write h^{(f)} = \underline{h}^{k′}, for \underline{h} an irreducible polynomial. Then χ_{p^{f}}^{tr} = \underline{h}^{kk′}.

If one of the integers k and k′ is even then assertion ii) is true. Thus, assume from now on that k and k′ are both odd. By Proposition 3.4, \underline{h}^{kk′} = χ_{p^{f}}^{tr} splits into two factors conjugate over ℚ_{l}(\sqrt{d}), for every l that is not inert in ℚ(\sqrt{d}). As kk′ is odd, the same is true for \underline{h}.

In particular, for every prime 𝔏 lying above (l) in the field ℚ[Z]/(\underline{h}), one has that f(𝔏|(l)) is even for l inert in ℚ(\sqrt{d}) and that e(𝔏|(l)) is even for l ramified in ℚ(\sqrt{d}). [30, Chapter VII, Proposition 13.9] implies that ℚ(\sqrt{d}) ⊆ ℚ[Z]/(\underline{h}).

Let now x_{0} ∈ \overline{ℚ} be an element having h as its minimal polynomial. Then, clearly, ℚ(x_{0}^{f}) ≅ ℚ[Z]/(\underline{h}). Altogether, ℚ(\sqrt{d}) ⊆ ℚ(x_{0}^{f}) ⊆ ℚ(x_{0}). But, according to Lemma 3.12 below, this is equivalent to h being reducible over ℚ(\sqrt{d}). It must split into two conjugate factors. □

Remark 3.6. In general, for an irreducible polynomial h ∈ ℚ[Z], the polynomial h^{(f)} may not factor otherwise as into the power of an irreducible polynomial. In fact, Gal(\overline{ℚ}/ℚ) permutes the roots x_{1}, …, x_{r} of h transitively. Therefore, it does the same to x_{1}^{f}, …, x_{r}^{f}.

Remarks 3.7. i) Let h be an irreducible polynomial such that χ_{p}^{tr} = h^{k} and consider Gal(h) as a permutation group on the roots of h. As such, it has an obvious block structure 𝔅 := \{\{z, \overline{z}\} | h(z) = 0\} into blocks of size two. Indeed, h is a real polynomial without real roots and every root is of absolute value p. Thus, \overline{z} = \frac{p^{2}}{z}, such that the pairs are respected by the operation of the Galois group. ii) Assume that k is odd and d > 0, i.e., that there is real multiplication. We claim that this causes a second block structure.

Suppose first that variant i) of Theorem 3.5 is true. Then there is the block structure 𝔅′ := \{\{z | g(z) = 0\}, \{z | g^{σ}(z) = 0\}\} into two blocks of size \frac{deg h}{2}. As g and g^{σ} are real polynomials, the blocks in 𝔅′ are non-minimal. Each is a union of some of the blocks in 𝔅.

If option ii) of Theorem 3.5 happens to be true then there is the block structure 𝔅′′, the blocks in which are formed by the roots of h having their f-th power in common. The mutual refinement of 𝔅′′ and 𝔅 is trivial. As k is assumed odd, the blocks are of even size. Thus, the block structure generated by 𝔅′′ and 𝔅 consists of blocks of a size that is a multiple of 4.

Corollary 3.8. Suppose that d > 0. Then, for every good prime p, deg χ_{p}^{tr} is divisible by 4.

Proof. Write χ_{p}^{tr} = h^{k}. As seen in Remark 3.7.i), deg h is even, which implies the claim as long as k is even. When k is odd, the observations made in Remark 3.7.ii) show in both cases that deg h must be divisible by 4. □

Corollary 3.9. Suppose that d > 0. Then, for every good prime p ≥ 5, we have

rk Pic((X_{p})_{\overline{𝔽}_{p}}) ≡ 2 (mod 4) .

Proof. The characteristic polynomial of Frob on H_{ét}^{2}((X_{p})_{\overline{𝔽}_{p}}, ℚ_{l}) has exactly 22 − deg χ_{p}^{tr} zeroes of the form p times a root of unity. Further, the Tate conjecture is known to be true for K3 surfaces in characteristic ≥ 5, cf. [5, Corollary 2] and [25]. □

Corollary 3.10. Suppose that d > 0 and let p ≥ 5 be a good prime number that is inert in E = ℚ(\sqrt{d}).

i) Then X_{p} is not ordinary. ii) Suppose that dim_{E} T ≤ 3. Then, either rk Pic((X_{p})_{\overline{𝔽}_{p}}) = 22 or χ_{p^{f}}^{tr} is the square of an irreducible quadratic polynomial.

Proof. i) X_{p} being ordinary would mean that χ_{p^{f}}^{tr} has exactly one zero that is a p-adic unit. Indeed, we have dim_{ℂ} H^{0,2} = 1 and ordinarity means that the Newton polygon coincides with the Hodge polygon [21, Définition IV.4.12], cf. [19, pages 48f].

By Theorem 3.5, in any case, we can say that there is a factorization χ_{p^{f}}^{tr} = \underline{g}\underline{g}^{σ}, for some \underline{g} ∈ 𝒪_{E}[Z]. Assume without restriction that the zero being a p-adic unit is a root of \underline{g}^{σ}. Then, for the coefficients of the polynomial

\underline{g}(Z) = Z^{n} + a_{n−1}Z^{n−1} + … + a_{0} ,

one has that ν_{p}(a_{j}) > 0, for every j. But, p being inert, the same is true for \underline{g}^{σ}. In particular, ν_{p}(a_{n−1}^{σ}) > 0. This shows that it is impossible for \underline{g}^{σ} to have exactly one root that is a p-adic unit. ii) The second variant comes from option ii) of Theorem 3.5. The quadratic polynomial appearing is irreducible, as its zeroes are non-reals.

Otherwise, there is a factorization χ_{p}^{tr} = gg^{σ}, for some g ∈ E[Z]. We have to show that deg g = 0. Assume the contrary. Then deg g = 2, since dim_{E} T ≤ 3 implies that deg g ≤ 3. Further, from Corollary 3.8, we know that deg g is even.

Write g(Z) = Z^{2} + aZ ± p^{2} = (Z − x_{1})(Z − x_{2}). Then

ν_{p}(x_{1}) + ν_{p}(x_{2}) = ν_{p}(x_{1}x_{2}) = ν_{p}(±p^{2}) = 2

and min(ν_{p}(x_{1}), ν_{p}(x_{2})) ≤ ν_{p}(x_{1} + x_{2}) = ν_{p}(−a), with equality obviously being true when ν_{p}(x_{1}) ≠ ν_{p}(x_{2}). However, a ∈ ℚ(\sqrt{d}) implies that ν_{p}(−a) is an integer and ν_{p}(x_{i}) ≥ 0 is well-known. Thus, there are only two cases. We will show that they are both contradictory.

If ν_{p}(x_{1}) = ν_{p}(x_{2}) = 1 then, as p is inert, the same is true for x_{1}^{σ} and x_{2}^{σ}. Further, since x_{1}/p, x_{2}/p, x_{1}^{σ}/p, and x_{2}^{σ}/p are known to be l-adic units for every l ≠ p, they must be roots of unity. This is a contradiction to the definition of χ_{p}^{tr}.

If, without restriction, ν_{p}(x_{1}) = 0 and ν_{p}(x_{2}) = 2 then ν_{p}(x_{1}^{σ}) = 0, too. We obtain a contradiction to the classical result that the Newton polygon always runs above the Hodge polygon [28], cf. [3, Theorem 8.39]. □

Corollary 3.11. Suppose that dim_{E} T ≤ 3. If χ_{p^{f}}^{tr} is the square of a quadratic polynomial, but χ_{p}^{tr} is not, then Gal(χ_{p}^{tr}) ≅ ℤ/2ℤ × ℤ/2ℤ.

Proof. The assumption implies that χ_{p}^{tr} = h is irreducible of degree four. Further, Gal(h) has two different block structures, both into blocks of size two. The only transitive subgroup of S_{4} having this property is the Klein four group.

□

Lemma 3.12. Let K be any field, K(\sqrt{d})/K a quadratic field extension, and h ∈ K[Z] an irreducible polynomial. Then h splits in K(\sqrt{d}) if and only if K(\sqrt{d}) ⊆ K[Z]/(h).

Proof. “⟸” Let z_{0} ∈ K[Z]/(h) be a zero of h. As [ℚ(z_{0}):ℚ(\sqrt{d})] = \frac{deg h}{2}, the minimal polynomial over ℚ(\sqrt{d}) is of degree \frac{deg h}{2} and a factor of h.

“⟹” Write h = gg^{σ}. Then the extension fields K[Z]/(h) and K(\sqrt{d})[Z]/(g) both contain a zero of g and have the same degree over K. Hence, they must be isomorphic to each other. □

4. Efficient algorithms to test a K3 surface for real multiplication

Generalities. According to the Lefschetz trace formula [35, Exposé XII, 6.3 and Exemple 7.3], a K3 surface Y over 𝔽_{p} is non-ordinary if and only if #Y (𝔽_{p}) ≡ 1 (mod p). In particular, non-ordinarity may be tested by counting points only over 𝔽_{p}.

Further, we expect naively that a K3 surface X over ℚ has non-ordinary reduction at p with a probability of \frac{1}{p}. Thus, the number of non-ordinary primes ≤ N should be of the order of log log N. However, for K3 surfaces with real multiplication by ℚ(\sqrt{d}), we expect approximately half the primes to be non-ordinary.

This suggests to generate a huge sample of K3 surfaces over ℚ, each having geometric Picard rank ≥ 16, and to execute the following statistical algorithm on all of them.

Algorithm 4.1 (Testing a K3 surface for real multiplication–statistical version).

i) Let p run over all primes p ≡ 1 (mod 4) between 40 and 300. For each p, count the number #X_{p}(𝔽_{p}) of 𝔽_{p}-rational points on the reduction of X modulo p. If #X_{p}(𝔽_{p}) ≡ 1 (mod p) for not more than five primes then terminate immediately. ii) Put p_{0} to be the smallest good and ordinary prime for X. iii) Determine the characteristic polynomial of Frob on H_{ét}^{2}((X_{p_{0}})_{\overline{𝔽}_{p_{0}}}, ℚ_{l}). For this, use the strategy described in [11, Examples 27 and 28]. It involves to count, in addition, the points on X_{p_{0}} defined over 𝔽_{p_{0}^{2}} and, possibly, those defined over 𝔽_{p_{0}^{3}}. Factorize the polynomial obtained to determine the factor χ_{p_{0}}^{tr}. If deg χ_{p_{0}}^{tr} ≠ 4 then terminate. If Gal(χ_{p_{0}}^{tr}) ≅ ℤ/2ℤ × ℤ/2ℤ or χ_{p_{0}}^{tr} is the square of a quadratic polynomial then raise p_{0} to the next good and ordinary prime and iterate this step. iv) Now, one has deg χ_{p_{0}}^{tr} = 4. Further, χ_{p_{0}^{f}}^{tr} is certainly irreducible, although the value of f is not known to us. Determine the Galois group of χ_{p_{0}}^{tr} and the quadratic subfields of its splitting field. Only one real quadratic field may occur. Put d to be the corresponding radicand.

v) Let p run over all good primes below 300, starting from the lowest. If #X_{p}(𝔽_{p}) ≢ 1 (mod p) for a prime inert in ℚ(\sqrt{d}) then terminate. vi) Output a message saying that X is highly suspicious to have real multiplication by ℚ(\sqrt{d}).

Remarks 4.2. i) The reason for restricting in step i) to primes 1 mod 4 is that, otherwise, too many surfaces are found that are suspicious to have complex multiplication by ℚ(\sqrt{−1}). ii) For small primes p, it happens too often that #X_{p}(𝔽_{p}) ≡ 1 (mod p), independently of whether or not X has real multiplication. That is why we restricted the calculations to primes p > 40. iii) Algorithm 4.1 is, of course, only statistically correct. There is a certain, very low but positive, probability that a K3 surface with real multiplication is thrown away in step i). iv) On the other hand, the algorithm is extremely efficient. The point is that, for the lion’s share of the surfaces, it terminates directly after step i). Only for a negligible percentage of the surfaces, the more time-consuming steps ii)-v) have to be carried out. This shows, in particular, that step i) is the only time-critical one. An efficient algorithm for point counting over relatively small prime fields is asked for.

When testing surfaces for real multiplication by a particular field ℚ(\sqrt{d}), the following modification of Algorithm 4.1 may be used.

Algorithm 4.3 (Testing a K3 surface for real multiplication–deterministic version).

o) In an initialization step, list the first ten primes that are inert in ℚ(\sqrt{d}).

i) Let p run over the list. For each p, count the numbers #X_{p}(𝔽_{p}) of 𝔽_{p}-rational points on the reduction X_{p}. If one of them turns out not to be congruent to 1 modulo p then terminate immediately. ii) Let p run over all good primes below 300, starting from the lowest. If #X_{p}(𝔽_{p}) ≢ 1 (mod p) for a prime inert in ℚ(\sqrt{d}) then terminate. iii) Output a message saying that X is highly suspicious to have real multiplication by ℚ(\sqrt{d}).

Remarks 4.4. i) This algorithm finds every surface in the sample having real multiplication by ℚ(\sqrt{d}). It could, in principle, find a few others, too. ii) One might be afraid that, for some of the surfaces X having real multiplication by ℚ(\sqrt{d}), there exists an inert prime p of bad reduction, for which #X_{p}(𝔽_{p}) ≢ 1 (mod p). These surfaces would be thrown away in step i). However, this does not happen, at least not in our samples.

Lemma 4.5. Let X be a double cover of P_{ℚ}^{2} , branched over the union of six lines. Suppose there is a quadratic number field ℚ(\sqrt{d}) such that #X_{q}(𝔽_{q}) ≡ 1 (mod q) for every good prime q that is inert in ℚ(\sqrt{d}). Then #X_{p}(𝔽_{p}) ≡ 1 (mod p), too, for every bad prime p that is inert.

Proof. If at least two of the six lines coincide modulo p then X_{p} is a rational surface and #X_{p}(𝔽_{p}) ≡ 1 (mod p) is automatic. Thus, let us assume the contrary.

We fix an auxiliary prime number l that is split in ℚ(\sqrt{d}) and let p be a bad, inert prime. For every prime q inert in ℚ(\sqrt{d}), choose an absolute Frobenius element Frob_{q} ∈ Gal(\overline{ℚ}/ℚ). By Cebotarev, the elements σ^{−1} Frob_{q} σ ∈ Gal(\overline{ℚ}/ℚ), for σ running through Gal(\overline{ℚ}/ℚ), are dense in the coset Gal(\overline{ℚ}/ℚ) \ Gal(\overline{ℚ}/ℚ(\sqrt{d})), to which Frob_{p} belongs. The same is still true when restricting to the primes q, at which X has good reduction.

For those, we have the congruence Tr Frob_{H_{ét}^{2}((X_{q})_{\overline{𝔽}_{q}},ℚ_{l})} ≡ 0 (mod q). In other words,

Tr \frac{1}{q}Frob_{H_{ét}^{2}((X_{q})_{\overline{𝔽}_{q}},ℚ_{l})} = Tr \frac{1}{q}Frob_{q,H_{ét}^{2}(X\overline{ℚ} ,ℚ_{l})} = Tr Frob_{q,H_{ét}^{2}(X\overline{ℚ} ,ℚ_{l}(1))}

is an integer, necessarily within the range [−22, 22]. As this condition defines a Zariski closed subset of GL(H_{ét}^{2}(X\overline{ℚ} , ℚ_{l}(1))), one has

Tr Frob_{H_{ét}^{2}(X_{\overline{ℚ} _{p}},ℚ_{l})} = Tr Frob_{p,H_{ét}^{2}(X\overline{ℚ} ,ℚ_{l})} ≡ 0 (mod p) ,  (2)

too, cf. [34, Exposé XVI, Corollaire 1.6]

Further, the eigenvalues of Frob on H_{ét}^{2}(X_{\overline{ℚ} _{p}}, ℚ_{l}) are the same as those on H_{ét}^{2}(X_{\overline{ℚ} _{p}}, ℚ_{p}) [31, Theorem 3.1]. In addition, a main result of p-adic Hodge theory [14, Theorem III.4.1] implies, as X is K3, that not more than one of the eigenvalues of Frob on H_{ét}^{2}(X_{\overline{ℚ} _{p}}, ℚ_{p}) may be a p-adic unit, the others being of strictly positive p-adic valuation. Under these circumstances, the congruence (2) expresses the property that, among the eigenvalues, there is actually no p-adic unit.

For comparison with the cohomology H_{ét}^{2}((X_{p})_{\overline{𝔽}_{p}}, ℚ_{l}) of the singular fiber, the theory of vanishing cycles [36, Exposés I, XIII, and XV] applies, as X_{p} has only isolated singularities [20, Corollaire 2.9]. In our case, it states that H_{ét}^{2}((X_{p})_{\overline{𝔽}_{p}}, ℚ_{l}) naturally injects into H_{ét}^{2}(X_{\overline{ℚ} _{p}}, ℚ_{l}). In particular, the eigenvalues of Frob on H_{ét}^{2}((X_{p})_{\overline{𝔽}_{p}}, ℚ_{l}) form a subset of the 22 eigenvalues of Frob on H_{ét}^{2}(X_{\overline{ℚ} _{p}}, ℚ_{l}).

This shows that all eigenvalues on H_{ét}^{2}((X_{p})_{\overline{𝔽}_{p}}, ℚ_{l}) are of strictly positive p-adic valuation. Further, using the Leray spectral sequence together with the proper base change theorem [34, Exposé XII, Corollaire 5.2.iii)], one sees that blow-ups do not affect the transcendental part T_{l} ⊂ H_{ét}^{2}((X_{p})_{\overline{𝔽}_{p}}, ℚ_{l}). Hence, #X̃_{p}(𝔽_{p}) ≡ 1 (mod p), for X̃_{p} the minimal resolution of singularities. The same is true for X_{p}. □

Counting points on degree-2 K3-surfaces.

4.6 (Structure of our samples). K3 surfaces that are given as desingularizations of the double covers of the projective plane, branched over the union of six lines. Our reason is that this family offers computational advantages, too.

We do not ask all lines to be defined over ℚ, however, as this seems to be too restrictive. At least, this is what we learned during our experiments. A compromise is as follows.

The lines are allowed to form three Galois orbits, each of size two. Assuming the three ℚ-rational points of intersection not to be collinear, we may suppose them to be the standard base of P^{2}. The equation of the surface then takes the form

w^{2} = q_{1}(y, z)q_{2}(x, z)q_{3}(x, y) .

This representation is unique up to action of the monomial group. I.e., up to permutation and scaling of the variables.

Algorithm 4.7 (Counting points on one surface). In order to determine the number of 𝔽_{q}-rational points on one surface, we count the points over the q affine lines of the form (1:u:⋆) and the affine line (0:1:⋆) and sum up these numbers. Finally, we add 1, as, on each of our surfaces, there is exactly one point lying above e_{3}.

Remark 4.8 (Counting points above one line). It is easy to count the number of points above the affine line L_{x,y}: A^{1} → P^{2}, given by t ↦ (x: y: t). Observe that q_{3} is constant on this line. Thus, we get a quadratic twist of an elliptic curve. The number of points on it is q + χ(q_{3}(x, y))λ_{x,y}, for

λ_{x,y} := ∑_{t∈𝔽_{q}} χ(q_{1}(y, t)q_{2}(x, t))  (3)

and χ the quadratic character of 𝔽_{q}.

Strategy 4.9 (Treating a sample of surfaces). Our samples are given by three lists of quadratic forms. One list for q_{1}, another for q_{2}, and third one for q_{3}. In the case that we want to count the points on all surfaces, given by the Cartesian product of the three lists, we perform as follows.

i) For each quadratic form q_{3}, compute the values of χ(q_{3}(1, ⋆)) and χ(q_{3}(0, 1)) and store them in a table. ii) Run in an iterated loop over all pairs (q_{1}, q_{2}). For each pair, do the following.

• Using formula (3), compute λ_{1,⋆} and λ_{0,1}.

• Run in a loop over all forms q_{3}. Each time, calculate

S_{q_{1},q_{2},q_{3}} := ∑_{⋆} χ(q_{3}(1, ⋆))λ_{1,⋆} ,

using the precomputed values. The number of points on the surface, corresponding to (q_{1}, q_{2}, q_{3}), is then q^{2} + q + 1 + χ(q_{3}(0, 1))λ_{0,1} + S_{q_{1},q_{2},q_{3}}.

Remarks 4.10. i) (Complexity and performance). In the case that the number of quadratic forms is bigger than q, the costs of building up the tables are small compared to the final step. Thus, the complexity per surface is essentially reduced to (q + 1) table look-ups for the quadratic character and (q + 1) look-ups in the small table, containing the values λ_{1,⋆} and λ_{0,1}.

ii) We are limited by the memory transfer generated by the former table access. We store the quadratic character in an 8-bit signed integer variable. This doubles the speed compared to a 16-bit variable.

Remark 4.11 (Detecting real multiplication). We used the point counting algorithm, in the version described in 4.9, within the deterministic Algorithm 4.3, in order to detect K3 surfaces having real multiplication by a prescribed quadratic number field. This allowed us to test more than 2.2·10^{7} surfaces per second on one core of a 3.40 GHz Intel^{(R)}Core^{(TM)}i7-3770 processor. The code was written in plain C.

The results. i) A run of Algorithm 4.1 over all triples (q_{1}, q_{2}, q_{3}) of coefficient height ≤ 12, using the method described in 4.9 for point counting, found the first five surfaces suspicious to have real multiplication by ℚ(\sqrt{5}). Observe that a sample of more than 10^{11} surfaces was necessary to bring these examples to light.

Analyzing the examples, we observed that the product of the discriminants of the three binary quadratic forms was always a perfect square.

ii) We added this restriction to our search strategy, which massively reduces the number of surfaces to be inspected. Doing so, we could raise the search bound up to 80. This resulted in more surfaces with probable real multiplication by ℚ(\sqrt{5}) and one example suspicious for real multiplication by ℚ(\sqrt{2}).

From the results, we observed that the square class of one of the three discriminants always coincided with the discriminant of the field of real multiplication.

iii) This restriction led to a further reduction of the search space. Further, it clearly suggests to use the deterministic version of the algorithm.

At a final stage, we could raise the search bound to 200 for real multiplication by ℚ(\sqrt{2}), ℚ(\sqrt{5}), ℚ(\sqrt{13}), and ℚ(\sqrt{17}). We found many more examples for ℚ(\sqrt{2}) and ℚ(\sqrt{5}), one example for ℚ(\sqrt{13}), but none for ℚ(\sqrt{17}).

Remark 4.12. The final sample for ℚ(\sqrt{17}) consisted of about 4.18·10^{13} surfaces and required about 24 days of CPU time. The computations were executed in parallel on two machines, making use of two cores on each machine. The other samples were comparable in size.

In the cases of ℚ(\sqrt{2}) and ℚ(\sqrt{5}), the examples found were sufficient to guess 1-parameter families. To summarize, our experiments let us suspect the following.

Observation–Theorem 4.13. Let t ∈ ℚ be arbitrary and X^{(2,t)} be the K3 surface given by

w^{2} = [(\frac{1}{8}t^{2}− \frac{1}{2}t+ \frac{1}{4})y^{2} + (t^{2}−2t+2)yz + (t^{2}−4t+2)z^{2}]

[(\frac{1}{8}t^{2}+ \frac{1}{2}t+ \frac{1}{4})x^{2} + (t^{2}+2t+2)xz + (t^{2}+4t+2)z^{2}][2x^{2} + (t^{2}+2)xy + t^{2}y^{2}] .

Then #X_{p}^{(2,t)}(𝔽_{p}) ≡ 1 (mod p) for every prime p ≡ 3, 5 (mod 8).

Proof. The case p = 3 is elementary. For p ≠ 3, we shall prove this result below in Theorem 5.3, under some additional restrictions on t. For the cases left out there, similar arguments work. Cf. Remark 5.4 for a few details. □

Observation–Conjecture 4.14. i) Let t ∈ ℚ be arbitrary and X^{(5,t)} be the K3 surface given by

w^{2} = [y^{2} + tyz + (\frac{5}{16}t^{2}+ \frac{5}{4}t+ \frac{5}{4})z^{2}][x^{2} + xz + (\frac{1}{320}t^{2}+ \frac{1}{16}t+ \frac{5}{16})z^{2}][x^{2} + xy + \frac{1}{20}y^{2}] .

Then #X_{p}^{(5,t)}(𝔽_{p}) ≡ 1 (mod p) for every prime p ≡ 2, 3 (mod 5). ii) Let X^{(13)} be the K3 surface given by

w^{2} = (25y^{2} + 26yz + 13z^{2})(x^{2} + 2xz + 13z^{2})(9x^{2} + 26xy + 13y^{2}) .

Then #X_{p}^{(13)}(𝔽_{p}) ≡ 1 (mod p) for every prime p ≡ 2, 5, 6, 7, 8, 11 (mod 13).

Remark 4.15. We verified the congruences above for all primes p < 1000. This concerns X^{(13)} as well as the X^{(5,t)}, for any residue class of t modulo p. There is further evidence, as we computed the characteristic polynomials of Frob_{p} for X^{(13)} as well as for X^{(5,t)} and several exemplary values of t ∈ ℚ, for the primes p below 100. It turns out that indeed they show the very particular behaviour, described in Theorem 3.5 and its corollaries.

5. The proof for real multiplication in the case of the ℚ(\sqrt{2})-family

Lemma 5.1. Let a, D ∈ ℤ be such that gcd(a, D) = 1 and X a K3 surface over ℚ. Suppose that #X_{p}(𝔽_{p}) ≡ 1 (mod p) for every good prime p ≡ a (mod D). Then X has real or complex multiplication.

Proof. For each prime p, choose an absolute Frobenius element Frob_{p} ∈ Gal(\overline{ℚ}/ℚ). By Cebotarev’s density theorem, the elements σ^{−1} Frob_{p} σ ∈ Gal(\overline{ℚ}/ℚ), for the good primes p ≡ a (mod D) and σ ∈ Gal(\overline{ℚ}/ℚ), are topologically dense in the coset of Gal(\overline{ℚ}/ℚ) modulo Gal(\overline{ℚ}/ℚ(ζ_{D})), they belong to. Thus, there are finitely many elements σ_{1}, …, σ_{k} ∈ Gal(\overline{ℚ}/ℚ) such that

\{ σ_{i}σ^{−1} Frob_{p} σ | i = 1, …, k, p ≡ a (mod D), p good for X, σ ∈ Gal(\overline{ℚ}/ℚ) \}

is dense in Gal(\overline{ℚ}/ℚ).

Now choose any prime l ≢ a (mod D), put T_{l} ⊂ H_{ét}^{2}(X\overline{ℚ} , ℚ_{l}) to be the transcendental part of l-adic cohomology, and write r := dim T_{l}. Then, for every good prime p ≡ a (mod D), one has Tr Frob_{p,T_{l}} = kp, for −22 < −r ≤ k ≤ r < 22, and det Frob_{p,T_{l}} = ±p^{r}. Hence,

(Tr Frob_{p,T_{l}})^{r} = ±k^{r} det Frob_{p,T_{l}} ,

which defines a Zariski closed subset I ⫋ GO(T_{l}, ⟨. , .⟩), invariant under conjugation. As GO(T_{l}, ⟨. , .⟩) is irreducible, the union σ_{1}I ∪…∪σ_{k}I cannot be the whole group. Consequently, the image of Gal(\overline{ℚ}/ℚ) → GO(T_{l}, ⟨. , .⟩) is not Zariski dense. In view of Theorem 3.1, this is enough to imply real or complex multiplication. □

Theorem 5.2. Let t ∈ ℚ be such that ν_{17}(t − 1) > 0 and ν_{23}(t − 1) > 0. Then the K3 surface X^{(2,t)} has geometric Picard rank 16 and real multiplication by ℚ(\sqrt{2}).

Proof. We will prove #X_{p}^{(2,t)}(𝔽_{p}) ≡ 1 (mod p) for all primes p ≡ 3, 5 (mod 8), p > 3, in Theorem 5.3, below. By Lemma 5.1, this guarantees that X^{(2,t)} has real or complex multiplication by a number field E.

Further, all the surfaces X^{(2,t)} considered coincide modulo 17 and modulo 23, these two primes being good. Counting points, one finds #X_{17}^{(2,t)}(𝔽_{17^{i}}) = 313, 83 881, and 24 160 345, as well as #X_{23}^{(2,t)}(𝔽_{23^{i}}) = 547, 280 729, and 148 114 771, for i = 1, 2, 3. The characteristic polynomials of Frob_{17} and Frob_{23} turn out to be

χ_{17}^{tr}(Z) = Z^{4} + 28Z^{3} + 646Z^{2} + 8092Z + 83521  and
χ_{23}^{tr}(Z) = Z^{4} + 52Z^{3} + 1702Z^{2} + 27508Z + 279841 ,

both being irreducible. In particular, rk Pic(X_{\overline{𝔽}_{17}}^{(2,t)}) = rk Pic(X_{\overline{𝔽}_{23}}^{(2,t)}) = 18. Applications of the Artin-Tate formula [29, Theorem 6.1] show

disc Pic(X_{\overline{𝔽}_{17}}^{(2,t)}) ∈ (2 mod (ℚ^{∗})^{2}) and disc Pic(X_{\overline{𝔽}_{23}}^{(2,t)}) ∈ (14 mod (ℚ^{∗})^{2}) .

From this information, one deduces that rk Pic(X(2,t) ; \overline{ℚ}) = 16 or 17. If the rank was 17 then [4, Theorem 1, together with Remark 2] shows that

rk Pic(X(2,t) ; \overline{ℚ}) ≤ rk Pic(X_{\overline{𝔽}_{17}}^{(2,t)}) − [E : ℚ] ,

a contradiction as the right hand side is at most 16.

Our next assertion is that [E : ℚ] = 2. As dim T = 6, the potential alternative degrees would be 3 or 6. In the first case, E is certainly totally real. In the second case, it must be CM. In both cases, there is a totally real, cubic number field E′, contained in End(T).

For l a prime that is inert in E′, T_{l} carries the structure of a vector space over the field E′⊗_{ℚ} ℚ_{l}. Further, there is a constant f such that (Frob_{p})^{f} is an E′⊗_{ℚ} ℚ_{l}-linear map, for every prime p ≠ l. This, however, implies that the number of eigenvalues of (Frob_{p})^{f}, considered as a ℚ_{l}-linear map, that are roots of unity multiplied by p, is a multiple of 3. The calculations shown above for p = 17 and p = 23 clearly disagree with that.

It remains to determine the quadratic number field E exactly. For this, an easy computation reveals that the Galois group of

χ_{17}^{tr}(Z) = Z^{4} + 28Z^{3} + 646Z^{2} + 8092Z + 83521

is cyclic of order four. In particular, variant i) of Theorem 3.5 applies, showing that χ_{17}^{tr} splits over E into two conjugate factors. But ℚ(\sqrt{disc χ_{17}^{tr}}) is the only quadratic subfield of the splitting field of χ_{17}^{tr}. A direct calculation yields, finally, that disc χ_{17}^{tr} = 2^{29}·17^{6}. □

Theorem 5.3 (The point count). Let 𝔽_{q} be a finite field of characteristic ≠ 2, 3 such that 2 is a non-square in 𝔽_{q} and V the singular surface given by w^{2} = q_{1}(y, z)q_{2}(x, z)q_{3}(x, y), for t ∈ 𝔽_{q} and

q_{1}(y, z) := (\frac{1}{8}t^{2}− \frac{1}{2}t+ \frac{1}{4})y^{2} + (t^{2}−2t+2)yz + (t^{2}−4t+2)z^{2} ,
q_{2}(x, z) := (\frac{1}{8}t^{2}+ \frac{1}{2}t+ \frac{1}{4})x^{2} + (t^{2}+2t+2)xz + (t^{2}+4t+2)z^{2} ,
q_{3}(x, y) := (x + y)(2x + t^{2}y) .

Suppose that t ≠ 0 and t^{2} ≠ −2. Then #V (𝔽_{q}) = q^{2} + q + 1.

Proof. We will prove this result in several steps. First step. Preparations. We will count fiber-wise using the fibration, given by y : x = l, for l ∈ P^{1}(𝔽_{q}). This will yield a result by q too large, as the point lying over (0 : 0 : 1) will be counted (q + 1) times.

The fiber V_{l} is the curve, given by w^{2} = (1+l)(2+t^{2}l)x^{2}q_{1}(lx, z)q_{2}(x, z). A partial resolution is provided by C_{l}: w^{2} = (1 + l)(2 + t^{2}l)q_{1}(lx, z)q_{2}(x, z), which defines an elliptic fibration.

We claim that ∑_{l} #C_{l}(𝔽_{q}) = ∑_{l} #V_{l}(𝔽_{q}). Indeed, the two fibrations differ only over the line “x = 0”. Since V ramifies over this line, V_{x} has exactly (q + 1) points. On the other hand, the curve C_{x} is given by w^{2} = (t^{4} −12t^{2} +4)·(1 + l)(2 + t^{2}l). Here, the constant t^{4} − 12t^{2} + 4 is non-zero, as 32 is not a square. Thus, C_{x} is a double cover of P^{1}, ramified at (−1) and (−\frac{2}{t^{2}} ). But −1 ≠ −\frac{2}{t^{2}} , since 2 is a non-square. In other words, C_{x} is a conic, which has exactly (q + 1) points. Second step. Singular fibers. There are four singular fibers, at l = −1, −\frac{2}{t^{2}} , 0, and ∞. In fact, for the first two, the coefficient is zero, while, for the others, one of the quadratic forms has a double zero. We claim that these are the only singular 𝔽_{q}-rational fibers.

To see this, we first observe that q_{1} is of discriminant

(t^{2}−2t+2)^{2}−4(\frac{1}{8}t^{2}− \frac{1}{2}t+ \frac{1}{4})(t^{2}−4t+2) = (t^{2}+2t+2)^{2}− \frac{1}{2}(t^{2}+4t+2)^{2} = \frac{1}{2}(t^{2}−2)^{2}

and the same for q_{2}. This term does not vanish, for any value of t. Therefore, q_{1} and q_{2} always define two lines each, never a double line. Consequently, for l ≠ 0, ∞, neither of the two quadratic factors q_{1}(lx, z) and q_{2}(x, z) may have a double zero.

To exclude a common zero, one has to compute the resultant, which turns out to be

\frac{1}{64}(t^{4} − 12t^{2} + 4)^{2}(l^{2} + \frac{−6t^{4}+8t^{2}−24}{t^{4}−12t^{2}+4}l + 1)(l^{2} + \frac{−2t^{4}−8t^{2}−8}{t^{4}−12t^{2}+4}l + 1) .

Here, t^{4} − 12t^{2} + 4 ≠ 0, as 32 is a non-square. Further, the quadratic polynomials in l are of the discriminants \frac{32(t^{2}−2)^{2}(t^{2}+2)^{2}}{(t^{4}−12t^{2}+4)^{2}} and \frac{128t^{2}(t^{2}−2)^{2}}{(t^{4}−12t^{2}+4)^{2}} , which are non-squares in 𝔽_{q}, because of t ≠ 0 and t^{2} ≠ ±2. Thus, the resultant does not vanish for any value of l, as long as t is admissible. Third step. Points on the singular fibers. The curves C_{−1} and C_{−\frac{2}{t^{2}}} are part of the ramification locus and therefore degenerate to lines. They have (q + 1) points each.

On the other hand, the fibers C_{0} and C_{∞} are given by w^{2} = 2(t^{2}−4t+2)z^{2}q_{2}(x, z) and w^{2} = t^{2}(t^{2}+4t+2)z^{2}q_{1}(y, z). Both are conics with the points over z = 0 unified into a double point. The corresponding points on the non-singular conics C_{0}^{ns} and C_{∞}^{ns} satisfy w^{2} = \frac{1}{4}(t^{4} − 12t^{2} + 4), and w^{2} = \frac{t^{2}}{8} (t^{4} −12t^{2} +4), respectively. The two equations differ by a factor of \frac{t^{2}}{2} , which is a non-square. Hence, one of the curves C_{0}^{ns} and C_{∞}^{ns} has two points such that z = 0, the other none. Accordingly, one of the singular curves C_{0} and C_{∞} has q points, the other (q + 2).

It therefore remains to show that ∑_{l,C_{l} smooth} #C_{l}(𝔽_{q}) = (q − 3)(q + 1). Fourth step. The classical invariants c_{4} and c_{6}. The invariants c_{4} and c_{6} of the family of binary quartic forms defining C are polynomials in l and t. They may easily be written down, but the formulas become quite lengthy. The discriminant Δ turns out to be

Δ = \frac{1}{1024}t^{12}(t^{2} − 2)^{4}(t^{4} − 12t^{2} + 4)^{4}l^{2}(l + \frac{2}{t^{2}} )^{6}(l + 1)^{6}

(l^{2} + \frac{−6t^{4}+8t^{2}−24}{t^{4}−12t^{2}+4}l + 1)^{2}(l^{2} + \frac{−2t^{4}−8t^{2}−8}{t^{4}−12t^{2}+4}l + 1)^{2} .

The arguments given in the second step show that Δ ≠ 0, except for l = −1, −\frac{2}{t^{2}} , 0, and ∞.

By Lemma 5.6, I_{l}: w^{2} = x^{3} − 27c_{4}(l)x − 54c_{6}(l) is isomorphic to the Jacobian Jac C_{l}, for l ≠ −1, −\frac{2}{t^{2}} , 0, ∞. This implies #C_{l}(𝔽_{q}) = #(Jac C_{l})(𝔽_{q}) = #I_{l}(𝔽_{q}), since genus-one curves over finite fields always have points.

We have to prove that ∑_{l,C_{l} smooth} #I_{l}(𝔽_{q}) = (q − 3)(q + 1). I.e., that the (q−3) smooth fibers of I have, on average, exactly (q + 1) points. Fifth step. l versus \frac{1}{l}. For the j-invariant j = \frac{c_{4}^{3}}{Δ} , one observes that j(\frac{1}{l}) = j(l). More precisely,

c_{4}(\frac{1}{l}) = K^{2}c_{4}(l) and c_{6}(\frac{1}{l}) = K^{3}c_{6}(l) ,

for K := \frac{2l+t^{2}}{l^{4}(t^{2}l+2)}.

In other words, the elliptic curves I_{l} and I_{\frac{1}{l}} are geometrically isomorphic to each other. They are quadratic twists, according to the extension 𝔽_{q}(\sqrt{K})/𝔽_{q}. Consequently, if \frac{2l+t^{2}}{t^{2}l+2} ∈ 𝔽_{q} is a non-square then I_{l} and I_{\frac{1}{l}} together have exactly 2(q + 1) points. Sixth step. Reparametrization. We reparametrize according to the Möbius transformation P^{1} →P^{1}, l ↦ s:= \frac{2l+t^{2}}{t^{2}l+2} . This is not a constant map, for any value of t. Indeed, the determinant of the corresponding 2 × 2-matrix is 4 − t^{4} = (2 − t^{2})(2 + t^{2}) ≠ 0. The inverse transformation is given by s ↦ l := \frac{−2s+t^{2}}{t^{2}s−2} .

Write I′ for the fibration, defined by I′_{s} := I_{l}. Then the bad fibers are located at s = −1, ∞, \frac{t^{2}}{2} , \frac{2}{t^{2}} . The correspondence l ↦ \frac{1}{l} goes over into

s = \frac{2l+t^{2}}{t^{2}l+2} ↦ \frac{\frac{2}{l} +t^{2}}{\frac{t^{2}}{l} +2} = \frac{2+t^{2}l}{t^{2}+2l} = \frac{1}{s} .

Thus, for s ≠ −1, \frac{t^{2}}{2} , \frac{2}{t^{2}} ∈ 𝔽_{q}^{∗} a non-square, the fibers I′_{s} and I′_{\frac{1}{s}} together have exactly 2(q + 1) points. Seventh step. Pairing the squares I. It remains to consider the fibers for s ∈ 𝔽_{q}^{∗}, s ≠ −1, a square and for s = 0. For these, I′_{s} ≅ I′_{\frac{1}{s}}, except for s = 0. There are 4n + 2 such fibers, for q = 8n + 3 as well as for q = 8n + 5.

It turns out that j′(s_{2}) = j′(s_{1}), for s_{1} = a^{2} and s_{2} = \frac{(a−1)^{2}}{(a+1)^{2}} . More precisely,

c′_{4}(s_{2}) = F^{2}c′_{4}(s_{1}) and c′_{6}(s_{2}) = F^{3}c′_{6}(s_{1}) ,

for

F := 8 \frac{(a+1)^{2}(a^{2}− \frac{2}{t^{2}} )^{4}}{(a^{2}− \frac{2t^{2}+4}{t^{2}−2} a+1)^{4}} .

We observe here that the denominator never vanishes for t ≠ 0, In fact, the discriminant of the quadratic polynomial is equal to \frac{32t^{2}}{(t^{2}−2)^{2}} , which is always a non-square.

As 8 is a non-square, we see that F is a non-square as long as F ≠ 0, which happens to be true for a ≠ −1.

In other words, for a ≠ −1, the elliptic curves I′_{s_{1}} and I′_{s_{2}} are non-trivial quadratic twists of each other. This shows that #I′_{s_{1}}(𝔽_{q}) + #I′_{s_{2}}(𝔽_{q}) = 2(q + 1). Eighth step. Pairing the squares II. In particular, we have #I′_{1}(𝔽_{q}) + #I′_{0}(𝔽_{q}) = 2(q + 1). For the other 4n fibers, we argue as follows. The group V := ℤ/2ℤ × ℤ/2ℤ operates on P^{1}(𝔽_{q}) via e_{1}·a := −a and e_{2}·a := \frac{1}{a}. The orbits are of size four, except for \{0, ∞\}, \{1, −1\}, and, possibly, \{i, −i\}. The map I: P^{1}(𝔽_{q}) → P^{1}(𝔽_{q}), a ↦ \frac{a−1}{a+1}, is compatible with the operation of V in the sense that e_{1}·I(a) = I(e_{2}·a) and e_{2}·I(a) = I(e_{1}·a).

Therefore, I defines a mapping \overline{I}: P^{1}(𝔽_{q})/V → P^{1}(𝔽_{q})/V from the orbit set to itself. One easily sees that I(I(a)) = e_{1}e_{2} ·a. I.e., \overline{I} is actually an involution. Solving the equations \frac{a−1}{a+1} = ±a and \frac{a−1}{a+1} = ±\frac{1}{a}, utilizing the fact that 2 is a nonsquare, we find that I has no fixed points, except for the possible orbit {i, −i}.

Accordingly, J: a^{2} ↦ (\frac{a−1}{a+1})^{2} defines an involution of the squares in P^{1}(𝔽_{q}) modulo the equivalence relation generated by x ∼ \frac{1}{x}. The only possible fixed point of J is {−1}. Further, J({0, ∞}) = {1}.

As a consequence, we see that the squares x ∈ 𝔽_{q}^{∗}, different from ±1, decompose into sets \{a^{2}, \frac{1}{a^{2}} , (\frac{a−1}{a+1})^{2}, (\frac{a+1}{a−1})^{2}\} of exactly four elements. The assertion follows immediately from this. □

Remark 5.4. If t^{2} = −2 then the same result is true. For t = 0, however, one has #V (𝔽_{q}) = q^{2} + 2q + 1, while, for t = ∞, #V (𝔽_{q}) = q^{2} + 1. Only minor modifications of the argument are necessary. The case t^{2} = −2 is actually simpler, as then K = −1 is constant and easily seen to be a non-square. In each case, there are exactly four singular 𝔽_{q}-rational fibers.

Remarks 5.5. i) Elliptic K3 surfaces generally have 24 singular fibers. In our case, I_{−1} and I_{−\frac{2}{t^{2}}} each have have multiplicity six, while the other six singular fibers, four of which are defined only over 𝔽_{q^{2}}, each have multiplicity two. ii) The symmetry under l ↔ \frac{1}{l} is enforced by the construction. In fact, consider the double cover of P^{2}, branched over the union of the four lines z = a_{1}x, z = a_{2}x, z = b_{1}y, and z = b_{2}y. The fiber for y: x = l has branch points at a_{1}, a_{2}, b_{1}l, b_{2}l, which is a quadruple projectively equivalent to a_{1}, a_{2}, \frac{Kb_{1}}{l} , \frac{Kb_{2}}{l}, for K := \frac{a_{1}a_{2}}{b_{1}b_{2}} . For our fibration, independently of the parameter t, we have K = \frac{q_{2}(1,0)}{q_{2}(0,1)} : \frac{q_{1}(1,0)}{q_{1}(0,1)} = 1.

The twist factor is q_{3}(1, \frac{K}{l})/q_{3}(1, l), which would be fractional-quadratic, in general, is fractional-linear in our case. iii) We found the second symmetry, which allowed us to pair the squares, by looking at the factorizations of the rational functions j(l) − C. It seems to be very specific for the particular fibrations, occurring in the proof of Theorem 5.3.

Lemma 5.6. Let C: w^{2} = F_{4}(x, y, l) be a family of smooth genus-one curves, parametrized by l ∈ B, for B an integral scheme in characteristic ≠ 2 or 3, c_{4}(l) and c_{6}(l) its classical invariants, and Δ := \frac{c_{4}^{3}(l)−c_{6}^{2}(l)}{1728}. Then, over the open subscheme D(Δ) ⊆ B,

I: w^{2} = x^{3} − 27c_{4}(l)x − 54c_{6}(l)

defines a family of elliptic curves, fiber-wise isomorphic to the relative Jacobian of C.

Proof. The existence of the relative Jacobian 𝒥 follows from [16, Exposé 232, Théorème 3.1]. This is a family of elliptic curves. I is a family of elliptic curves, too, as −16[4(−27c_{4}(l))^{3} + 27(−54c_{6}(l))^{2}] = 6^{12}Δ(l) ≠ 0.

Further, the generic fiber I_{η} is isomorphic to the Jacobian of C_{η} [15, Proposition 2.3]. Thus, over D(Δ), we have two families of elliptic curves that coincide over the generic point η ∈ D(Δ). The assertion follows from this, since the moduli stack of elliptic curves is separated [22, First main Theorem 5.1.1, together with 2.2.11].

□

Appendix A. Dimension counting

Proposition A.1. Let T be a ℚ-vector space of dimension six, equipped with a non-degenerate symmetric, bilinear pairing ⟨. , .⟩: T × T → ℚ of discriminant (1 mod (ℚ^{∗})^{2}) and φ: T → T be a self-adjoint endomorphism such that φ∘φ = [d]. Then d ∈ ℚ is a sum of two rational squares.

Proof. Assume the contrary. Then, in particular, d is a non-square. The assumptions on φ imply that φ_{ℚ(\sqrt{d})} is diagonalizable. For the eigenvalues ±\sqrt{d}, the eigenspaces, which we will denote by T_{+} and T_{−}, both must be three-dimensional. As φ is self-adjoint, they are perpendicular to each other.

In particular, the pairings ⟨. , .⟩|_{T_{+}} and ⟨. , .⟩|_{T_{−}} are non-degenerate, too. We may choose an orthogonal system {x_{1}, x_{2}, x_{3}} ⊂ T_{+} such that ⟨x_{i}, x_{i}⟩ =: a_{i} ≠ 0, for i = 1, 2, 3. Then the real conjugates x′_{1}, x′_{2}, x′_{3} ∈ T_{−} also form an orthogonal system, and one has ⟨x′_{i}, x′_{i}⟩ = a′_{i} ≠ 0.

From this, one finds an orthogonal decomposition T = T_{1} ⊕ T_{2} ⊕ T_{3}, defined over ℚ, when putting

T_{i} := span (x_{i} + x′_{i}, \sqrt{d} (x_{i} − x′_{i})) .

The discriminant of T_{i} is in the class of

det(\begin{matrix}a_{i} + a′_{i} & \sqrt{d} (a_{i} − a′_{i}) \\ \sqrt{d} (a_{i} − a′_{i}) & d(a_{i} + a′_{i}) \end{matrix})= d[(a_{i}+a′_{i})^{2}−(a_{i}−a′_{i})^{2}] = 4da_{i}a′_{i} = 4dN_{ℚ(\sqrt{d})/ℚ} (a_{i})

modulo squares. Consequently, disc T = ((4d)^{3}N_{ℚ(\sqrt{d})/ℚ} (a_{1}a_{2}a_{3}) mod (ℚ^{∗})^{2}). By our assumption about disc T, this implies that d is a norm from ℚ(\sqrt{d}).

As (−d) is clearly a norm, we conclude that (−1) must be a norm from ℚ(\sqrt{d}), too. I.e., −1 = a^{2} − db^{2} for suitable a, b ∈ ℚ, such that d is a sum of two squares.

□

Remark A.2 (cf. [17, Example 3.4]). Suppose T ≅ ℚ^{6} and that ⟨. , .⟩ is the bilinear form defined by the matrix diag(1, 1, −1, −1, −1, −1). Then, for every d ∈ ℚ being a sum of two squares, there exists a self-adjoint endomorphism φ: T → T such that φ ∘ φ = [d].

Indeed, decompose T orthogonally as ℚ^{2} ⊕ ℚ^{2} ⊕ ℚ^{2} such that, on each summand, the bilinear form is given by either diag(1, 1) or diag(−1, −1). Then define φ component-wise by taking the matrix ({u v \atop v −u}), for d = u^{2}+v^{2}, three times. The symmetry of the matrix implies that φ is self-adjoint and φ ∘ φ = [d] is a obvious.

Theorem A.3. Let d ∈ ℚ be a non-square.

i) If d is not a sum of two squares then there is no weight-2 Hodge structure of dimension six, having a polarization of discriminant (1 mod (ℚ^{∗})^{2}) and an endomorphism algebra containing ℚ(\sqrt{d}). ii) Suppose that d is a sum of two squares. Then there exists a one-dimensional family of polarized, six-dimensional weight-2 Hodge structures of K3 type, having the underlying quadratic space (ℚ^{6}, diag(1, 1, −1, −1, −1, −1)) and real multiplication by ℚ(\sqrt{d}).

Proof. i) follows immediately from Proposition A.1. Cf. [42, Theorem 1.6.a) and Theorem 1.5.1].

ii) To convert T := (ℚ^{6}, diag(1, 1, −1, −1, −1, −1)) into a weight-2 Hodge structure of K3 type, one has to select a one-dimensional isotropic subspace H^{2,0} ⊂ T_{ℂ} such that \overline{H^{2,0}} is not perpendicular to H^{2,0}. This will automatically fix H^{0,2} := \overline{H^{2,0}} and H^{1,1} := (H^{2,0} + \overline{H^{2,0}})^{⊥}.

In addition, we choose the endomorphism φ: T → T constructed in Remark A.2. By construction, φ_{ℂ} commutes with complex conjugation on T_{ℂ} . Furthermore, as φ_{ℂ} is self-adjoint and fulfills φ_{ℂ} ∘ φ_{ℂ} = [d], it respects orthogonality. Therefore, φ_{ℂ} (H^{2,0}) ⊆ H^{2,0} alone will be sufficient for φ to cause real multiplication.

To ensure this, let us take H^{2,0} ⊂ T_{ℂ,+} . The eigenspace T_{ℂ,+} has a real basis, given by e_{i} − \frac{u−\sqrt{d}}{v}e_{i+1}, for i = 1, 3, 5. In this basis, the pairing ⟨. , .⟩|_{T_{ℂ,+}} is given by the non-degenerate matrix diag(1 + (\frac{u−\sqrt{d}}{v})^{2}, −1 − (\frac{u−\sqrt{d}}{v})^{2}, −1 − (\frac{u−\sqrt{d}}{v})^{2}), which is indefinite. Consequently, on P(T_{ℂ,+} ) ≅ P^{2}, the condition ⟨x, x⟩ = 0 defines a conic C and, on this conic, ⟨x, \overline{x}⟩ ≠ 0 is fulfilled on a dense open subset. □

Remark A.4. Consider the four-dimensional family of the K3 surfaces that are given as desingularizations of the double covers of P^{2}, branched over the union of six lines. In this case, rk Pic(𝔛) ≥ 16 and we are particularly interested in those surfaces, for which equality occurs.

Anyway, the pull-back of a general line and the 15 exceptional curves generate a sub-Hodge structure P′ of dimension 16. The symmetric, bilinear form on P′ is given by the matrix diag(2, −2, …, −2). Indeed, the exceptional curves have self-intersection number (−2) [2, Proposition VIII.13.i)]. According to [32, Ch. IV, Theorem 9], P′ ≅ (ℚ^{16}, diag(1, −1, …, −1)).

Corollary A.5. Let d ∈ ℚ be a non-square being the sum of two squares. Then there exists a one-dimensional family of K3 surfaces over ℂ, the generic member of which has Picard rank 16 and real multiplication by ℚ(\sqrt{d}).

Proof. As a quadratic space, H = H^{2}(𝔛, ℚ) is the same for all K3 surfaces. One has H ≅ (ℚ^{22}, diag(1, 1, 1, −1, …, −1)). By [1, Corollary 14.2], cf. [37, Ch. IX, Theorem 4], there exists a complex-analytic K3 surface 𝔛 for every choice of a onedimensional subspace span(x) ⊂ H_{ℂ} fulfilling ⟨x, x⟩ = 0 and ⟨x, \overline{x}⟩ > 0.

We choose P′ ⊂ H_{ℂ} as in Remark A.4, put T′ := (P′)^{⊥}, and restrict considerations to subspaces span(x) ⊂ T′ ⊂ H_{ℂ} . By the classification of the quadratic forms over ℚ [32, Ch. IV, §3], we have T′ ≅ (ℚ^{6}, diag(1, 1, −1, −1, −1, −1)).

Therefore, Theorem A.3 guarantees the existence of a one-dimensional family of subspaces span(x) ⊂ T′ such that ⟨x, x⟩ = 0 and ⟨x, \overline{x}⟩ ≠ 0. The construction given shows that the first condition actually defines a conic C and that ⟨x, \overline{x}⟩ > 0 is satisfied on a non-empty open subset of C.

We still have to show that, generically, rk Pic(𝔛) = 16. For this observe, by the Lefschetz theorem on (1, 1)-classes, Picard rank 16 is equivalent to ℚ^{6} ∩ H^{1,1} = 0.

To investigate this condition, let 0 ≠ v = (v_{1}, …, v_{6}) ∈ ℚ^{6} be any vector. The inclusion v ∈ H^{1,1}, for a particular choice of x, implies that v ∈ span(x)^{⊥}. I.e.,

v_{1}x_{1} + v_{2}x_{2} − v_{3}x_{3} − … − v_{6}x_{6} = 0 .

This hyperplane meets the conic C in at most two points. Indeed, the plane P(T_{ℂ,+} ) is not contained in any ℚ-rational hyperplane, as an inspection of the base vectors given above immediately shows. In total, there are only countably many exceptions, for which rk Pic(𝔛) > 16. □

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School of Mathematics and Statistics F07, University of Sydney, NSW 2006, Sydney,

Australia, http://www.staff.uni-bayreuth.de/∼bt270951/
   E-mail address: stephan@maths.usyd.edu.au

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

Germany, http://wwwuser.gwdg.de/~jjahnel
   E-mail address: ~jjahnel@mathematik.uni-siegen.de