ON THE DISTRIBUTION OF THE PICARD RANKS OF THE REDUCTIONS OF A K3 SURFACE

EDGAR COSTA^{}, ANDREAS-STEPHAN ELSENHANS, AND JÖRG JAHNEL

Abstract. We report on our results concerning the distribution of the geometric Picard ranks of K3 surfaces under reduction modulo various primes. In the situation that rk Pic S\overline{K} is even, we introduce a quadratic character, called the jump character, such that rk Pic S_{\overline{𝔽}_{𝔭}} rk Pic S\overline{K} for all good primes at which the character evaluates to 1 .

1. Introduction

Let S be a K3 surface over a number field K. It is a well-known fact that the geometric Picard rank of S may not decrease under reduction modulo a good prime 𝔭 of S. I.e., one always has

rk Pic S_{\overline{𝔽}_{𝔭}} rk Pic S\overline{K} .  (1)

It would certainly be interesting to understand the sequence rk Pic S_{\overline{𝔽}_{𝔭}} _{𝔭}, or at least the set of jump primes

Π_{jump} S : 𝔭 prime of K 𝔭 good for S, rk Pic S_{\overline{𝔽}_{𝔭}} rk Pic S\overline{K} ,

for a given surface. In an ideal case, one would be able to give a precise reason why the geometric Picard rank jumps at a given good prime. There are two well-known such reasons.

i) According to the Tate conjecture [KM], the left hand side is always even. Thus, in the case that rk Pic S\overline{K} is odd, inequality (1) is always strict and every good prime is a jump prime. ii) Generalising this, if S has real multiplication (RM) by an endomorphism field E and the integer 22 rk Pic S\overline{K} E : ℚ is odd, then again every good prime is a jump prime [Ch14, Theorem 1(2)]. It is known due to F. Charles [Ch14, Theorem 1] that these are the only cases in which every good prime is a jump prime.

The first author was partially supported by the Simons Collaboration Grant #550033.

Date: June 1, 2020.

2010 Mathematics Subject Classification. Primary 14J28; Secondary 14F20, 11G35, 14G25.

Key words and phrases. Characteristic polynomial of the Frobenius, functional equation, K3 surface, Picard rank.

Acknowledgements. The authors would like to thank J. Voight (Dartmouth College), who alerted them of an error in early version of this article, and the anonymous referees, whose comments turned out to be extremely helpful.

In this article, we describe a third reason for a prime to jump, the jump character. It was observed experimentally by the first author together with Yu. Tschinkel [CT] that, in the even rank case, one seems to have lim inf_{B} γ S, B \frac{1}{2}, for

γ S, B : \frac{# 𝔭 Π_{jump} S 𝔭 B}{# 𝔭 B}.

Theorem A shows that this observation is indeed true, except for some corner cases, which are simple to describe. Moreover, Π_{jump} S contains an entirely regular subset of exact density one half.

In some sense, this result is complementary to those of F. Charles [Ch14]. In fact, [Ch14, Theorem 1] shows, except for the cases i) and ii), that there are always infinitely many non-jump primes.

Theorem A (Theorem 2.4.4). Let K be a number field and S a K3 surface over K. Moreover, let 𝔭 𝒪_{K} be a prime of good reduction and residue characteristic 2.

a) Then there are two quantities, the discriminant Δ_{H^{2}} S of S and the discriminant of the Picard representation or algebraic part of the discriminant Δ_{Pic} S of S (cf. 2.2.2.ii) and 2.3.2.a.ii) for precise definitions), such that the following equations hold,

det Frob_{𝔭}: H_{ét}^{2} S\overline{K}, ℚ_{l} 1  \frac{Δ_{H^{2}} S}{𝔭}  and
det Frob_{𝔭}: T  \frac{Δ_{H^{2}} S Δ_{Pic} S}{𝔭} .

Here, T : H_{alg}^{}H_{ét}^{2} S\overline{K}, ℚ_{l} 1 denotes the transcendental part of the cohomology, and \frac{.}{𝔭} is the quadratic residue symbol modulo 𝔭 [Ne, Chapter V, §3].

b) If rk Pic S\overline{K} is even then

\frac{Δ_{H^{2}} S Δ_{Pic} S}{𝔭}  1  rk Pic S_{\overline{𝔽}_{𝔭}} rk Pic S\overline{K} 2 .  (2)

In other words, if K \overline{Δ_{H^{2}} S Δ_{Pic} S} K is indeed a quadratic extension then

𝔭 𝔭 inert in K \overline{Δ_{H^{2}} S Δ_{Pic} S} K Π_{jump} S .

The quadratic character τ_{S}, given by

𝔭 \frac{Δ_{H^{2}} S Δ_{Pic} S}{𝔭}

might be called the transcendental character of the K3 surface S. Nevertheless, having implication (2) in mind, we prefer to call it the jump character of S, at least in the even rank case. It may happen that the quadratic extension, and hence the jump character, are trivial. We provide particular surfaces of this kind, defined over ℚ, in Examples 2.6.10.a) and 2.6.12. These are the corner cases mentioned above.

One might think about the jump character also as follows. The ℚ_{l}-vector space T is equipped with the non-degenerate cup product pairing and acted upon by Gal \overline{K} K . Moreover, the action is orthogonal with respect to the pairing, so that one has a continuous group homomorphism

τ: Gal \overline{K} K O T .

There is, however, no reason for im τ to be contained in SO T , in general, so the group homomorphism det τ: Gal \overline{K} K 1, 1 is usually non-trivial. Moreover, det τ turns out to be independent of l, cf. Proposition 2.2.1.b). The article thus in essence describes the effects of det τ being non-trivial.

The criterion for non-triviality. Due to its construction, the jump character is unramified at every prime of good reduction, cf. Corollary 2.5.7.a). On the other hand, it may ramify at bad primes. We show this to be always the case when the singular reduction is of the mildest possible type.

Theorem B (Corollary 2.5.7.b)). Let K be a number field and S a K3 surface over K. Moreover, let 𝔭 𝒪_{K} be a prime of residue characteristic different from 2. Suppose that S has a regular, projective model \underline{S} over 𝒪_{K,𝔭}, the geometric fibre \underline{S}\overline{𝔭} of which has exactly one singular point, and assume this to be an ordinary double point. Then the jump character τ_{S} \frac{Δ_{H^{2}} S Δ_{Pic} S}{}ramifies at 𝔭.

In addition, we present algorithms to compute the two characters for a given surface S over ℚ, a deterministic one for Δ_{H^{2}} S and a statistical one for the jump character. We can deterministically compute the jump character in two particular situations, which together cover many, but not all examples. These situations occur when Corollary 2.5.7.b) applies to at least a single bad prime, and when Pic S\overline{ℚ} is known as a Galois module to such an extent that ^{max} NS S\overline{K} _{ℤ} ℚ can be determined. When neither of these circumstances occurs, the statistical algorithm may still be used.

An application. Rational curves on even rank K3 surfaces. It has been a long standing conjecture that every K3 surface over an algebraically closed field contains infinitely many rational curves. In full generality, it has been settled only recently by X. Chen, F. Gounelas, and C. Liedtke [CGL]. As an application of our results, we show that the existence of infinitely many rational curves may be obtained rather easily in the situation that the jump character is non-trivial and the surface is otherwise generic.

Theorem C (Theorem 3.1). Let K be a number field and S a K3 surface over K. Assume that rk Pic S\overline{K} is even, that S\overline{K} has neither real nor complex multiplication, and that Δ_{H^{2}} S Δ_{Pic} S is a non-square in K. Then S\overline{K} contains infinitely many rational curves.

Notation. i) Abstract Algebra, Algebraic Geometry, and Algebraic Number Theory. For K a field, we write \overline{K} for a fixed algebraic closure of K. When K is a number field then by 𝒪_{K} we denote the ring of algebraic integers in K. For 𝔭 𝒪_{K} a prime ideal, we let K_{𝔭} be the completion of K with respect to 𝔭, 𝒪_{K,𝔭} the ring of integers in K_{𝔭}, and 𝔽_{𝔭} the residue field.

Moreover, we always let Frob Gal \overline{𝔽}_{q} 𝔽_{q} be the geometric Frobenius automorphism x x^{1 q}, cf. [DeWI, (1.15)]. Similarly, when K is a number field and 𝔭 𝒪_{K} a prime ideal, Frob_{𝔭} Gal \overline{K} K denotes an arbitrary lift of the geometric Frobenius. We write Frob and Frob_{𝔭}, too, for the automorphisms of schemes and their cohomology groups, induced by Frob, respectively Frob_{𝔭}, via functoriality. Let us note that Frob: P_{𝔽_{q}}^{N} P_{𝔽_{q}}^{N} maps the geometric point x_{0} : : x_{N} to x_{0}^{q} : : x_{N}^{q} .

Perhaps deviating from a certain standard, we say that a proper variety S over K has good reduction at 𝔭 when there exists a proper model of S over 𝒪_{K} that has good reduction at 𝔭 in the usual sense. ii) Quadratic extensions. Let K be a field of characteristic 2 and L K an at most quadratic field extension. Then, according to Kummer theory, there exists a unique class Δ_{L} K^{}K^{}^{2} such that L K \overline{u} for any u Δ_{L}. In this situation, we shall also write K \overline{Δ_{L}} for K \overline{u} .

Assume that K is a number field and 𝔭 𝒪_{K} a prime ideal of residue characteristic 2, at which L K is unramified. Then the quadratic residue symbol \frac{u}{𝔭} [Ne, Chapter V, Proposition (3.5)] is independent of the choice of a 𝔭-adic unit u Δ_{L}. We will therefore write \frac{Δ_{L}}{𝔭} instead of \frac{u}{𝔭} . iii) Characters. By a character, we always mean a continuous homomorphism from a topological group to a discrete abelian group. A quadratic character is a character to 1, 1 .

We often describe a quadratic character χ: Gal \overline{K} K 1, 1 , when K is a number field, in the form

𝔭 \frac{Δ}{𝔭} ,

or simply \frac{Δ}{}, for Δ K^{}K^{}^{2}. This is supposed to mean that χ Frob_{𝔭} \frac{Δ}{𝔭} for every prime ideal 𝔭 𝒪_{K} of residue characteristic 2 of the kind that Δ is representable by a 𝔭-adic unit. Note that, 1, 1 being abelian, χ Frob_{𝔭} is well defined for every quadratic character. Moreover, the values at the Frobenii Frob_{𝔭} determine χ uniquely, due to the Chebotarev density theorem.

When V is a one-dimensional vector space over a field F, equipped with the discrete topology and acted upon continuously by a topological group G, we denote by V : G F^{}the character given by V g a, for a F^{}the scalar satisfying g v av for every v V. This notation is due to T. Saito [Sa].

Computations. All computations are done using magma [BCP], sage [St], and C++, including the libraries FLINT [HJP] and NTL [Sh]. For point counting on the examples being quartic surfaces, we used the software developed by the first author, which is publicly

available at https://github.com/edgarcosta/controlledreduction.

2. The jump character

2.1. The determinant of Frob and the relationship with the sign in the functional equation. Let S be a smooth, proper variety over a finite field 𝔽_{q} of characteristic p 0. Then Frob acts linearly on the l-adic cohomology modules H_{ét}^{i} S_{\overline{𝔽}_{q}}, ℤ_{l} j . The characteristic polynomial Φ_{j}^{i} of Frob is independent of the choice of l p and has in fact rational coefficients [DeWI, Théorème (1.6)]. In particular, the determinant of Frob is a rational number and independent of l p.

In this section, we discuss the behaviour of det Frob. The theorem below, which is essentially a summary of results of P. Deligne and J. Suh, shows that each statement on det Frob may be translated into a statement about the sign in the functional equation.

Theorem 2.1.1 (Deligne, Suh). a) The polynomial Φ_{j}^{i} ℚ T fulfils the functional equation

T^{N}Φ q^{i 2j} T  q^{\frac{N}{2} i 2j} Φ T ,  (3)

for N : rk H_{ét}^{i} S_{\overline{𝔽}_{q}}, ℤ_{l} j .

b) The sign in the functional equation (3) is that of

det Frob: H_{ét}^{i} S_{\overline{𝔽}_{q}}, ℚ_{l} j  1 ^{N} det Frob: H_{ét}^{i} S_{\overline{𝔽}_{q}}, ℚ_{l} j  .

This is a rational number, the sign of which is independent of the Tate twist, i.e. of the choice of j. c.i) If i is even, then det Frob: H_{ét}^{i} S_{\overline{𝔽}_{q}}, ℚ_{l} i 2 is either 1 or 1 . In other words, the determinant gives the sign in (3) exactly. ii) If i is odd then N is even and in (3), the plus sign always holds.

Proof. a) and b) Let us write Φ for Φ_{j}^{i}. The polynomials on both sides of (3) then have the same roots as, with z, the number \overline{z} \frac{q^{i 2j}}{z} is a root of Φ, too [DeWII, Corollaire (3.3.9)], and has the same multiplicity. To show that they perfectly agree, let us adopt the convention that Φ is monic. Then the leading coefficient of the polynomial on the left hand side is equal to the constant term of Φ. By definition, this is the determinant of Frob on H_{ét}^{i} S_{\overline{𝔽}_{q}}, ℚ_{l} j , which is known to be a rational number and of absolute value q^{\frac{N}{2} i 2j} [DeWII, Corollaire (3.3.9)]. Thus, a) follows, together with the first assertion of b). The final claim is clear, since

det Frob: H_{ét}^{i} S_{\overline{𝔽}_{q}}, ℚ_{l} j q^{Nj} det Frob: H_{ét}^{i} S_{\overline{𝔽}_{q}}, ℚ_{l} .

c.i) As det Frob 1 ^{N}Φ 0 for Φ monic, this can be read off the functional equation T^{N}Φ 1 T Φ T . ii) If S is projective then, by Poincaré duality and the hard Lefschetz theorem [DeWII, Théorème (4.1.1)], there is a non-degenerate pairing

H_{ét}^{i} S_{\overline{𝔽}_{q}}, ℚ_{l} j H_{ét}^{i} S_{\overline{𝔽}_{q}}, ℚ_{l} j ℚ_{l} 2j i

that is compatible with the action of Frob. It is, moreover, alternating since i is supposed to be odd, cf. [Sp, Chapter 5, Section 6, §11]. The assertion follows directly from this [DeWI, (2.6)]. Cf. the remarks after [DeWII, Corollaire (4.1.5)]. The proper non-projective case has only recently been settled by J. Suh [Su, Corollary 2.2.3 and Corollary 3.3.5]. □

Remark 2.1.2. The polynomials Φ_{j}^{i} ℚ T occurring as characteristic polynomials of Frob have remarkable properties, which were established mainly by P. Deligne, B. Mazur, and A. Ogus. Details are summarised in the article [EJ15] of the second and third authors. In the proof above, the only property that was used is that every complex root of Φ_{j}^{i} is of absolute value q^{i 2 j} . This was first proven by P. Deligne in [DeWI, Théorème (1.6)] for the projective case and later in [DeWII, Corollaire (3.3.9)], in general. The assertion had been formulated by A. Weil as a part of his famous conjectures.

2.2. The discriminant of the H^{i}-representation. Let us start by recalling some facts on l-adic cohomology.

Proposition 2.2.1. Let K be a field and S a smooth and proper K-scheme.

a) Then, for all prime numbers l char K and all integers i and j, associated with the one-dimensional ℚ_{l}-vector space ^{max}H_{ét}^{i} S\overline{K}, ℚ_{l} j , there is the character

det H^{i} S\overline{K}, ℚ_{l} j : Gal \overline{K} K ℚ_{l}^{}

of the absolute Galois group of K.

b) Suppose that i is even and that S is pure of dimension i. Then the character det H^{i} S\overline{K}, ℚ_{l} i 2 has values in 1, 1 ℚ_{l}^{}and is independent of l.

Proof. a) This follows from the functoriality of l-adic cohomology, together with the fact that every σ Gal \overline{K} K induces an automorphism of schemes of S\overline{K}.

b) By Poincaré duality [SGA4, Exp. XVIII, Théorème 3.2.5], there is a canonical non-degenerate pairing s: H_{ét}^{i} S\overline{K}, ℚ_{l} i 2 H_{ét}^{i} S\overline{K}, ℚ_{l} i 2 ℚ_{l} that is compatible with the action of Gal \overline{K} K . Here, i is assumed even, so the pairing s is symmetric. According to a standard fact from linear algebra [War, Def. 2.9], s induces another symmetric pairing

max s : ^{max}H_{ét}^{i} S\overline{K}, ℚ_{l} i 2  ^{max}H_{ét}^{i} S\overline{K}, ℚ_{l} i 2  ℚ_{l}

that is again non-degenerate. The action of Gal \overline{K} K is orthogonal with respect to s, which implies that the character det H^{i} S\overline{K}, ℚ_{l} i 2 must have values in 1, 1 ℚ_{l}^{}.

For the case that K is a number field, independence of l is easily reduced to the Weil conjectures, proven by P. Deligne [DeWII, Corollaire (3.3.9)], using the Chebotarev density theorem together with the smooth specialisation theorem for cohomology groups [SGA4, Exp. XVI, Corollaire 2.3]. In general, the result has been established by T. Saito [Sa, Corollary 3.3].

N.B. In the notation for the characters, we write det instead of ^{max}. This convention follows [Sa]. □

Definition 2.2.2. i) In the situation of part b), we denote by L_{S} the extension field of K that corresponds to ker det H^{i} S\overline{K}, ℚ_{l} i 2 under the Galois correspondence. By construction, L_{S} K is an at most quadratic extension.

ii) If char K 2 then we denote the class in K^{}K^{}^{2} that yields the field extension L_{S} K by Δ_{H^{i}} S and call it the discriminant of the H^{i}-representation of S.

Lemma 2.2.3. Let K be a number field and S a smooth and proper K-scheme. Moreover, let 𝔭 𝒪_{K} be a prime at which S has good reduction.

a) If l is a prime different from the residue characteristic of 𝔭 then, for every j ℤ, the Gal \overline{K} K -representation H_{ét}^{i} S\overline{K}, ℚ_{l} j is unramified at 𝔭.

b) Suppose that S is pure of dimension i, for an even integer i. Then the quadratic character \frac{Δ_{H^{i}} S}{}det H^{i} S\overline{K}, ℚ_{l} i 2 is unramified at 𝔭. Equivalently, the splitting field L_{S} is unramified at 𝔭.

Proof. a) For this, it suffices to consider the restriction of the representation to the decomposition group D_{𝔭} Gal \overline{K}_{𝔭} K_{𝔭} . This coincides with the natural action of Gal \overline{K}_{𝔭} K_{𝔭} on H_{ét}^{i} S_{\overline{K}_{𝔭}}, ℚ_{l} j H_{ét}^{i} S\overline{K}, ℚ_{l} j , according to invariance of étale cohomology under extensions of separably closed fields [SGA4, Exp. XII, Corollaire 5.4]. Moreover, by the smooth specialisation theorem for cohomology groups [SGA4, Exp. XVI, Corollaire 2.2], H_{ét}^{i} S_{\overline{K}_{𝔭}}, ℚ_{l} j H_{ét}^{i} S_{\overline{𝔽}_{𝔭}}, ℚ_{l} j , which shows that this cohomology vector space is acted upon via the quotient Gal \overline{𝔽}_{𝔭} 𝔽_{𝔭} D_{𝔭} I_{𝔭} . I.e., the inertia group I_{𝔭} fixes H_{ét}^{i} S\overline{K}, ℚ_{l} j pointwise, as required.

b) This is a direct consequence of a). □

Remarks 2.2.4. i) In particular, for every exponent k ℕ, the splitting field of H_{ét}^{i} S\overline{K}, ℤ l^{k}ℤ j is unramified at every prime ideal 𝔭 of good reduction.

ii) If the residue characteristic is 2 then one has

det Frob_{𝔭}: H_{ét}^{i} S\overline{K}, ℚ_{l} i 2  \frac{Δ_{H^{i}} S}{𝔭} .  (4)

iii) For any number field K^{}K, one has Δ_{H^{i}} S_{K^{}}Δ_{H^{i}} S K^{}^{}^{2} K^{}K^{}^{}^{2}.

iv) The name “discriminant” has not been chosen at random. Indeed, Proposition 2.2.1 allows a generalisation to families π: F X over a general base scheme. The quadratic field extension then goes over into a twofold covering ϱ_{π}: Y X, ramified at most over the discriminant locus. If π is sufficiently reasonable then ϱ_{π} is given by w^{2} Δ, for Δ a normalised version of the discriminant of the family. Furthermore, for every non-singular member S F_{x}, the value Δ x belongs to the class Δ_{H^{i}} S .

We plan to report about the interpretation of the quantities Δ_{H^{i}} S as actual discriminants, as well as some applications thereof, in a forthcoming paper.

2.3. Surfaces—The discriminant of the Néron–Severi representation.

Proposition 2.3.1. Let K be a field and S a smooth projective surface over K.

a) Then, associated with the one-dimensional ℚ-vector space ^{max} NS S\overline{K} _{ℤ} ℚ , there is the character

det NS S\overline{K} _{ℤ} ℚ : Gal \overline{K} K ℚ^{}

of the absolute Galois group of K.

b) The character det NS S\overline{K} _{ℤ} ℚ takes values only in 1, 1 ℚ^{}.

Proof. a) This follows from the functoriality of the Néron–Severi group, together with the fact that every σ Gal \overline{K} K induces an automorphism of schemes of S\overline{K}.

b) Every σ Gal \overline{K} K induces an automorphism of the Néron–Severi group NS S\overline{K} , in particular one of NS S\overline{K} _{tors}, and consequently one of the torsion-free ℤ-module NS S\overline{K} NS S\overline{K} _{tors}. As that is a full rank lattice in NS S\overline{K} _{ℤ} ℚ, it induces a lattice in the one-dimensional vector space ^{max} NS S\overline{K} _{ℤ} ℚ , which must be respected by the action of Gal \overline{K} K . The assertion immediately follows from this. □

Definition 2.3.2. a.i) In the situation of part b), we denote by L_{S,alg} the extension field of K that corresponds to ker det NS S\overline{K} _{ℤ} ℚ under the Galois correspondence. ii) If char K 2 then we denote the class in K^{}K^{}^{2} that yields the field extension L_{S,alg} K by Δ_{NS} S and call it the discriminant of the Néron–Severi representation or the algebraic part of the discriminant of S.

b) For surfaces such that H^{1} S\overline{K}, ℚ_{l} 0, one has Pic S\overline{K} _{ℤ} ℚ NS S\overline{K} _{ℤ} ℚ. In this case, one may write Δ_{Pic} S instead of Δ_{NS} S and speak of the discriminant of the Picard representation. Similarly, let us then write det Pic S\overline{K} _{ℤ} ℚ instead of det NS S\overline{K} _{ℤ} ℚ .

Remark 2.3.3. The algebraic part of the discriminant of S should not be confused with the discriminants of NS S and NS S\overline{K} as lattices. Instead, one might think about it as follows.

Since NS S\overline{K} _{ℤ} ℚ is a finite-dimensional ℚ-vector space, there is a smallest finite field extension of K, over which all elements of NS S\overline{K} _{ℤ} ℚ are defined, the splitting field L of NS S\overline{K} _{ℤ} ℚ. Then Gal \overline{K} K acts on NS S\overline{K} _{ℤ} ℚ via its quotient Gal L K and the action of this quotient is faithful.

Therefore, the action of Gal \overline{K} K on ^{max} NS S\overline{K} _{ℤ} ℚ factors via Gal L K , too. On the other hand, ^{max} NS S\overline{K} _{ℤ} ℚ is one-dimensional, so the action of the finite group Gal L K must factor via ℚ^{}_{tors} 1, 1 . The stabiliser is of index at most 2 in Gal \overline{K} K and hence the splitting field of ^{max} NS S\overline{K} _{ℤ} ℚ is an at most quadratic extension K \overline{Δ_{NS} S} L_{S,alg} L .

Lemma 2.3.4. Let K be a number field and S a smooth projective surface over K. Moreover, let 𝔭 𝒪_{K} be a prime at which S has good reduction. a) Then the splitting field of NS S\overline{K} _{ℤ} ℚ is unramified at 𝔭.

b) The character \frac{Δ_{NS} S}{}det NS S\overline{K} _{ℤ} ℚ is unramified at 𝔭. Equivalently, the splitting field L_{S,alg} is unramified at 𝔭.

Proof. a) The first Chern class homomorphism factors via the Néron–Severi group, i.e. via algebraic equivalence,

\xymatrix{Pic S\overline{K} _{ℤ} ℚ_{l} \ar[r]^{c_{1}} \ar@{->>}[d] & H_{ét}^{2} S\overline{K}, ℚ_{l} 1 \\ NS S\overline{K} _{ℤ} ℚ_{l} . \ar@{-->}[ur]}

Indeed, c_{1} factors via numerical equivalence, since the intersection pairing on Pic S\overline{K} _{ℤ} ℚ_{l} is compatible with the cup product pairing on H_{ét}^{2} S\overline{K}, ℚ_{l} 1 . Moreover, Matsusaka’s theorem [Ma, Theorem 4], cf. [An, paragraph 3.2.7], shows that algebraic equivalence coincides with numerical equivalence, already on Pic S\overline{K} _{ℤ} ℚ.

Now write L for the splitting field of NS S\overline{K} _{ℤ} ℚ and assume that L would ramify at 𝔭. By definition, Gal L K acts faithfully on NS S\overline{K} _{ℤ} ℚ . Choose a prime 𝔮 of L lying above 𝔭 and a non-trivial element σ Gal L_{𝔮} K_{𝔭}^{n} Gal L_{𝔮} K_{𝔭} , for K_{𝔭}^{n} the maximal unramified subfield of L_{𝔮}. Then σ acts non-trivially on the image of the first Chern class homomorphism

c_{1} : NS S\overline{K} _{ℤ} ℚ_{l} H_{ét}^{2} S\overline{K}, ℚ_{l} 1 .

This, however, is in contradiction with the smooth specialisation theorem for cohomology groups, as seen before.

b) This follows immediately from a). □

Remarks 2.3.5. i) If the residue characteristic is not 2 then one has

det Frob_{𝔭}: NS S\overline{K} _{ℤ} ℚ  \frac{Δ_{NS} S}{𝔭} .  (5)

ii) If K^{}K is a number field extending K, then one has

Δ_{NS} S_{K^{}}Δ_{NS} S K^{}^{}^{2} K^{}K^{}^{}^{2}.

iii) It is worthwhile to observe that, for K3 surfaces, the assertion of Lemma 2.3.4 is still true when S has bad reduction at 𝔭 of the mildest possible form. Cf. Corollary 2.5.4, below.

2.4. K3 surfaces. There is a strong relation, which is established for K3 surfaces, but relies on Tate’s and Serre’s conjectures in general, between the Galois action on l-adic cohomology and the variation of the geometric Picard ranks under reduction modulo various primes [EJ12b, CT]. From our point of view, this is in fact the main application of the constructions presented so far.

Facts 2.4.1. Let S be a K3 surface over a base field K. a) Then Pic S\overline{K} is a free abelian group of rank at most 22. If K is of characteristic zero then the rank is at most 20. If K is finite then the rank is even.

b) If K is finite then rk Pic S\overline{K} is equal to the number [counted with multiplicities] of all eigenvalues of Frob on H_{ét}^{2} S\overline{K}, ℚ_{l} 1 that are roots of unity.

Proof. a) The first statement is found, e.g., in [Hu, Chapter 17, first formula of Section 2]. The second one is [Hu, Chapter 17, formula (1.1)] in the case that K ℂ, while the general case follows from this in view of [Hu, Chapter 17, Lemma 2.2]. The final claim is a direct consequence of b).

b) See [Hu, Chapter 17, Corollary 2.9 and the arguments given before]. Note that this result is an application of the Tate conjecture, which has been shown for K3 surfaces over finite fields by to the combined work of several people, most notably F. Charles [Ch13], M. Lieblich, D. Maulik, and A. Snowden [LMS], K. Madapusi Pera [MP], as well as W. Kim and K. Madapusi Pera [KM]. □

For K an arbitrary field and l char K a prime number, there is a canonical orthogonal decomposition

H_{ét}^{2} S\overline{K}, ℚ_{l} 1 H_{alg} T .  (6)

Here, H_{alg} c_{1} Pic S\overline{K} _{ℤ} ℚ_{l} is clearly Gal \overline{K} K -invariant. Moreover, T : H_{alg}^{} is Gal \overline{K} K -invariant, too, as the Galois action is orthogonal.

In the particular case that K is a number field, let 𝔭 𝒪_{K} be any prime of good reduction and of residue characteristic different from l. Then Frob_{𝔭} Gal \overline{K} K is determined only up to conjugation. But this suffices to have well-defined eigenvalues and a well-defined determinant of Frob_{𝔭}, associated with any vector space being acted upon by Gal \overline{K} K . In particular,

det Frob_{𝔭}: T  \frac{det Frob_{𝔭}: H_{ét}^{2} S\overline{K}, ℚ_{l} 1}{det Frob_{𝔭}: Pic S\overline{K}} .  (7)

Our main theoretical observation on the distribution of the Picard ranks of the reductions is then as follows.

Proposition 2.4.2 (Rank jumps). Let S be a K3 surface over a number field K and 𝔭 𝒪_{K} a prime of good reduction. Assume that rk Pic S\overline{K} is even. Then the following is true: If det Frob_{𝔭}: T 1, then rk Pic S_{\overline{𝔽}_{𝔭}} rk Pic S\overline{K} 2. Proof. Choose a prime number l, different from the residue characteristic of 𝔭. Then one has H_{ét}^{2} S\overline{K}, ℚ_{l} 1 H_{ét}^{2} S_{\overline{K}_{𝔭}}, ℚ_{l} 1 [SGA4, Exp. XII, Corollaire 5.4], as well as H_{ét}^{2} S_{\overline{K}_{𝔭}}, ℚ_{l} 1 H_{ét}^{2} S_{\overline{𝔽}_{𝔭}}, ℚ_{l} 1 [SGA4, Exp. XVI, Corollaire 2.2]. Consequently, by transport of structure, the orthogonal decomposition (6) carries over into

H_{ét}^{2} S_{\overline{𝔽}_{𝔭}}, ℚ_{l} 1 H_{alg} T .  (8)

Note that, as a consequence of its construction, T may well contain algebraic classes.

Moreover, under the first isomorphism, the action of Gal \overline{K}_{𝔭} K_{𝔭} is compatible with that of the decomposition group D_{𝔭} Gal \overline{K} K , while the second isomorphism shows that Gal \overline{K}_{𝔭} K_{𝔭} acts via its quotient Gal K_{𝔭}^{nr} K_{𝔭} . In particular, the action of Frob Gal \overline{𝔽}_{𝔭} 𝔽_{𝔭} on H_{ét}^{2} S_{\overline{𝔽}_{𝔭}}, ℚ_{l} 1 agrees with that of any Frob_{𝔭} Gal \overline{K} K on H_{ét}^{2} S\overline{K}, ℚ_{l} 1 . For instance, Frob and Frob_{𝔭} have the same eigenvalues on H_{alg}, as well as on T.

By Lemma 2.3.4.a), the splitting field of Pic S\overline{K} is a number field unramified at 𝔭. Therefore, Gal K_{𝔭}^{nr} K_{𝔭} acts on H_{alg} via a finite quotient group. In particular, there exists an integer e 0 such that Frob^{e} acts trivially. Consequently, all eigenvalues of Frob on H_{alg} are roots of unity.

In view of Fact 2.4.1.b), we need to show that Frob acts on T with at least two eigenvalues being roots of unity. For this, let us observe that each eigenvalue is of absolute value 1, so that those different from 1 and 1 come in pairs z, \overline{z} of complex conjugates. As z\overline{z} 1 and det Frob_{𝔭}: T 1, one of the eigenvalues must be equal to 1 . Finally, as dim T 22 rk Pic S\overline{K} is even, a further eigenvalue 1 is enforced. This completes the proof. □

Remarks 2.4.3. i) The proof given above, shows that, in addition to the specialisations of the invertible sheaves from Pic S\overline{K}, the Picard group of Pic S_{\overline{𝔽}_{𝔭}} has (at least) two further generators. One of them may be chosen to be defined over 𝔽_{𝔭} , the other over its quadratic extension. ii) Without the hypothesis on the determinant of the Frobenius, the argument simply reproves the standard fact that rk Pic S_{\overline{𝔽}_{𝔭}} rk Pic S\overline{K}.

Theorem 2.4.4. Let K be a number field and S a K3 surface over K. Moreover, let 𝔭 𝒪_{K} be a prime of residue characteristic 2 and good reduction.

a) Then the following two equations hold,

det Frob_{𝔭}: H_{ét}^{2} S\overline{K}, ℚ_{l} 1  \frac{Δ_{H^{2}} S}{𝔭}  and
det Frob_{𝔭}: T  \frac{Δ_{H^{2}} S Δ_{Pic} S}{𝔭} .

b) If rk Pic S\overline{K} is even, then

\frac{Δ_{H^{2}} S Δ_{Pic} S}{𝔭}  1  rk Pic S_{\overline{𝔽}_{𝔭}} rk Pic S\overline{K} 2 .

In other words, if Δ_{H^{2}} S Δ_{Pic} S is not a square in K then

𝔭 𝔭 inert in K \overline{Δ_{H^{2}} S Δ_{Pic} S} Π_{jump} S .

Proof. a) The first formula is a particular case of formula (4). The second one is a consequence of the first together with formulae (5) and (7).

b) This follows from a), together with Proposition 2.4.2. □

Corollary 2.4.5. Let K be a number field and S a K3 surface over K. Assume that Δ_{H^{2}} S Δ_{Pic} S is a non-square in K. Then

lim inf_{B} γ S, B \frac{1}{2} .

Definition 2.4.6. For K a number field and S a K3 surface over K, we call the quadratic character

τ_{S} : det H^{2} S\overline{K}, ℚ_{l} 2  det Pic S\overline{K} _{ℤ} ℚ : Gal \overline{K} K  1, 1

the jump character of S.

Remarks 2.4.7. i) The jump character τ_{S} is given by

𝔭 \frac{Δ_{H^{2}} S Δ_{Pic} S}{𝔭} ,

for all good primes 𝔭. ii) Proposition 2.4.2 shows that, for S a K3 surface of even geometric Picard rank, τ_{S} 𝔭 1 implies rk Pic S_{\overline{𝔽}_{𝔭}} rk Pic S\overline{K} 2.

In this section, the assumption on the surface to be of type K3 was used only in referring to the Tate conjecture. We actually showed the following.

Theorem 2.4.8. Let K be a number field and S a smooth and proper surface over K, for which the Tate conjecture holds. Moreover, let 𝔭 𝒪_{K} be a prime of good reduction and suppose that the Tate conjecture holds for S_{𝔽_{𝔭}} , too. Then, in the situation that rk NS S\overline{K} dim H_{ét}^{2} S\overline{K}, ℚ_{l} 1 mod 2 , one has

det H^{2} S\overline{K}, ℚ_{l} 2 det Pic S\overline{K} _{ℤ} ℚ 𝔭 1

\frac{Δ_{H^{2}} S Δ_{NS} S}{𝔭}  1  rk NS S_{\overline{𝔽}_{𝔭}} rk NS S\overline{K} 2 .
□

2.5. The criterion for non-triviality.

Lemma 2.5.1. Let p 2 be a prime number and M a free ℤ_{p}-module of finite rank r, equipped with a non-degenerate symmetric pairing. Let an orthogonal map V : M M be given with characteristic polynomial T 1 ^{r_{1}} T 1 ^{r_{2}}. Then M is the orthogonal direct sum of the two generalised eigenspaces,

M ker 1 V ^{r_{1}} ker 1 V ^{r_{2}} .

Proof. It is well known that the generalised eigenspaces for two distinct eigenvalues only have the zero element in common. Moreover, \frac{1 V}{2} \frac{1 V}{2} id yields

_{k 0}^{r} _{k}^{r} \frac{1 V}{2} ^{k} \frac{1 V}{2} ^{r k} x x ,

for every x M. According to Cayley–Hamilton, the summands for k r_{2} are contained in ker 1 V ^{r_{1}}, while those for k r_{2}, i.e. r k r_{1}, are contained in ker 1 V ^{r_{2}}. Thus, the sum is the whole of M. Finally, it is a classical result for orthogonal maps that the generalised eigenspaces for two eigenvalues λ_{1}, λ_{2} with λ_{1}λ_{2} 1 are perpendicular. An argument is given, e.g., in [Wal, Proposition 10.4.1].

□

Theorem 2.5.2 (The vanishing cycle). Let K be a number field and S a K3 surface over K. Moreover, let 𝔭 𝒪_{K} be a prime of residue characteristic 2 such that S has a regular, projective model \underline{S} over 𝒪_{K,𝔭}, the geometric fibre \underline{S}\overline{𝔭} of which has exactly one singular point 𝔷. Assume 𝔷 to be an ordinary double point. Then, for every prime number l, different from the residue characteristic of 𝔭, the vanishing cycle [SGA7, Exp. XV, Théorème 3.4.(i)] associated with 𝔷, fulfils

δ_{𝔷,l} H_{alg,l}^{}.

Proof. First step. Generalities. Let us denote the residue characteristic of 𝔭 by p. On the scheme S\overline{K}, there is a monodromy automorphism [SGA7, Exp. XV, Proposition 3.2.1.(ii)], which is induced by a particular non-trivial element ν I_{𝔭} Gal \overline{K} K of the inertia group. For every prime number l, including l p, the induced map on l-adic cohomology is called the monodromy operator V : H_{ét}^{2} S\overline{K}, ℤ_{l} 1 . This is an orthogonal map with respect to the cup product pairing. By a slight abuse of notation, we denote the map induced by ν on Pic S\overline{K} by V, too.

If l p then the action of V is described by the Picard–Lefschetz formula [SGA7, Exp. XV, Théorème 3.4.(iii)]

V c c c, δ_{𝔷,l} δ_{𝔷,l} .  (9)

Note here that V, being induced by an element from the inertia group, acts trivially on ℤ_{l} 1 itself. The class δ_{𝔷,l} H_{ét}^{2} S\overline{K}, ℤ_{l} 1 is the so-called vanishing cycle. It is known that δ_{𝔷,l}, δ_{𝔷,l} 2 [SGA7, Exp. XV, Théorème 3.4.(i)]. In particular, the Picard–Lefschetz formula shows V δ_{𝔷,l} δ_{𝔷,l}. Moreover, the operator V acts with characteristic polynomial T 1 ^{21} T 1 .

When the action on H_{ét}^{2} S\overline{K}, ℤ_{p} 1 is concerned, the characteristic polynomial is the same [Oc, Theorem 3.1], cf. [Fo, §§2.3 and 2.4]. It seems, however, not to be known whether the action of V is semisimple. In particular, no Picard– Lefschetz formula is available. Second step. If, for some l_{0} p, δ_{𝔷,l_{0}} H_{alg,l_{0}}^{}then V acts non-trivially on Pic S\overline{K}. The assumption means that c, δ_{𝔷,l_{0}} 0, for a certain class c H_{alg,l_{0}}. Then formula (9) immediately implies that

δ_{𝔷,l_{0}} \frac{1}{c, δ_{𝔷,l_{0}}} V c c H_{alg,l_{0}} .

I.e., the operator V acts on H_{alg,l_{0}} Pic S\overline{K} _{ℤ} ℚ_{l_{0}} non-trivially with one eigenvector 1 , while all others are equal to 1. As 1 ℚ, the eigenspaces are defined in Pic S\overline{K} _{ℤ} ℚ already, which implies the claim.

Third step. Let d Pic S\overline{K} Pic S\overline{K} _{ℤ} ℚ be a generator of the 1 -eigenspace, which is minimal, i.e. not divisible by any integer 1. Then d, d 2. For every prime number l, the inclusion Pic S\overline{K} l ^{\overline{c}_{1}} H_{ét}^{2} S\overline{K}, ℤ_{l} 1 l coming from the Kummer sequence shows that c_{1} d H_{ét}^{2} S\overline{K}, ℤ_{l} 1 is not divisible by l. Hence, c_{1} d is actually a generator of the 1 -eigenspace in H_{ét}^{2} S\overline{K}, ℤ_{l} 1 .

Claim: d, d is an l-adic unit, for every prime l 2. Indeed, for l 2, p, the ℤ_{p}-module H_{ét}^{2} S\overline{K}, ℤ_{l} 1 is the orthogonal direct sum of the 1- and 1 -eigenspaces, since \frac{1}{2} ℤ_{l} and every cohomology class c may be written in the form \frac{1}{2} c V c \frac{1}{2} c V c . For l p, the 1 -eigenspace is a direct summand, too, due to Lemma 2.5.1. As the pairing on the total space is perfect, the same is true for the direct summands, which implies the claim.

Thus, d, d 2^{k} for some non-negative integer k. On the other hand, c_{1} d H_{ét}^{2} S\overline{K}, ℤ_{2} 1 is a generator of the 1 -eigenspace and δ_{𝔷,2} is another. Hence, c_{1} d u δ_{𝔷,2}, for a certain unit u ℤ_{2}^{}. Consequently,

d, d c_{1} d , c_{1} d u^{2} δ_{𝔷,2}, δ_{𝔷,2} 2u^{2} ,

which immediately shows that k 1. Moreover, the minus sign is correct, since 1 is not a square in ℚ_{2}.

Fourth step. Conclusion. In the particular case of a K3 surface, it is well known that, for a class d Pic S\overline{K} with d, d 2, either d or d is represented by an effective divisor [BHPV, Chap. VIII, Proposition 3.7]. But then V cannot interchange the two, a contradiction. □

Remarks 2.5.3. i) The regularity of the model \underline{S} implies that the singular point on \underline{S}\overline{𝔭} does not lift to a 𝒪_{K_{𝔭}}^{nr}-rational point on \underline{S}.

ii) When there are two singular points instead of one, then the argument above then only shows that a non-trivial linear combination of δ_{𝔷_{1}} and δ_{𝔷_{2}} lies in H_{alg}^{}. The splitting field of Pic S\overline{K} _{ℤ} ℚ may well ramify then. Cf. Corollary 2.5.4.a), below.

iii) There does not seem to be an obvious generalisation to other types of surfaces. For example, for rational surfaces one has H_{alg}^{}0 , but δ_{𝔷,l} is clearly nonzero. Also, the argument heavily relies on the fact that, for d Pic S\overline{K} with d, d 2, either d or d is effective, which seems to be rather specific for K3 surfaces.

Corollary 2.5.4. Let K be a number field and S a K3 surface over K. Moreover, let 𝔭 𝒪_{K} be a prime of residue characteristic 2 such that S has a regular, projective model \underline{S} over 𝒪_{K,𝔭}, the geometric fibre \underline{S}\overline{𝔭} of which has exactly one singular point 𝔷. Assume 𝔷 to be an ordinary double point.

a) Then the splitting field of Pic S\overline{K} _{ℤ} ℚ is unramified at 𝔭.

b) The character \frac{Δ_{NS} S}{}det NS S\overline{K} _{ℤ} ℚ is unramified at 𝔭. Equivalently, the splitting field L_{S,alg} is unramified at 𝔭.

Proof. a) Choose a prime number l, different from the residue characteristic of 𝔭. Then there is the short exact sequence [SGA7, Exp. XV, Théorème 3.4.(ii)]

0 H_{ét}^{2} \underline{S}\overline{𝔭}, ℚ_{l} 1  H_{ét}^{2} S\overline{K}, ℚ_{l} 1  ℚ_{l} 0 ,
c  c, δ_{𝔷,l} ,

provided by the theory of vanishing cycles. Together with the result of Theorem 2.5.2, it shows that every invertible sheaf on S\overline{K} extends to \underline{S}\overline{𝔭}. In other words, the splitting field of Pic S\overline{K} _{ℤ} ℚ is contained in K_{𝔭}^{nr}.

b) This is a direct consequence of a). □

Proposition 2.5.5 (Reduction to one ordinary double point). Let K be a number field and S a proper K-scheme that is pure of even dimension i. Moreover, let 𝔭 𝒪_{K} be a prime of residue characteristic 2 such that S has a regular, projective model \underline{S} over 𝒪_{K,𝔭}, the geometric fibre \underline{S}\overline{𝔭} of which has exactly one singular point 𝔷. Assume 𝔷 to be an ordinary double point.

a) Then, for any prime l different from the residue characteristic of 𝔭, the Gal \overline{K} K -representation H_{ét}^{i} S\overline{K}, ℚ_{l} i 2 is tamely ramified at 𝔭. The 𝔭-adic valuation of its conductor is equal to 1.

b) The quadratic character \frac{Δ_{H^{i}} S}{}det H^{i} S\overline{K}, ℚ_{l} i 2 is ramified at 𝔭. Equivalently, the splitting field L_{S} is ramified at 𝔭.

Proof. a) In this generality, the short exact sequence provided by the theory of vanishing cycles reads [SGA7, Exp. XV, Théorème 3.4.(ii)]

0 H_{ét}^{i} \underline{S}\overline{𝔭}, ℚ_{l} i 2  H_{ét}^{i} S\overline{K}, ℚ_{l} i 2  ℚ_{l} 0 ,
c  c, δ_{𝔷,l} ,

with δ_{𝔷,l}, δ_{𝔷,l} 1 ^{i 2} 2. The 𝔭-adic valuation of the conductor is determined by the restriction of the representation to the decomposition group D_{𝔭} Gal \overline{K}_{𝔭} K_{𝔭} . The exact sequence shows that the subspace δ_{𝔷,l}^{}H_{ét}^{i} S\overline{K}, ℚ_{l} i 2 is acted upon via the quotient Gal \overline{𝔽}_{𝔭} 𝔽_{𝔭} D_{𝔭} I_{𝔭} . I.e., the inertia group I_{𝔭} fixes δ_{𝔷,l}^{}pointwise. As one has V δ_{𝔷,l} δ_{𝔷,l} for V I_{𝔭}, due to the Picard–Lefschetz formula, this yields

H_{ét}^{i} S\overline{K}, ℚ_{l} i 2 ^{I_{𝔭}} δ_{𝔷,l}^{}.

Moreover, the action of I_{𝔭} respects orthogonality and cup product pairing, so δ_{𝔷,l} can be mapped only to δ_{𝔷,l}. Thus, there is a subgroup I^{}I_{𝔭} of index two acting trivially. Since the residue characteristic of 𝔭 is 2, this yields tameness.

In this case, the 𝔭-adic valuation of the conductor is given by [Se, formulae (11) and (8)]

dim_{ℚ_{l}} H_{ét}^{i} S\overline{K}, ℚ_{l} i 2 dim_{ℚ_{l}} H_{ét}^{i} S\overline{K}, ℚ_{l} i 2 ^{I_{𝔭}}
dim_{ℚ_{l}} H_{ét}^{i} S\overline{K}, ℚ_{l} i 2 dim_{ℚ_{l}} δ_{𝔷,l}^{}1 .

b) This is very easily shown directly. The monodromy operator V I_{𝔭} fixes all cohomology classes perpendicular to δ_{𝔷,l} and sends δ_{𝔷,l} to δ_{𝔷,l} . Therefore, det V : H_{ét}^{2} S\overline{K}, ℚ_{l} 1 1. In particular, ker det H^{2} S\overline{K}, ℚ_{l} 1 does not include all of I_{𝔭}, and hence the field corresponding under the Galois correspondence to ker det H^{2} S\overline{K}, ℚ_{l} 1 is not contained in K_{𝔭}^{nr}. □

Remark 2.5.6. Suppose that, for some prime 𝔭, S has a model of bad reduction of the kind as described above. Then S does not have a model of good reduction at 𝔭. Indeed, the conclusions of Lemma 2.2.3.b) and Proposition 2.5.5.b) are independent of the model. The existence of a model of good reduction implies that \frac{Δ_{H^{i}} S}{}is unramified, while the existence of a model of the type above enforces ramification.

Corollary 2.5.7 (The jump character). Let K be a number field and S a K3 surface over K. Moreover, let 𝔭 𝒪_{K} be a prime of residue characteristic different from 2.

a) If S has good reduction at 𝔭, then τ_{S} \frac{Δ_{H^{2}} S Δ_{Pic} S}{}is unramified at 𝔭.

b) Suppose that S has a regular, projective model \underline{S} over 𝒪_{K,𝔭}, the geometric fibre \underline{S}\overline{𝔭} of which has exactly one singular point, and assume this to be an ordinary double point. Then the jump character τ_{S} \frac{Δ_{H^{2}} S Δ_{Pic} S}{}ramifies at 𝔭. Proof. a) is clear from Lemmata 2.2.3.b) and 2.3.4.b), while the assertion of part b) follows from Corollary 2.5.4.b) together with Proposition 2.5.5.b). □

2.6. Examples and experimental results.

Algorithm 2.6.1 (Computing Δ_{H^{2}} S ). Given a proper surface S over ℚ, the set q_{1}, …, q_{m} of all bad primes of S, and an oracle for det Frob_{p}: H_{ét}^{2} S\overline{ℚ} , ℚ_{l} 1 for any p q_{j}, this algorithm computes Δ_{H^{2}} S .

i) Add q_{0} : 1 to the list of bad primes. ii) Build a matrix A, the entries of which are the Legendre symbols \frac{q_{j}}{p_{i}} , for good primes p_{i} chosen at random. Keep adding rows until the matrix has rank m 1 over 1, 1 ℤ 2ℤ. iii) Put b_{i} det Frob_{p_{i}}: H_{ét}^{2} S\overline{ℚ} , ℚ_{l} 1 and solve the linear system Ax b of equations. If the solution vector is x_{0}, …, x_{m} ℤ 2ℤ ^{m} then Δ_{H^{2}} S is the class of 1 ^{e_{0}}q_{1}^{e_{1}} q_{m}^{e_{m}} in ℚ^{}ℚ^{}^{2}, for e_{i} 0, 1 ℤ representing the residue class x_{i} ℤ 2ℤ.

Remarks 2.6.2. i) The oracle for det Frob_{p}: H_{ét}^{2} S\overline{ℚ} , ℚ_{l} 1 is, of course, provided by counting the points on S that are defined over 𝔽_{p} and some of its extensions. ii) Dirichlet’s Theorem on primes in arithmetic progressions ensures that there exist primes so that the matrix A has rank m 1. iii) (An improvement.) In the case that 2 is a good prime, step i) of Algorithm 2.6.1 may be omitted. The sign of Δ_{H^{2}} S is then determined by the condition that the character be unramified at 2. iv) (A further improvement.) Assume that the surface S is K3. Then, for some or many of its bad primes p 2, it may happen that Proposition 2.5.5.b) applies. At such a prime, the jump character necessarily ramifies, which means that Δ_{H^{2}} S must be of odd p-adic valuation. Thus, the solution vector is bound to have a component 1 ℤ 2ℤ at the corresponding coordinates.

If S has a model in some P_{ℤ}^{N} that is given by explicit equations, one may compute the set of all bad primes using Gröbner bases and integer factorisation, and finally analyse the singular points. Having found several primes to which Proposition 2.5.5.b) applies, this information may be used in order to get by with a linear system of equations of smaller size. In other words, less point counting is necessary. It is our experience that this improvement of Algorithm 2.6.1 often leads to an enormous gain for a “random” surface, while for the constructed examples, which we present below, it would not help much.

v) There is an obvious modification of Algorithm 2.6.1 to directly determine the jump character.

Algorithm 2.6.3 (Statistical algorithm computing the jump character). Given a K3 surface S over ℚ of even geometric Picard rank, the set q_{1}, …, q_{m} of all bad primes of S, and a list of good non-jump primes, this algorithm determines a finite subgroup, containing the jump character, of the group of all characters of Gal \overline{ℚ} ℚ with values in 1, 1 .

i) Add q_{0} : 1 to the list of bad primes. ii) Build a matrix A, the entries of which are the Legendre symbols \frac{q_{j}}{p_{i}} , for the non-jump primes p_{i}. iii) Determine the kernel of A. From each kernel vector, calculate a candidate for the jump character in the same way as in Algorithm 2.6.1.iii).

Remarks 2.6.4. i) If the kernel is the zero space, then this proves the jump character to be trivial. If the kernel is one-dimensional, then there are two possible answers. A non-trivial character, which is directly computed from a kernel vector, and the trivial one. ii) If the kernel is still one-dimensional when the system of equations is rather overdetermined, then this gives strong evidence for the jump character to be non-trivial. In practice, we work with at least 4 m 1 non-jump primes. iii) The trivial character is unramified at every prime. Thus, as soon as it applies, Corollary 2.5.7.b) excludes the trivial character, and therefore makes the outcome of Algorithm 2.6.3 usually unique. Corollary 2.5.7.b) is useful as well to accelerate the calculations.

Example 2.6.5. Let S be the diagonal quartic in P_{ℚ}^{3} , given by X_{0}^{4} X_{1}^{4} X_{2}^{4} X_{3}^{4} 0. Then the geometric Picard rank of S is 20 and the jump character is given by \frac{1}{}. Proof. The model 𝒮 of S that is given in P_{ℤ}^{3} by the same equation has bad reduction only at 2. Hence, Δ_{H^{2}} S 1 or 2. Counting points on the reductions S_{𝔽_{3}} and S_{𝔽_{5}} , one finds that det Frob_{p}: H_{ét}^{2} S\overline{ℚ} , ℚ_{l} 1 1 for both p 3 and 5. Thus, Algorithm 2.6.1 shows that Δ_{H^{2}} S 1.

On the other hand, it is classically known that the 48 lines on S\overline{ℚ} generate the geometric Picard group, which is of rank 20. In particular, Pic S\overline{ℚ} is defined over ℚ ζ_{8} ℚ i, \overline{2} . Moreover, [Br, Appendix A, Examples A62, B33, C27, and D27] show that the Galois representation Pic S\overline{ℚ} _{ℤ} ℂ splits into characters as

χ_{triv}^{5} χ_{ℚ i}^{3} χ_{ℚ \overline{2}}^{6} χ_{ℚ \overline{2}}^{6} .

Here, for K a quadratic number field, χ_{K}: Gal \overline{ℚ} ℚ 1, 1 denotes the character that becomes trivial after restriction to Gal \overline{ℚ} K and defines the non-trivial quadratic character on Gal K ℚ . Consequently,

max Pic S _{ℤ} ℂ χ_{triv}^{5} χ_{ℚ i}^{3} χ_{ℚ \overline{2}}^{6} χ_{ℚ \overline{2}}^{6} χ_{ℚ i} ,

Δ_{Pic} S 1 and Δ_{H^{2}} S Δ_{Pic} S 1. □

Remark 2.6.6. It is known at least since 1963 [T] that, in this example, there are no rank jumps, except for those explained by the jump character. I.e., one has rk Pic S_{\overline{𝔽}_{p}} 20 for all primes p 1 mod 4 . In fact, the eigenvalues of Frob_{p} on H_{ét}^{2} S\overline{ℚ} , ℚ_{l} 1 may be determined using Jacobi sums [IR, Chapter 8, Theorem 5] and it turns out that two of them are \frac{π^{2}}{p} and its conjugate, for p π\overline{π} a factorisation in ℚ i . Cf. [PS, particularly formulae (12) and (13)] for more details.

Example 2.6.7. Let S be the double cover of P_{ℚ}^{2} , given by w^{2} X_{0}^{6} X_{1}^{6} X_{2}^{6}. Then the geometric Picard rank of S is 20 and the jump character is given by \frac{3}{}. Proof. The double cover 𝒮 of P_{ℤ}^{2} that is given by the same equation has bad reduction only at the primes 2 and 3. Hence, Δ_{H^{2}} S 1 , 2 , 3 or 6 . Furthermore, counting points on the reductions S_{𝔽_{5}} , S_{𝔽_{7}} , and S_{𝔽_{13}} , one finds that det Frob_{p}: H_{ét}^{2} S\overline{ℚ} , ℚ_{l} 1 1, for both p 5 and p 13, and 1 for p 7. Thus, Algorithm 2.6.1 shows Δ_{H^{2}} S 1.

The ramification sextic allows 18 tritangent lines of the type X_{i} ζ_{12}^{m}X_{j} 0, for m odd. Furthermore, the 18 conics of type X_{i}X_{j} \frac{ζ_{6}^{m}}{^{3} \overline{2}}X_{k}^{2} 0 are six times tangent to the ramification sextic. The irreducible components of the preimages of these curves together generate the geometric Picard group up to finite index. Working with the tritangents alone, one would end up with a sublattice that is not of full rank.

We implemented in magma a function to compute intersection numbers on S and, starting with 14 tritangent lines and six conics being six times tangent, found a nondegenerate 20 20 intersection matrix. Using this, it turns out that the splitting field of Pic S\overline{ℚ} is in fact ℚ ζ_{3}, ^{3} \overline{2}, i , having Galois group S_{3} ℤ 2ℤ. Moreover, the Galois representation Pic S\overline{ℚ} _{ℤ} ℂ splits into irreducible components as

χ_{triv}^{4} χ_{ℚ i}^{4} χ_{ℚ \overline{3}}^{3} χ_{ℚ \overline{3}}^{3} V^{3} ,

where the characters are defined as above and V denotes the irreducible twodimensional representation of the factor group Gal ℚ ζ_{3}, ^{3} \overline{2} ℚ S_{3} . Consequently,

max Pic S\overline{ℚ} _{ℤ} ℂ χ_{triv}^{4} χ_{ℚ i}^{4} χ_{ℚ \overline{3}}^{3} χ_{ℚ \overline{3}}^{3} χ_{ℚ \overline{3}}^{3} χ_{ℚ \overline{3}} ,

which implies that Δ_{Pic} S 3 and Δ_{H^{2}} S Δ_{Pic} S 3. □

Remark 2.6.8. Again, there are no rank jumps, except for those explained by the jump character. I.e., one has rk Pic S_{\overline{𝔽}_{p}} 20 for all primes p 1 mod 3 . The eigenvalues of Frob_{p} may again be determined using Jacobi sums. Here, it turns out that two of them are J ω, ω, ω p [IR, Proposition 8.5.1] and its conjugate, for ω a primitive sextic character on 𝔽_{p}^{}. A short calculation, using [IR, Chapter 8, Theorem 3] and [BE, Theorems 3.1 and 3.4)] shows that these quantities evaluate to \frac{1}{p} \frac{π^{2}}{p} and its conjugate, for π a primary element [IR, Proposition 9.3.5] in ℤ ζ_{3} of norm p.

Remarks 2.6.9. i) The surfaces described in Examples 2.6.5 and 2.6.7 are K3 surfaces defined over ℚ of geometric Picard rank 20, and, as such, very particular objects. Due to the work of R. Livné [Lv], it is known that they are modular. Moreover, there is a formula describing the determinant of Frobenius on the transcendental part of cohomology in terms of the discriminant of the Picard lattice alone [Lv, Example 1.6]. Our calculations presented above are supposed to illustrate our method to compute the jump character in a situation where the Picard group is completely known. ii) The surface from Example 2.6.7 is, up to isogeny, the Kummer surface associated with a product of an elliptic curve with itself. (Over ℂ, this is classically known, cf. [SI, Proof of Theorem 4]. Over ℚ, an argument is given, e.g., in [BCFNW, Proposition 2.3].) Its jump character may also be determined that way.

Example 2.6.10. Let K be a number field and S the Kummer surface of an abelian surface over K that geometrically splits into a product E_{1} E_{2} of elliptic curves. Assume that rk Pic S\overline{K} 18. Then there are two cases.

a) If the elliptic curves E_{1} and E_{2} are defined over K then the jump character of S is trivial.

b) If the elliptic curves E_{1} and E_{2} are defined over a quadratic extension K \overline{d} and conjugate to each other, then the jump character of S is \frac{d}{.}

Proof. a) The transcendental part T H_{ét}^{2} S\overline{K}, ℚ_{l} 1 is isomorphic to

T H_{ét}^{1} E_{1}, ℚ_{l} H_{ét}^{1} E_{2}, ℚ_{l} 1 ,

hence

maxT  ^{2}H_{ét}^{1} E_{1}, ℚ_{l} ^{2}  ^{2}H_{ét}^{1} E_{2}, ℚ_{l} 1 ^{2}
H_{ét}^{2} E_{1}, ℚ_{l} ^{2} H_{ét}^{2} E_{2}, ℚ_{l} 2 ^{2}
H_{ét}^{2} E_{1}, ℚ_{l} 1 ^{2} H_{ét}^{2} E_{2}, ℚ_{l} 1 ^{2} ,

and both factors are acted upon trivially by Gal \overline{K} K .

b) Let σ Gal \overline{K} K be any automorphism that changes the sign of \overline{d}. Then σ interchanges the components of H_{ét}^{1} E_{1} E_{2}, ℚ_{l} H_{ét}^{1} E_{1}, ℚ_{l} H_{ét}^{1} E_{2}, ℚ_{l} . I.e., σ acts with eigenvalues 1 and 1, both of multiplicity 2. Hence, on H_{ét}^{2} E_{1} E_{2}, ℚ_{l} ^{2}H_{ét}^{1} E_{1} E_{2}, ℚ_{l} , one has the eigenvalues 1 , of multiplicity 4, and 1, of multiplicity 2.

However, under σ, the two algebraic classes in H_{ét}^{2} E_{1}, ℚ_{l} H_{ét}^{0} E_{2}, ℚ_{l} and H_{ét}^{0} E_{1}, ℚ_{l} H_{ét}^{2} E_{2}, ℚ_{l} are interchanged, so that the eigenvalues 1 and 1 occur on the algebraic part. Therefore, the eigenvalues on T are 1 , with multiplicity 3, and 1, with multiplicity 1. Hence, every σ Gal \overline{K} K as chosen above acts as 1 on ^{max}T, which is enough to imply the claim. □

Example 2.6.11. In [CT, Examples 3.3, 3.4, and 3.5], Yu. Tschinkel and the first author reported numerical evidence for lim inf_{B} γ S, B 1 2, in the case of three K3 surfaces over ℚ of geometric Picard rank two.

This indeed follows from Corollary 2.4.5, once one proves that Δ_{H^{2}} S Δ_{Pic} S is not a square in ℚ. For each of the examples, one has Pic S\overline{ℚ} Pic S and therefore Δ_{Pic} S 1 ℚ^{}ℚ^{}^{2} . Moreover, Algorithm 2.6.1 determines Δ_{H^{2}} S to, in this order,

5 151 22490817357414371041

387308497430149337233666358807996260780875056740850984213276970343278935342068889706146733313789 ,

53 2624174618795407 512854561846964817139494202072778341

1215218370089028769076718102126921744353362873

6847124397158950456921300435158115445627072734996149041990563857503 ,

and

47 3109 4969 14857095849982608071 445410277660928347762586764331874432202584688016149 658652708525052699993424198738842485998115218667979560362214198830101650254490711 .

Each of the factors listed is reported as being prime by magma, version 2.21.8.

Example 2.6.12. Let S be the K3 surface over ℚ, given by the equation

X_{3}^{4} f_{2} X_{0}, X_{1}, X_{2} X_{3}^{2} f_{4} X_{0}, X_{1}, X_{2} 0 ,  (10)

for

f_{2} X_{0}, X_{1}, X_{2} : X_{0}^{2} X_{0}X_{1} X_{0}X_{2} X_{1}X_{2} and
f_{4} X_{0}, X_{1}, X_{2} : X_{0}^{3}X_{2} X_{0}X_{1}^{2}X_{2} X_{1}^{4} X_{2}^{4} .

Then the geometric Picard rank of S is 8 and the jump character of S is trivial.

Proof. First of all, a space quartic S of the form (10) is of geometric Picard rank at least 8. Indeed, the surface S comes equipped with a finite morphism p: S S^{}, which is generically 2:1, to an underlying degree two del Pezzo surface S^{}. The induced homomorphism p^{}: Pic S ; \overline{K} Pic S\overline{K} doubles all intersection numbers. As, on a degree two del Pezzo surface, there are no non-trivial invertible sheaves that are numerically equivalent to zero, we see that p^{}is necessarily injective. The claim follows.

Thus, for the first assertion, it suffices to find a prime p of good reduction such that rk Pic S_{\overline{𝔽}_{p}} 8 . For example, p 19, 43, 61, 101, 109, 139, 149, 151, 157, and 163 do the job, as is easily shown in the usual way, based on counting points. Cf. [CT] for more details and further references.

On the other hand, a calculation using Gröbner bases shows that the model 𝒮 of S given by the same equation in P_{ℤ}^{3} has bad reduction only at the primes 2, 3, 47, and 431. Using Algorithm 2.6.3, one then proves the triviality of the jump character. In fact, only the first five non-jump primes 19, 43, 61, 101 and 109 are needed in order to do this. □

Remark 2.6.13. This example, and several others of the same kind, were found by a systematic inspection of all space quartics of the form (10), with coefficients from 1, 0, 1 . This led to a sample of 183 098 318 non-singular surfaces in total, among which only a few hundred have trivial jump character, together with geometric Picard rank 8.

Example 2.6.14. Let S be the K3 surface over ℚ, given by the equation

X_{3}^{4} f_{4} X_{0}, X_{1}, X_{2} 0 ,  (11)

for

f_{4} X_{0}, X_{1}, X_{2} : X_{0}^{4} X_{0}^{3}X_{1} 2X_{0}^{3}X_{2} X_{0}^{2}X_{1}X_{2} X_{0}X_{1}^{2}X_{2} X_{1}^{4} X_{2}^{4} .

Then the geometric Picard rank of S is 8 and the jump character of S is \frac{1}{.} .

Proof. Again, for the first claim, it suffices to find a prime p of good reduction such that rk Pic S_{\overline{𝔽}_{p}} 8 . For example, p 5, 13, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, and 197 do the job.

Moreover, a calculation using Gröbner bases shows that the model 𝒮 of S given in P_{ℤ}^{3} by equation (11) has bad reduction only at the primes 2, 7, 6449, and 39 870 353. For the obvious integral model of the underlying degree two del Pezzo surface S^{}, the same is true. Algorithm 2.6.1 then proves that Δ_{H^{2}} S 2 7 6449 39 870 353 and that Δ_{H^{2}} S^{}2 7 6449 39 870 353. The seven good primes up to 23 are in fact sufficient.

As, on a del Pezzo surface, every cohomology class is algebraic, we conclude that Δ_{NS} S^{}Δ_{H^{2}} S^{}. Furthermore, the linear map

p^{}_{ℤ} ℚ: NS S^{}_{ℤ} ℚ Pic S^{}_{ℤ} ℚ Pic S _{ℤ} ℚ

is an isomorphism, since it is injective and either ℚ-vector space is of dimension 8. Therefore, Δ_{Pic} S Δ_{NS} S^{}, which implies the claim. □

Remark 2.6.15. Note that, for the surface above, Algorithm 2.6.3 would only prove that the jump character is \frac{1}{.} or trivial. Moreover, the non-triviality criterion, given in Corollary 2.5.7.b), could not resolve the ambiguity either.

2.7. Interaction of jumps. As is well known, the geometric Picard rank always jumps under reduction when rk Pic S\overline{K} is odd. The same is true, when there is real multiplication by some field E and one has an odd quotient 22 rk Pic S\overline{K} E:ℚ [Ch14, Theorem 1(2)]. One might speculate in these cases, whether the jump character causes the jumps to be even larger. This does, however, not happen, as is shown by the examples below.

Lemma 2.7.1 (K3 surfaces having a non-singular degree two model). Let K be a number field and S a K3 surface over K, given by w^{2} f_{6} X_{0}, X_{1}, X_{2} , for f_{6} a homogeneous form of degree 6. Write

S_{λ}: λw^{2} f_{6} X_{0}, X_{1}, X_{2}

for the quadratic twist by λ K^{}. Then

Δ_{H^{2}} S_{λ} λΔ_{H^{2}} S and Δ_{Pic} S_{λ} λ^{rk Pic S\overline{K} 1} Δ_{Pic} S .

Proof. Let 𝔭 be a good prime of S such that λ is a 𝔭-adic unit. Then, for the reductions mod 𝔭, one has that S_{λ} _{𝔭} is a non-trivial twist of S_{𝔭} in the case that λ is a non-square modulo 𝔭, and S_{𝔭} S_{λ} _{𝔭}, otherwise. The assertion therefore follows from [EJ10, Fact 25]. □

Remark 2.7.2 (The odd rank case). Assume that rk Pic S\overline{K} 1. Then, for any prime 𝔭 of good reduction, there exists some 𝔭-adic unit λ K^{}such that Δ_{H^{2}} S_{λ} Δ_{Pic} S_{λ} is a non-square modulo 𝔭. If the effect of the odd rank added up with that of the transcendental character, then this would imply

rk Pic S_{\overline{𝔽}_{𝔭}} rk Pic S_{λ} _{\overline{𝔽}_{𝔭}} 4 .

There are, however, explicit degree 2 K3 surfaces known of geometric Picard rank 1 that reduce to geometric Picard rank 2 at certain primes [vL, Theorem 3.1], cf. [EJ11, Example 5.1.1].

Example 2.7.3 (The case of real multiplication). Let S be the minimal desingularisation of the double cover of P_{ℚ}^{2} , given by w^{2} X_{0}X_{1}X_{2} f_{3} X_{0}, X_{1}, X_{2} , for

f_{3} X_{0}, X_{1}, X_{2} : X_{0}^{3} 3X_{0}^{2}X_{1} 2X_{0}^{2}X_{2} 5X_{0}X_{1}^{2} X_{0}X_{2}^{2} 3X_{1}^{3}

2X_{1}^{2}X_{2} 3X_{1}X_{2}^{2} 2X_{2}^{3} .

There is strong evidence that S has real multiplication by ℚ \overline{3} . Indeed, S is the surface V_{1,1}^{3} from [EJ16, Conjectures 5.2]. Its model 𝒮 being the double cover of P_{ℤ}^{2} given by the same equation has bad reduction only at 2, 3, and 5. Modulo all other primes p 1000, the reduction 𝒮_{𝔽_{p}} is of geometric Picard rank 18, except for p 263, where the geometric Picard rank is 22. On the other hand, a sublattice of Pic S\overline{ℚ} of rank 16 may be explicitly given. Altogether, taking real multiplication for granted, one concludes that rk Pic S\overline{ℚ} 16 .

Concerning Pic S\overline{ℚ} , there are 13 obvious generators, given by the pull-back of a general line on P_{ℚ}^{2} and the exceptional curves obtained by blowing up the twelve singular points of the ramification locus. Ten of these singular points are defined over ℚ, the other two over ℚ \overline{2} . Hence, this part of Pic S\overline{ℚ} _{ℤ} ℂ splits into irreducible components as χ_{triv}^{12} χ_{ℚ \overline{2}} . Further generators are formed by a line and two conics in P_{ℚ}^{2} , the preimages of which split in S. From these altogether, one calculates that

Pic S\overline{ℚ} _{ℤ} ℂ χ_{triv}^{12} χ_{ℚ \overline{2}} χ_{ℚ \overline{2}} χ_{ℚ \overline{6}} χ_{ℚ \overline{6}}

and, consequently, Δ_{Pic} S 1. The splitting field of Pic S\overline{ℚ} is ℚ i, \overline{2}, \overline{3} . On the other hand, Algorithm 2.6.1 yields Δ_{H^{2}} S 3, so the jump character is given by \frac{3}{}.

If the effect of real multiplication added up with that of the transcendental character then this would imply rk Pic S_{\overline{𝔽}_{p}} 18 for every prime p such that \frac{3}{p} 1 , a contradiction.

3. Infinitely many rational curves

It has since long been conjectured that every K3 surface S over an algebraically closed field K contains infinitely many rational curves. The problem has been settled only recently by X. Chen, F. Gounelas, and C. Liedtke [CGL]. Many particular cases had been known before, most notably, that of odd Picard rank ([LL], based on the ideas of [BHT], cf. [Ben15]). Other sufficient conditions included those that S has infinitely many automorphisms, that S is elliptic [BT], or that K is of characteristic zero and S cannot be defined over \overline{ℚ} [BHT, Theorem 3].

As an application of Theorem 2.4.4, we show that the existence of infinitely many rational curves may be obtained rather easily in the situation that S is defined over \overline{ℚ} , the jump character is non-trivial and the surface is otherwise generic. Our result is as follows.

Theorem 3.1. Let K be a number field and S a K3 surface over K. Assume that rk Pic S\overline{K} is even, that S\overline{K} has neither real nor complex multiplication, and that Δ_{H^{2}} S Δ_{Pic} S is a non-square in K. Then S\overline{K} contains infinitely many rational curves.

Remark 3.2. The transcendental part of T H^{2} S_{ℂ} , ℚ , considered as a pure weight-2 Hodge structure, has an endomorphism algebra End_{Hs} T that may only be a totally real field or a CM field [Za, Theorem 1.6.(a) and Theorem 1.5.1]. Our assumption concerning real and complex multiplication just means that End_{Hs} T ℚ, which is fulfilled as long as X is sufficiently general.

Lemma 3.3. Let K be a number field and S a K3 surface over K. Assume that S\overline{K} has neither real nor complex multiplication. Then, for every quadratic field extension L K, there are infinitely many inert primes 𝔭 𝒪_{K} such that the reduction S_{\overline{𝔽}_{𝔭}} is non-supersingular.

Proof. The case that rk Pic S\overline{K} 20 is degenerate of the kind that T contains no 1, 1 -classes. It is known in this situation that S\overline{K} automatically has complex multiplication [SI, Theorem 4]. We may therefore assume that r : dim T 3.

We choose a prime l and put T_{l} H_{ét}^{2} S\overline{ℚ} , ℚ_{l} to be the transcendental part of l-adic cohomology. Then GO T_{l}, . , . _{\overline{ℚ} _{l}} is an algebraic group over ℚ_{l}. One has GO T_{l}, . , . _{\overline{ℚ} _{l}} GO_{r,\overline{ℚ} _{l}}, so that there are two connected components when r is even, while the group is connected for odd r.

As End_{Hs} T ℚ, we know that the image of the canonical continuous representation ϱ_{l}: Gal \overline{K} K GO T_{l}, . , . is Zariski dense either in GO T_{l}, . , . or in the neutral component GO^{0} T_{l}, . , . . Indeed, this follows from the Mumford–Tate conjecture, proven by S. G. Tankeev [Ta90, Ta95], together with Yu. G. Zarhin’s explicit description of the Mumford–Tate group in the case of a K3 surface [Za, Theorem 2.2.1].

Now, let us assume, to the contrary, that for all but finitely many inert primes 𝔭, the reduction S_{\overline{𝔽}_{𝔭}} were supersingular. We put

M : 𝔭 𝒪_{K} prime ideal

𝔽_{𝔭} is a prime field, #𝔽_{𝔭} l, 𝔭 inert in L, 𝔭 good for S, S_{\overline{𝔽}_{𝔭}} supersingular .

Then M I, for I the set of all inert primes, and the difference I M is of analytic density zero. Indeed, the prime ideals 𝔭 such that 𝔽_{𝔭} is a prime field form a set of density 1.

For every prime ideal 𝔭 𝒪_{K}, we choose a geometric Frobenius automorphism Frob_{𝔭} Gal \overline{K} K . According to the Chebotarev density theorem, the elements σ^{1} Frob_{𝔭} σ Gal \overline{K} K , for 𝔭 M and σ Gal \overline{K} K , are topologically dense in the non-trivial coset of Gal \overline{K} K modulo Gal \overline{K} L . Thus, there are two elements σ_{1}, σ_{2} Gal \overline{K} K such that

σ_{j}σ^{1} Frob_{𝔭} σ j 1, 2, 𝔭 M, σ Gal \overline{K} K

is dense in Gal \overline{K} K .

On the other hand, for 𝔭 M one has, due to supersingularity, p Tr Frob_{𝔭,T_{l}}, when writing p : #𝔽_{𝔭}. Moreover, det Frob_{𝔭,T_{l}} p^{r} . As Tr Frob_{𝔭,T_{l}} rp, this shows that

Tr Frob_{𝔭,T_{l}} ^{r} k^{r} det Frob_{𝔭,T_{l}} ,

for 22 r k r 22. Accordingly, let C_{k} GO T_{l}, . , . be the closed subscheme, defined by the equation Tr A ^{r} k^{r} det A, and put C : _{k r}^{r} C_{k}. Then C GO T_{l}, . , . is a closed subscheme and invariant under conjugation.

As GO T_{l}, . , . _{\overline{ℚ} _{l}} GO_{r,\overline{ℚ} _{l}}, for r 3, it is easily seen that C cannot include a complete component of GO T_{l}, . , . _{\overline{ℚ} _{l}}. I.e., one has dim C dim GO T_{l}, . , . . Thus, the union σ_{1}C σ_{2}C cannot be the whole group. Consequently, the image of Gal \overline{K} K GO T_{l}, . , . is Zariski dense neither in GO T_{l}, . , . , nor in GO^{0} T_{l}, . , . , a contradiction. □

Proposition 3.4 (Li–Liedtke). Let K be a number field and S P_{K}^{N} a K3 surface. Assume that Pic S Pic S\overline{K} and that there is an infinite set J of primes such that rk Pic S_{\overline{𝔽}_{𝔭}} rk Pic S for 𝔭 J. Then there exist a sequence without repetitions 𝔭_{j} _{j ℕ} of primes from J and a sequence D_{𝔭_{j}} _{j ℕ} of rational curves D_{𝔭_{j}} S_{\overline{𝔽}_{𝔭_{j}}} such that the following two conditions are satisfied. The class D_{𝔭_{j}} Pic S_{\overline{𝔽}_{𝔭_{j}}} does not lie in the image of Pic S\overline{K} under specialisation, for any j, and lim_{j}deg D_{𝔭_{j}} .

Proof. This is [LL, Proposition 4.2]. □

Proposition 3.5 (Li–Liedtke). Let K be a number field and S P_{K}^{N} a K3 surface. Assume that there exist a sequence without repetitions 𝔭_{j} _{j ℕ} of primes and a sequence D_{𝔭_{j}} _{j ℕ} of rational curves D_{𝔭_{j}} S_{\overline{𝔽}_{𝔭_{j}}} satisfying the following conditions.

i) Each S_{\overline{𝔽}_{𝔭_{j}}} is non-supersingular, ii) D_{𝔭_{j}} does not lie in the image of the Picard group Pic S\overline{K} of the generic fibre under specialisation, for any j, and iii) lim_{j}deg D_{𝔭_{j}} .

Then, for every j 0, there exists a rational curve D_{j} S\overline{K} such that its specialisation to S_{\overline{𝔽}_{𝔭_{j}}} is reducible, containing D_{𝔭_{j}} as one of its components. In particular, deg D_{j} deg D_{𝔭_{j}} .

Proof. This is shown in the proof of [LL, Theorem 4.3]. □

3.6. Proof of Theorem 3.1. As Δ_{H^{2}} S Δ_{Pic} S is a non-square in K, the field L : K \overline{Δ_{H^{2}} S Δ_{Pic} S} is indeed a quadratic extension. By Lemma 3.3, we have an infinite set J of inert primes such that S_{\overline{𝔽}_{𝔭}} is non-supersingular for every 𝔭 N. Moreover, rk Pic S_{\overline{𝔽}_{𝔭}} rk Pic S\overline{K} according to Theorem 2.4.4.b).

Let now K^{}K be the splitting field of Pic S\overline{K}. For each 𝔭 J, there is at least one prime 𝔭^{}𝒪_{K^{}}lying above 𝔭. This yields an infinite set J^{}of primes in 𝒪_{K^{}}, to which Proposition 3.4 applies. It provides a sequence 𝔭_{j} _{j ℕ} of primes in J^{}without repetitions and rational curves D_{𝔭_{j}} S_{\overline{𝔽}_{𝔭_{j}}} , not lying in the image of Pic S\overline{K} under specialisation, such that lim_{j} deg D_{𝔭_{j}} . Knowing this, Proposition 3.5 yields a sequence D_{j} _{j ℕ} of rational curves on S\overline{K} of degrees tending towards infinity. This completes the proof. □

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E-mail address: edgarc@mit.edu
URL: https://math.mit.edu/~edgarc/

Institut für Mathematik, Universität Würzburg, Emil-Fischer-Straße 30, D-97074 Würzburg, Germany

E-mail address: stephan.elsenhans@mathematik.uni-wuerzburg.de
URL: https://www.mathematik.uni-wuerzburg.de/computeralgebra/team/elsenhans-step
han-prof-dr/

Department Mathematik, Univ. Siegen, Walter-Flex-Str. 3, D-57068 Siegen, Germany

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