¯ Isogenies between K3 Surfaces over Fp

Ziquan Yang

Abstract We generalize Mukai and Shafarevich’s definitions of isogenies between K3 surfaces over C to an arbitrary perfect field and describe how to construct isogenous K3 surfaces ¯ over Fp by prescribing linear algebraic data when p is large. The main step is to show that isogenies between Kuga-Satake abelian varieties induce isogenies between K3 surfaces, in the context of integral models of Shimura varieties. As a byproduct, we show that every K3 surface of finite height admits a CM lifting under a mild assumption on p.

1 Introduction

Efforts to define the notion of isogeny between K3 surfaces have a long history. We refer the reader to [33] for a summary. Shafarevich defined an isogeny between two complex algebraic ∼ K3 surfaces X,X0 to be a Hodge isometry H2(X0, Q) → H2(X, Q) (cf. [42]). Mukai reserved the term for those Hodge isometries that are induced by correspondences (cf. [34]). Thanks to a recent result of Buskin [10, Theorem 1.1], now we know these two definitions coincide. Therefore, we make the following definition, which generalizes Mukai and Shafarevich’s:

Definition 1.1. Let k be a perfect field with algebraic closure k¯. Let X,X0 be two K3 surfaces over k with a correspondence (i.e., algebraic cycles on X × X0 over k with Q- 0 coefficients) f : X X .

2 0 ∼ • If char k = 0, we say f is an isogeny over k if the induced map Hét(Xk¯, Af ) → 2 Hét(Xk¯, Af ) is an isometry.

2 0 p ∼ • If char k = p, we say f is an isogeny over k if the induced maps Hét(Xk¯, Af ) → 2 p 2 0 ∼ 2 Hét(Xk¯, Af ) and Hcris(X /W (k))[1/p] → Hcris(X/W (k))[1/p] are isometries. Two K3 surfaces X and X0 over k are said to be isogenous if there exists an isogeny between them. Two isogenies are viewed as equivalent if their cohomological realizations mentioned above all agree.

Starting with a complex algebraic K3 surface, one can easily construct an isogenous one by prescribing a different Z lattice in its rational Hodge structure. More precisely, using Buskin’s result, the surjectivity of the period map and [4, IV Theorem 6.2], one easily deduces the following:

Theorem 1.2. Let X be a complex algebraic K3 surface. Let Λ be a quadratic lattice isomorphic to H2(X, Z). Then for each isometric embedding Λ ⊂ H2(X, Q), there exists

1 0 0 another complex algebraic K3 surface X together with an isogeny f : X X such that f ∗H2(X0, Z) = Λ.

We phrase and prove a partial analogue of Theorem 1.2 for quasi-polarized K3 surfaces in positive characteristic, using K3 crystals in the sense of Ogus and Zˆp-lattices to replace Z-lattices. K3 crystals are F-crystals with properties modeled on the second crystalline cohomology of K3 surfaces (cf. [37, Def. 3.1]):

Definition 1.3. Let k be a perfect field of characteristic p > 0 and let σ be the lift of Frobenius on W (k).A K3 crystal over k is a finitely generated self-dual quadratic lattice D over W (k) equipped with a σ-linear injection ϕ : D → D such that p2D ⊂ ϕ(D), rank ϕ⊗k = 1 and ϕ satisfies the following equation with the symmetric bilinear pairing h−, −i on D: For any x, y ∈ D, hϕ(x), ϕ(y)i = p2σ(hx, yi).

Recall that a quasi-polarized K3 surface is a pair (X, ξ) where X is a K3 surface and ξ is a big and nef line bundle on X. We call the self-intersection number of ξ the degree of 2 ⊥ 2 (X, ξ). Customarily, we write P∗ (X) := hch∗(ξ)i ⊂ H∗ (X) for the primitive cohomology of (X, ξ) (∗ = dR, cris, ét, etc). Our main theorem states: ¯ Theorem 1.4. Let (X, ξ) be a quasi-polarized K3 surface over Fp of degree 2d. Assume ¯ that p > 18d + 4. Let (Mp, ϕ) be a K3 crystal over Fp such that as a quadratic lattice Mp is 2 ¯ p 2 ˆp isomorphic to Pcris(X/W (Fp)) and let M be a quadratic lattice isomorphic to Pét(X, Z ). Then for each pair of isometric embeddings

2 ¯ (i) Mp ⊂ Pcris(X/W (Fp))[1/p] such that ϕ agrees with the Frobenius action

p 2 p (ii) M ⊂ Pét(X, Af )

0 0 ¯ there exists another quasi-polarized K3 surface (X , ξ ) of degree 2d over Fp with an isogeny 0 ∗ 0 ∗ 2 0 ¯ f : X X such that f ch∗(ξ ) = ch∗(ξ) for ∗ = cris, ét, f Pcris(X /W (Fp)) = Mp and ∗ 2 0 ˆp p f Pét(X , Z ) = M .

In section 6, we will propose a conjecture (cf. Conjecture 6.7) which is an exact ana- logue of Thm 1.2 and explain how Thm 1.4 is a partial result towards this conjecture (cf. Rmk 6.10). Moreover, we show that this conjecture is completely known for supersingular K3 surfaces in odd characteristic (cf. Prop. 6.12). Therefore, the paper is mainly concerned with the case when X is not supersingular. Our main idea is to apply Kisin’s results on the Langlands-Rapoport conjecture [22] to Shimura varieties which parametrize K3 surfaces, via the Kuga-Satake construction. We summarize the construction in the following diagram:

S (CSpin(Ld), Ω) A Siegel modular variety

Me 2d,Z(p) S (SO(Ld), Ω)

In this diagram, Ld is a certain quadratic lattice; S (CSpin(Ld), Ω) and S (SO(Ld), Ω) are the canonical integral models of the Shimura varieties associated to algebraic groups

CSpin(Ld), SO(Ld) and a certain period domain Ω; Me 2d,Z(p) is a moduli stack of primitively

2 quasi-polarized K3 surfaces of degree 2d with some additional structures. We have tem- porarily suppressed level structures, but we will introduce these objects in detail in Section ¯ 3. The work of Kisin clarifies the notion of isogeny classes on S (CSpin(Ld), Ω)(Fp). We can prove that, as one would expect, K3 surfaces with isogenous Kuga-Satake abelian varieties are themselves isogenous.

0 ¯ Proposition 1.5. Assume p - d and p ≥ 5. Let t, t ∈ Me 2d,Z(p) (Fp) correspond to quasi- 0 polarized K3 surfaces (Xt, ξt) and (Xt0 , ξt0 ) of degree 2d. If the images of the points t, t ¯ 0 ¯ in S (SO(Ld), Ω)(Fp) lift to points s, s ∈ S (CSpin(Ld), Ω)(Fp) in the same isogeny class, ∗ then there exists an isogeny Xt Xt0 such that f ch∗(ξt0 ) = ch∗(ξt) for ∗ = cris, ét. The precise form of this assertation will be stated in Prop. 5.2. Once we have Prop. 5.2, we can prove Theorem 1.4 by a group-theoretic computation. The condition on p arises when we apply the surjectivity of the period map, due to Matsumoto (cf. [30, Thm 4.1]). As a key intermediate step in [22], Kisin proved that in each isogeny class of mod p points of a Shimura variety of Hodge type, there exists a point which lifts to a special point. ¯ Prop. 1.5 implies that, under mild assumptions, K3 surfaces over Fp also admit CM liftings up to isogeny. It turns out that for K3 surfaces of finite height, one can propagate the property of having CM lifting within an isogeny class, so there is no need to pass to isogeny. ¯ Theorem 1.6. Let X/Fp be a K3 surface of finite height. Suppose X admits a quasi- ¯ polarization of degree p - d and p ≥ 5. Then there exists a finite extension V/W (Fp) and a lift XV of X to V such that for any V,→ C, the complex K3 surface XV ×C has commutative Mumford-Tate group. Moreover, the natural map Pic (XV ) → Pic (X) is an isomorphism.

While preparing this paper, the author noticed that K. Ito, T. Ito and T. Koshikawa released a new preprint [20] which contains the above result with no assumption on p. Finally, we remark that isogenies between K3 surfaces (in our sense) can also be given by constructing moduli of twisted sheaves. The moduli theory of twisted sheaves for complex K3’s was initiated by S. Mukai and its generalization to positive characteristic has been studied by Lieblich, Maulik, Olsson and Snowden (cf. [25], [26], [24]). Huybrechts has shown that in fact, over C, every isogeny can be realized by a sequence of such operations [18, ¯ Thm 0.1]. It would be interesting to figure out whether this is true over Fp.

Outline/Strategy of the Paper In section 2 and 3, we review the Kuga-Satake con- struction and the relevant Shimura varieties, mostly following [28] and [29]. In addition, we emphasize that an isogeny between Kuga-Satake abelian varieties should not be a bare isogeny between abelian varieties, but one which repects certain tensors. We call such iso- genies “CSpin-isogenies". This is an analogue of André’s notion of an isomorphism between Kuga-Satake packages (cf. [1, Def. 4.5.1, 4.7.1]). We also discuss special endomorphisms of Kuga-Satake abelian varieties, which correspond to line bundles on K3 surfaces. Special endomorphisms are preserved by CSpin-isogenies. In Section 4, we prove a lifting lemma, which we use in Section 5 to prove the theorems of the paper. The main idea is that a K3 surface of finite height deforms like an elliptic curve: There is a natural one-dimensional formal group of finite height attached to each such surface X, namely the formal Brauer group Brˆ X of X. Moreover, there is a natural map of

3 deformation functors Def → Def . Although this map is not an isomorphism, Nygaard X Brˆ X and Ogus constructed a natural section to this map. We make use of this section to show that away from the supersingular locus, CSpin-isogenies always lift to characteristic zero. This treats the finite height case for Prop. 1.5. We treat the supersingular case separately. In Section 6, we discuss how isogenies interact with ample cones, polarizations, and quasi-polarizations, and how Torelli theorems can be understood from the perspective of isogenies. Then we will discuss how Thm 1.4 is related to an exact analogue of Thm 1.2.

Notations and Conventions

• Zˆ denotes the profinite completion of Zˆ, and Zˆp denotes the prime-to-p part of Zˆ. We p write Af for the finite adeles and Af for the prime-to-p part. When k is a perfect field of characteristic p > 0, we denote by W (k) the Witt ring of vectors. The lift of Frobenius on W (k) is denoted by σ. If X is a smooth projective variety over k, the 2 σ-linear Frobenius action on Hcris(X/W (k)) is denoted by F .

• Suppose k is a perfect field of characteristic p > 0 and S is a scheme on which p is locally nilpotent over W (k). For any p-divisible group G over S, we let D(G ) denote its contravariant Dieudonné crystal on Cris(S/W (k)) and G ∗ denote its Cartier dual. When S = Spec k, we also write the W (k)-module D(G )W (k) simply as D(G ). • For a ring R and a finite free R-module M, we denote by M ⊗ the direct sum of the R-modules obtained from M by taking duals, tensor products, symmetric and exterior powers.

• For any field extension E/F and algebraic group G over E, we write Res E/F G for the

Weil restriction of G to F . S := Res C/RGm denotes the Deligne torus. • By a quadratic lattice over a ring R we always mean a finitely generated free module over R equipped with a symmetric bilinear pairing.

2 Hodge Theoretic Preparations

Hodge Structures of K3 Surfaces We collect some definitions and facts about Hodge structures of K3 type. The definition below differs from the usual definition (cf. [18, Ch. 3 Def. 2.3]) by a Tate twist.

Definition 2.1. A rational or integral Hodge structure V of weight 0 is said to be of K3 type if dim V −1,1 = 1 and V a,b = 0 if |a − b| > 2.

Let X be a complex K3 surface. The transcendental lattice of X, denoted by T (X), is Pic (X)⊥ ⊂ H2(X, Z(1)). H2(X, Z(1)) and T (X) are both Hodge structures of K3 type.

Moreover, if X is algebraic, then T (X)Q := T (X)⊗Q is polarizable and irreducible as a Hodge structure (cf. [17, Lemma 3.1]). We denote by MT(−) the Mumford-Tate group associated to a Hodge structure.

Theorem 2.2. Let V be a rational polarized irreducible Hodge structure which is of K3 type.

4 (a) E := EndHdg(V ) is either a totally real field or a CM field.

(b) MT(V ) is commutative if and only if dimE V = 1. In this case, E is a CM field and MT(V ) ⊂ GL(V ) is equal to

ker(Nm : Res E/QGm → Res F/QGm) (2.3)

where F is the maximal totally real subfield of E and Nm is the norm map. Note that by

definition E is a subalgebra of End(V ), so there is an embedding Res E/QGm ⊂ GL(V ).

(c) When E is a CM field, there exists a Hodge isometry τ such that E = Q(τ).

Proof. (a) and (b) are due to Zarhin (cf. [48, Thm 1.5.1, Rmk 1.5.3, Thm 2.3.1]). (c) can be found in [17, Thm 3.7], where the result is credited to Borcea [8].

Remark 2.4. X is said to have CM if MT(T (X)Q) is commutative. Some authors say that X has CM if E is a CM field. We emphasize that E being a CM field is not enough to ensure that MT(T (X)Q) is commutative. Here is an example: Take two elliptic curves E1 and E2. Suppose E1 has CM but E2 does not. Apply the Kummer construction to the abelian surface

E1 × E2. We obtain a K3 surface X with dim T (X)Q = 4 but E := EndHdg(T (X)Q) is a

CM field of dimension 2. The real points of the Hodge group Hdg(T (X)Q) form a unitary group of signature (1+, 1−) which is not commutative.

A Review of the Clifford Algebra Let L be a self-dual quadratic lattice over a com- mutative ring R. Let q denote the quadratic form attached to L. Assume that 2 is invertible in R. We can form the Clifford algebra

⊗n Cl(L) := (⊕n≥0L )/(v⊗v − q(v)).

+ − Cl(L) has a natural Z/2Z-grading and we denote by Cl (L) (resp. Cl (L)) the even (resp. odd) part. We define the group CSpin(L) by

CSpin(L) = {v ∈ Cl+(L)× : vLv−1 = L}.

We give CSpin(L) the structure of an algebraic group over R by defining CSpin(L)(R0) = 0 CSpin(LR0 ) for any R-algebra R . There is a norm map Nm : CSpin(L) → Gm given by v 7→ v∗ · v, where v∗ is the natural anti-involution of v. By letting CSpin(L) act on L by conjugation, we get the adjoint representation of CSpin(L), which fits into an exact sequence of algebraic groups ad 1 → Gm → CSpin(L) → SO(L) → 1. (2.5)

Let H be a free Cl(L) bi-module of rank 1. Note that H has a natural Z/2Z-grading. Left multiplication of Cl(L) on H gives us a spin representation sp : CSpin(L) → GL(H) and an embedding L,→ Cl(L) ,→ End(H) (2.6)

Equip End(H) with the metric (α, β) := 2−rank Ltr(α ◦ β). Then we have

5 Lemma 2.7. (a) The embedding (2.6) is an isometry with respect to q and (−, −).

(b) There exists a unique orthogonal projection π : End(H) → L.

(c) CSpin(L) is the stabilizer of π ∈ H(2,2) among all automorphisms of H as a Z/2Z- graded right Cl(L)-module.

Proof. See [29, 1.3].

The Kuga-Satake Construction Take R = Q and assume that (L, q) has signature (n+, 2−). The Kuga-Satake construction associates a Hodge structure eh on H of type {(1, 0), (0, 1)} to each Hodge structure h on L of type {(−1, 1), (0, 0), (1, −1)} which respects q. It can be packed in the following diagram

sp CSpin(LR) GL(HR) eh ad h S SO(LR)

Figure 1: Kuga-Satake construction for a Hodge structure of K3 type

For each h, there is a unique lift eh such that Nm ◦ eh gives the norm map S → Gm,R. The composition sp ◦ eh gives the desired Hodge structure of weight 1 on H (cf. [12, 4.2]). In fact, the induced Hodge structure on H has a simple description by that on L:

1 1 Fil HC = ker(Fil LC)

1 Here we are viewing L as a subspace of End(H) via left multiplication. Since Fil LC is 1 one-dimensional, it makes sense to talk about ker(Fil LC). The following simple observation makes the Kuga-Satake construction naturally suitable for studying complex multiplication:

Lemma 2.8. MT(eh) is commutative if and only if MT(h) is commutative.

Proof. We obtain the following exact sequence

1 → Gm,Q → MT(eh) → MT(h) → 1 by pulling back (2.5) along the inclusion MT(h) ,→ SO(L). If MT(h) is commutative, then by Thm 2.2(b) it is a torus. One easily deduces from [9, IV Cor. 11.5] that an extension of a torus by another torus is again a torus, so MT(eh) is commutative. The converse implication is clear.

Lemma 2.9. Every isometry g ∈ SO(L)(Q) preserving the Hodge structure h lifts to ge ∈ CSpin(L)(Q) preserving eh. In other words, the natural map of centralizers

CCSpin(L)(MT(eh))(Q) → CSO(L)(MT(h))(Q) is surjective.

6 Proof. Let ge ∈ CSpin(L)(Q) be any lift of g ∈ SO(L)(Q). In fact, since the kernel of ad lies in the center of CSpin(L), one lift works if and only if any other one does. Let ω be 1 1 any generator of Fil LC. Since g preserves the Hodge structure on L and Fil LC is one- −1 dimensional, we have gωe ge = λω, or equivalently gωe = λωg,e for some λ 6= 0 ∈ C. Now if a ∈ Fil1H , then ωga = λ−1gωa = 0, which implies that ga ∈ Fil1H . C e e e C

3 The Kuga-Satake Period Map

3.1 Shimura Varieties In this section, we review the theory of spinor and orthogonal Shimura varieties that we need.

3.1.1 Let L be a quadratic lattice of signature ((m − 2)+, 2−) with m ≥ 3 over Z. Let q be the quadratic form attached to L. Let p > 2 be a prime. Assume that L⊗Z(p) is ad self-dual. Let G (resp. G ) denote the reductive group over Z(p) given by CSpin(L⊗Z(p)) (resp. SO(L⊗Z(p))). Again let H denote Cl(L), viewed as a Cl(L)-bimodule. For every p ad compact open subgroup U of G(Af ), G(Af ), or G(Qp), we denote by U the image of U ad under the adjoint map G → G . Set Kp = G(Zp). Let KL ⊂ G(Af ) be the intersection ˆ × ad p ad p,ad p G(Af ) ∩ Cl(L⊗Z) . Then KL (resp. KL ) is of the form KpKL (resp. Kp KL ), where KL p,ad p ad p (resp. KL ) is a compact open subgroup of G(Af ) (resp. G (Af )). We fix a choice of a non-zero element δ ∈ det(L) = ∧mL. Since there is a natural embedding L,→ End(H) given by left multiplication, we can also view δ as an element of ∧mH⊗(1,1).

Let Ω be the space of oriented negative definite planes in LR. Then the pairs (GQ, Ω) and (Gad, Ω) define Shimura data with reflex field (cf. [29, 3.1]). We can choose a symplectic Q Q form ψ : H × H → Z such that the embedding G,→ GL(H) induced by the spin representa- tion factors through the general symplectic group GSp := GSp(H, ψ) (cf. [29, 3.5]). Let H be the Siegel half-spaces attached to (H, ψ). Then we have an embedding of Shimura data (G, Ω) ,→ (GSp, H).

Let KH ⊂ GSp( f ) be the maximal compact open subgroup which stabilizes Hˆ. Then A Z p p p KH is of the form KpKH with Kp ⊂ GSp(Qp) and KH ⊂ GSp(Af ). We have that Kp = Kp ∩ G(Qp).

3.1.2 We introduce the moduli interpretation of the Shimura variety ShK(GSp, H) (cf. [29, p 3.7], see also [22, 1.3.4]), assuming that K is of the form KpK , where Kp is the p-part of KH p p as above and K ⊂ GSp(Af ) is chosen to be sufficiently small. Let T be a Z(p)-scheme. Let A (T ) denote the category of abelian schemes over T and let A (T )⊗Z(p) be the category obtained by tensoring the Hom groups in A (T ) by Z(p). An object in A (T )⊗Z(p) is called an abelian sheme up to prime-to-p isogeny over T . An isomorphism in A (T )⊗Z(p) is called 0 ∗ a p -quasi-isogeny. Now let B be an object in A (T )⊗Z(p) and let B be its dual. By a weak polarization on B we mean an equivalence class of p0-quasi-isogenies λ : B → B∗ such that some multiple of λ is a polarization and two such λ’s are equivalent if they differ × by an element in Z(p). Let f : B → T be the structure morphism. For the pair (B, λ), p 1 p p ∼ 1 p denote by Isom(H⊗Af ,R fét∗Af ) the set of isomorphisms H⊗Af → R fét∗Af which are p × p compatible with the pairings induced by ψ and λ up to a (Af ) scalar. Here H⊗Af denotes

7 p the constant sheaf on T with coefficients H⊗Af .A K-level structure on (B,T ) is a section p 0 p 1 p p εK ∈ H (T, Isom(H⊗Af ,R fét∗Af )/K ). The functor which sends each T over Z(p) to p the set of isomorphism classes of triples (B, λ, εK) as above is representable by a scheme SK(GSp, H) over Z(p), whose fiber of Q is naturally identified with ShK(GSp, H). The above description of points on SK(GSp, H) or ShK(GSp, H) depends only on the

Z(p)-lattice HZ(p) . Since we have a chosen Z-lattice H inside HZ(p) and ψ induces an em- ∨ bedding H,→ H , there is a universal abelian scheme over SK(GSp, H) (cf. [21, 2.3.3]). More precisely, if d0 is the index of H in H∨ under the embedding induced by ψ, then p for each triple (B, λ, εK) as in the previous paragraph, there is a unique triple of the form q p 0 (A → T, µ, K) where A is an abelian scheme over T , µ is a polarization on A of degree d , p 0 ˆp 1 ˆp p and K ∈ H (T, Isom(H⊗Z ,R qét∗Z )/K ) such that when we view A only as an abelian p scheme up to prime to p isogeny and µ as a weak polarization on λ, the triple (A, µ, K⊗Q) p ˆp 1 ˆp is isomorphic to (B, λ, εK). Here the (pro)-étale sheaf Isom(H⊗Z ,R qét∗Z ) on T is de- ˆp ∼ 1 ˆp fined to be the set of the isomorphisms H⊗Z → R fét∗Z which are compatible with the pairings induced by ψ and λ up to a (Zˆp)× scalar. Note that since we have assumed p p p that K is sufficiently small, the triple (B, λ, εK), and hence (A, µ, K), have no nontrivial automorphisms. p p By [21, Lem. 2.1.2], for any choice of compact open subgroup K ⊂ G(Af ), there ex- p p p p ists a compact open subgroup K ⊂ GSp(Af ) which contains K such that for K := KpK p and K := KpK , the embedding of Shimura data (G, Ω) ,→ (GSp, H) induces an embed- p p ding ShK(G, Ω) ,→ ShK(GSp, H). When K and K are chosen to be sufficiently small, the canonical integral model SK(G, Ω) of ShK(G, Ω) over Z(p) is contructed by taking the normalization of the closure of ShK(G, Ω) inside SK(GSp, H) (cf. [21, Thm 2.3.8]), and the ad ad canonical integral model SKad (G , Ω) of ShKad (G , Ω) over Z(p) can be constructed out of SK(G, Ω) as a quotient (cf. [21, 2.3.9], see also [29, Thm 4.4]). What is important for us ad is that the natural morphism ShK(G, Ω) → ShKad (G , Ω) is a finite étale cover which is a torsor of the group (cf. [29, (3.2)])1

× × × p,× × p p,× ∆(K) = Af /Q (K ∩ Af ) = Af /Z(p)(K ∩ Af ) (3.1)

ad and this morphism extends to a pro-étale cover SK(G, Ω) → SKad (G , Ω) which is a torsor of the same group.

3.1.3 Next, we introduce the sheaves on spinor or orthogonal Shimura varieties. To begin with, the spin representation CSpin(L) → GL(H) (resp. adjoint representation • CSpin(L) → SO(L)) gives rise to a variation of Z-Hodge structures (HB, Fil HdR,C) (resp. • • (LB, Fil LdR,C)) on ShK(G, Ω)C (cf. [29, 3.4]). (HB, Fil HdR,C) carries in addition a Z/2Z- 0 ⊗(2,2) grading, right Cl(L)-action and a tensor πB ∈ H (ShK (G, Ω)C, HB ⊗Q) such that • ⊗(1,1) • (LB⊗Q, Fil LdR,C) can be identified with the image πB((HB⊗Q) , Fil HdR,C) and ∨ ⊗(1,1) LB ⊂ LB⊗Q is identified with πB(HB ) (cf. [29, 1.5]). Moreover, since CSpin(L) stabi- m ⊗(1,1) 0 lizes δ ∈ det(L) ⊂ (∧ H ), we have a global section δB ∈ H (ShK(G, Ω)C, det(LB)). p p p For the following, assume that K = KpK with K ⊂ KL sufficiently small. 1We have communicated to Madapusi Pera and confirmed that in [29, (3.2)] there should be no “> 0" sign in the formula of ∆(K).

8 Now let h : A → SK(G, Ω) be the pullback of the universal abelian scheme over ¯ SK(GSp, H). Let hQ (resp. hC, resp. h) denote the fiber of h over Q (resp. C, resp. Fp). Let HdR be the first relative de Rham cohomology of A over SK(G, Ω). For every rational prime `, write H for R1h and for ` 6= p set H = R1h . Let han be the `,Q Q,ét∗Z` ` ét∗Z` C morphism between complex analytic manifolds associated to hC. The variation of Z-Hodge structures given by the first relative Betti and de Rham cohomology of han can be identified C • with the aforementioned (HB, Fil HdR,C). Let Hcris denote the crystal of vector bundles on the the crystalline site Cris( (G, Ω) / ) given by R1h¯ O . SK Fp Zp cris∗ A⊗Fp/Zp • Since the variation of Z-Hodge structures (HB, Fil HdR,C) carries a Z/2Z-grading and a right Cl(L)-action, the abelian scheme AC over ShK(G, Ω)C carries a Z/2Z-grading and a left Cl(L)-action. The global section πB gives rise to a global section πdR,C of the vector 0 ⊗(2,2) ⊗(2,2) bundle Fil H and a global section π`, of H , where H`, is the étale local system dR,C C `,C C over ShK(G, Ω)C obtained by restricting H`. Similarly, δB gives rise a global section δ`,C of ∧mH⊗(1,1). `,C Now we explain how to equip A with “CSpin-structures”. Results here are taken from [21,

(2.2)] and [29, Prop. 3.11, Sect. 4.5]. The Z/2Z-grading and the left Cl(L)-action on AC descend to AQ and extend to A. The global sections π`,C and δ`,C descend to global sections ⊗ ⊗ π`, and δ`, of H and if ` 6= p, extend to global sections π` and δ` of H . Similarly, Q Q `,Q ` ⊗ ⊗ the de Rham section πdR, descend to a global section πdR, of H := H | C Q dR,Q dR ShK(G,Ω) ⊗ and extend to a global section πdR of HdR (cf. [21, Cor. 2.3.9]). Finally, we introduce the crystalline realization of π. Let ∇ be the Gauss-Manin connection on HdR. Since SK(G, Ω) is smooth, the crystal of vector bundles Hcris is completely determined by the restriction of the pair (HdR, ∇) to the completion of SK(G, Ω) along the special fiber. Since the global ⊗(2,2) section πdR of HdR is horizontal with respect to ∇, it gives rise to a global section πcris of Hcris (cf. [29, 4.14]).

Remark 3.2. The following can be extracted from the proof of [21, 2.3.5, 2.3.9], or [29, Prop. 4.7]: Let k be a perfect field of characteristic p. Let K be a totally ramified extension of W (k)[1/p], OK be the ring of integers of K and K¯ be an algebraic closure of K. Let s ∈ SK(G, Ω)(k) be a point. Let sK be a K-valued point on SK(G, Ω) and s¯K be the geometric point over sK associated to K¯ . Suppose that sK specializes to s via an OK - valued point. Under the de Rham comparison isomorphism

1 ∼ 1 ¯ HdR(As/K)⊗K BdR → Hét(AsK ⊗K, Qp)⊗Qp BdR

πdR,sK is sent to πp,s¯K by a result of Blasius [7, Thm 0.3] and Wintenberger. πcris,s is sent 1 ∼ 1 to πdR,sK via the Berthelot-Ogus isomorphism Hcris(As/W (k))⊗W (k)K = HdR(AsK /K). In loc. cit., Blasius restricted to considering abelian varieties which can be obtained via base change from the ones defined over Q¯ , but this condition can be removed (cf. [32, Thm 5.6.3]). The original reference for the de Rham comparison isomorphism in [7] is [15, Thm 8.1]. It is remarked in [20, Sect. 11.3] that for Blasius’ theoerem one can alternatively use the de Rham comparison isomorphism constructed by Scholze (cf. [?, Thm 8.4], see also [20, Sect. 11.1]). For our purposes, all that is important is that πdR,sK , and hence

πcris,s, are completely determined by πp,s¯K .

9 The Z/2Z-grading, Cl(L)-action and various realizations of the tensor π are the “CSpin- structure” on A. Note that we do not think of realizations of δ as part of the “CSpin- structure” because δ is uncessary in cutting out CSpin(L) inside GL(H). Later will only make use of δB and δ` in the construction of the integral period morphism.

3.2 Isogeny Classes Consider the limits SK (G, Ω) = lim SK Kp (G, Ω) and ShK (G, Ω) = p ←−Kp p p p p lim Sh p (G, Ω) where K varies over compact open subgroups of G( ). Similarly, take ←−Kp KpK Af the limit SKp (GSp, H). We explain the notion of isogeny classes of points on SKp (G, Ω) 2 under the assumption that ψ induces a self-dual pairing on HZ(p) .

0 Definition 3.3. Let k be a perfect field and s, s ∈ SK(G, Ω)(k). Let f : As → As0 be a ¯ quasi-isogeny over k which respects the Z/2Z-grading and Cl(L)-action on As, As0 . Let k be an aglebraic closure of k. If char k = 0, we say f is a CSpin-isogeny if it sends π`,s⊗k¯ to

π`,s0⊗k¯ for every prime `. If char k = p, we say f is a CSpin-isogeny if it sends π`,s⊗k¯ to

π`,s0⊗k¯ for every prime ` 6= p and πcris,s to πcris,s0 .

¯ 3.2.1 We now begin to explain the notion of isogeny classes for Fp points, following [22]. ¯ ¯ We write W for W (Fp) and K0 for W [1/p]. Let s be an Fp-point on SKp (G, Ω). There is p ¯ a triple (As, µs, s) attached to s, where (As, µs) is a polarized abelian variety over Fp and p ˆp 1 ˆp s is an isomorphism H⊗Z → Hét(As, Z ) which respects the pairings induced by ψ and p × p µs up to a (Af ) -multiple. Moreover, s respects the Z/2Z-grading and Cl(L)-action and sends π to π p (cf. [22, (1.3.6)]). Af ,s p By the extension property of the integral model SKp (G, Ω), the (right) G(Af )-action p p 0 p on ShKp (G, Ω) extends to SKp (G, Ω). Take g ∈ G(Af ). If s = s · g , then the triple p p p (As0 , µs0 , s0 ) is isomorphic to the triple (As, µs, s ◦g ) when we view (As0 , µs0 ) and (As, µs) as weakly polarized abelian schemes up to prime-to-p isogeny. In other words, there exists a 0 p × p -quasi-isogeny f : As → As0 which sends µs0 to a (Af ) -multiple of µs and the composition

p ⊗ ∗ p s0 Q 1 p f 1 p H⊗Af → Hét(As0 , Af ) → Hét(As, Af )

p p 1 p 1 p is equal to (s⊗Q) ◦ g . Such f is unique, as the induced map Hét(As0 , Af ) → Hét(As, Af ) is completely determined. We argue that f is a CSpin-isogeny. We only need to check that f sends πcris,s to πcris,s0 . This is easily done by a lifting argument: Lift s to a W -point sW 0 p 0 0 on SKp (G, Ω) and set sW := sW · g . Then sW is a W -point which lifts s . Moreover, there 0 0 is a p -quasi-isogeny f : A → A 0 which lifts f. Let s and s be the generic points W sW sW K0 K0 of s and s0 . Choose an algebraic closure K¯ of K and let s¯ , s¯0 be the corresponding W W 0 0 K0 K0 0 0 geometric points over sK0 , s . We argue that fW ⊗K0 : AsK → As sends πp,s¯K to K0 0 K0 0 0 0 0 ¯ ∼ πp,s¯ . Choose an isomorphism K0 = and let s , s be the -points given by s¯K0 , s¯ . K0 C C C C K0 It suffices to check that fW ⊗C : As → As0 sends πB,s to πB,s0 , but this is clear: By the C C smooth and proper base change theorem, fW ⊗C sends π p ,s to π p ,s0 as f sends π p ,s Af C Af C Af to π p 0 . One deduces that f sends πcris,s to πcris,s0 using that fW ⊗K0 sends πp,s¯ to Af ,s K0 πp,s¯0 (cf. Rmk 3.2). K0

2This assumption is imposed in [22, (1.3.3)]. We do not want to replace H by H⊗(4,4).

10 1 ∼ 3.2.2 We fix an isomorphism of Z/2Z-graded right Cl(L)-modules Hcris(As/W ) = H⊗W 1 which sends πcris,s to π (cf. [29, Prop. 4.7]). Then the Frobenius action on Hcris(As/W ) takes the form bσ for some b ∈ G(W ). By [22, Lem. 1.1.12], there exists a GW -valued cocharacter −1 1 ¯ ν such that b ∈ G(W )ν(p)G(W ) and σ (ν) gives the filtration on HdR(As/Fp). Now define

−1 Xp := {gp ∈ G(K0)/G(W ): gp bσ(gp) ∈ G(W )ν(p)G(W )}.

Let Gs be the p-divisible group of As. If gp ∈ Xp, then gp · D(Gs) is stable under Frobenius and satisfies the axioms of a Dieudonné module. Hence gp · D(Gs) corresponds to a p- divisible group Ggps naturally equipped with a quasi-isogeny Gs → Ggps. We denote by Agps the corresponding abelian variety, such that the p-divisible group of Agps is identified with

Ggps and there is a quasi-isogeny fgp : As → Agps which induces Gs → Ggps. We can view ¯ Agps as an object in A (Fp)⊗Z(p) and equip it with a weak polarization and a level structure ¯ using those on As via fgp . This gives rise to a point in SKp (GSp, H)(Fp), which we denote ¯ by gps. Thus we obtain a map Xp → SKp (GSp, H)(Fp) (cf. Sect. 3.1.2) by sending gp to ¯ gps. By [22, Prop. 1.4.4], this map lifts uniquely to a map ιp : Xp → SKp (G, Ω)(Fp) such that fgp : As → Aιp(gp) = Agps respects the Z/2Z-grading, Cl(L)-action and sends πcris,s to πcris,ιp(gp). Since the level structure on Aιp(gp) is induced from the one on As via fgp , fg sends π p to π p . Therefore, fg s is a CSpin-isogeny. p Af ,s Af ,ιp(gp) p p p p ¯ Set X = G(Af ). We construct a map ιs : Xp ×X → SKp (G, Ω)(Fp) by sending the pair p p (gp, g ) to [ιp(gp)] · g . The image of ιs is called the isogeny class of s. By [22, Prop. 1.4.15] and its proof, we have:

¯ 0 p p Proposition 3.4. Let s ∈ SKp (G, Ω)(Fp) be a point. If s = [ιp(gp)] · g for (gp, g ) ∈ p ∗ 1 Xp × X , then there exists a CSpin-isogeny f : As → As0 such that f Hcris(As0 /W ) = 1 ∗ 1 ˆp p 1 ˆp gp · Hcris(As/W ) and f Hét(As0 , Z ) = g · Hét(As, Z ).

ad 3.2.3 We now study the images on ad (G , Ω)(¯ ) of isogeny classes on (G, Ω)(¯ ). SKp Fp SKp Fp ¯ p ad ad,p Let s ∈ SKp (G, Ω)(Fp) and let Xp,X be as above. Let Xp and X be the quotients of p p Xp and X under the action of Gm(Qp) and Gm(Af ) respectively.

ad Lemma 3.5. ιs descends to a map ιs which fits into a commutative diagram

p ιs ¯ Xp × X SKp (G, Ω)(Fp)

ad ad ad,p ιs ad X × X ad (G , Ω)(¯ ) p SKp Fp

p p Proof. ιs is equivariant under the action of Gm(Qp) × Gm(Af ) ⊂ ZG(Qp) × G(Af ) (cf. [22, ad Cor. 1.4.13] ). By (3.1), (G, Ω) → ad (G , Ω) is a pro-étale cover which is a torsor SKp SKp p of the group ∆ = Gm(Af )/Gm(Z(p)). One may easily check, for example by examining the p generic fiber, that the action of Gm(Qp) × Gm(Af ) factors through ∆ via the quotient map

p p ∼ p Gm(Qp) × Gm(Af ) → [Gm(Qp)/Gm(Zp) × Gm(Af )]/Gm(Q) → Gm(Af )/Gm(Z(p)) = ∆

11 1 ∼ The isomorphism Hcris(As/W ) = H⊗W we fixed in section 3.2.2 gives us an isometry ∼ ad ad Lcris,s = L⊗W . Set ν := ad ◦ ν and b := ad ◦ b.

ad 0 ad ad 0 −1 ad 0 ad ad ad Lemma 3.6. Xp = {gp ∈ G (K0)/G (W ):(gp) b σ(gp) ∈ G (W )ν (p)G (W )}

0 ad ad Proof. Let gp ∈ G (K0) be any representative of an element of Xp and let gp ∈ G(K0) be 0 −1 any lift of gp. We want to show that gp bσ(gp) ∈ G(W )ν(p)G(W ). ad Set Ge := Gm × G . Consider the central isogeny

Nm × ad : G → Ge

Let T ⊂ GZp (resp. Te ⊂ GeZp ) be the centralizer of a maximal split torus and let ΩG (resp.

ΩG) be the associated Weyl group. We can arrange that T = Te ×G G p and ν ∈ X∗(T ). e eZp Z The Cartan decomposition gives us an isomorphism

∼ X∗(T )/ΩG → G(W )\G(K)/G(W ) defined by µ 7→ µ(p)

and an analogous one for Te ⊂ GeZp . −1 0 0 The element gp bσ(gp) ∈ G(W )ν (p)G(W ) for some cocharacter ν ∈ X∗(T ). As the central isogeny Nm × ad induces an injection X (T )/Ω → X (T )/Ω , it suffices to check ∗ G ∗ e Ge that (Nm × ad) ◦ ν and (Nm × ad) ◦ ν0 lie in one Ω -orbit. By assumption, Gad(W )(ad ◦ Ge ν0)(p)Gad(W ) = Gad(W )νad(p)Gad(W ), so it remains to check that

0 Gm(W )(Nm ◦ ν )(p)Gm(W ) = Gm(W )(Nm ◦ ν)(p)Gm(W ).

−1 This follows easily from the observation that valp(Nm(gp bσ(gp))) = valp(Nm(b)) for any p-adic valuation valp.

ad,p ad p Lemma 3.7. X = G (Af )

ad Proof. By Cartan-Dieudonné theory, G(Q`)/Gm(Q`) → G (Q`) is surjective for every Q Q ad ` 6= p, so we already know that `6=p G(Q`) → `6=p G (Q`) is surjective. We have em- ad beddings G,→ GL(HZ(p) ) (resp. G ,→ GL(LZ(p) ) given by the spin representation (resp. Q 0 0 standard representation). An element g = (g`)`6=p ∈ `6=p G(Q`) (resp. g = (g`)`6=p ∈ Q ad p ad p `6=p G (Q`)) lies in G(Af ) (resp. G (Af )) if and only if for all but finitely many ` 6= p, 0 g` ∈ GL(H⊗Z`) (resp. g` ∈ GL(L⊗Z`)). 0 ad It suffices to check that for all but finitely many ` 6= p, if g` ∈ G (Q`) ∩ GL(L⊗Z`), 0 g` can be lifted to an element g` in G(Q`) ∩ GL(H⊗Z`). We claim this is true for ev- ery odd ` 6= p such that L⊗ is self dual. In this case, the -groups G and Gad have Z` Q Q Q ad natural models G and G over (`) such that G ( `) = G( `) ∩ GL(H⊗ `) and Z(`) Z(`) Z Z(`) Z Q Z ad ad G ( `) = G ( `) ∩ GL(L⊗ `). Now by Cartan-Dieudonné theory again, the map Z(`) Z Q Z ad ad G ( `) → G ( `) is surjective. The map G → G is smooth because the ker- Z(`) F Z(`) F Z(`) Z(`) ad nel m, is smooth. Therefore, G ( `) → G ( `) is also surjective. G Z(`) Z(`) Z Z(`) Z

3.3 The Period Morphism

12 3.3.1 Let U denote the standard hyperbolic plane, i.e., the quadratic lattice over Z of rank 2 2 2 with a basis x, y such that x = y = 0 and hx, yi = 1. Let E8 be the unique unimodular positive definite even lattice of rank 8. As is customary in the study of K3 surfaces, we equip the second cohomology of K3 surfaces with the negative Poincaré pairing, so that the K3 ⊕3 8 lattice Λ is isomorphic to U ⊕E2. Fix a prime p > 2 and d ∈ Z>0 which is prime to p. Let e, f be a basis of the first copy of the hyperbolic plane U such that e2 = f 2 = 0, he, fi = 1 ⊥ and set Ld = he − dfi . Ld is an integral lattice of discriminant 2d and is abstractly 2 isomorphic to P (XC, Z) for any primitively quasi-polarized K3 surface (XC, ξC) of degree ξ2 = 2d over . Recall that a quasi-polarization is said to be primitive if it is not a positive C C power of another line bundle.

3.3.2 A K3 surface over a scheme S is a proper and smooth algebraic space f : X → S whose geometric fibers are K3 surfaces. A polarization (resp. quasi-polarization) of a K3 surface X → S is a section ξ ∈ Pic (X/S)(S) whose fiber at each geometric point is an ample (resp. big and nef) line bundle. A section ξ is called primitive if for all geometric points s → S, ξ(s) is primitive. (X → S, ξ) is said to be of degree 2d if for all geometric points s → S, ξ(s) has degree 2d. ◦ Let M2d (resp. M2d) be the moduli problem over Z[1/2] which sends each Z[1/2]-scheme S to the groupoid of tuples (f : X → S, ξ), where f : X → S is a K3 surface and ξ is ◦ a primitive quasi-polarization (resp. polarization) of X with deg(ξ) = 2d. M2d and M2d ◦ are Deligne-Mumford stacks of finite type over Z, M2d is separated, and the natural map ◦ M2d → M2d is an open immersion (cf. [40], [31, 2.1], [28, 3.1]). Let (X → M2d, ξ) be the universal object over M2d. For each scheme T → M2d, we 2 denote by (XT , ξT ) the pullback of (X , ξ). Let HB be the second relative Betti cohomology 2 of XC → M2d,C. For every `, let H` be the relative second étale cohomology of the restriction 2 of X to M2d,Z[1/2`] with coefficients in Z`. Let HdR be the vector bundle on M2d given by the second relative de Rham cohomology of X , which comes with a natural filtration Fil•. 2 2 For ∗ = B, `, dR, denote by P∗ the primitive part of H∗, i.e., the orthogonal complement to 2 the Chern class of ξ. We put together the relative `-adic cohomology sheaves H` to form 2 Q 2 2 Hˆp := `6=p H` and similarly we put together Chern classes to form chˆp (ξ) in Hˆp . Z Z Z For each scheme T → M2d,Fp , we can additionally consider the crystal of vector bundles 2 2 Hcris,T on Cris(T/Zp) given by the second crystalline cohomology of XT . Let Pcris,T be the primitive part of Hcris,T .

3.3.3 We use the notations of Sect. 3.1 by setting L = Ld. Let ψ : H × H → Z be the symplectic pairing constructed in [31, Sect. 5.1], so that ψ induces a perfect pairing on

H⊗Z(p). Again let δ ∈ det(L) be some fixed nonzero element and m be the rank of L. Let Me 2d → M2d be the étale 2-cover such that for every scheme T , a point in Me 2d(T ) 0 is given by a pair (T → M2d, β2,T ), where β2,T is a section in H (T, det(P2,T )) such that hβ2,T , β2,T i is the constant section hδ, δi. We call it the orientation cover. The universal ad ad property of the Shimura stack Sh ad (G , Ω) induces a map ρ : M2d, → Sh ad (G , Ω) KL C C e C KL C ad (cf. [28, Prop. 4.3, 5.2]). ρ descends to a map ρ : M2d, → Sh ad (G , Ω) defined over C Q e Q KL Q (cf. [28, Cor. 5.4], see also [41, 3.16]). By construction, there is an isomorphism α : ρ∗ L → B C B 2 PB(1). Deligne’s big monodromy argument in [12, 6.4] can be used to show that for each

13 ∗ 2 rational prime `, there exists a (necessarily unique) isomorphism α`, : ρ L`, → P of Q Q `,Q étale local systems over Me 2d,Q (cf. [28, Prop. 5.6].) which is compatible with αB via Artin’s comparison isomorphism. Let β be the pullback of the global section δ of L . Since `,Q `,Q `,Q M is normal (cf. [28, Cor. 3.9]), we can extend β to a global section β of det(P2) over e 2d `,Q ` ` −1 Me 2d,Z[(2`) ] for every `. p For our purposes it suffices to look at Me 2d,Z(p) . Let I be the étale sheaf over M2d,Z(p) which associates to each morphism S → M2d,Z(p) the set

ˆp ∼ 2 m {isometries η :Λ⊗ → Hˆp such that η(e − df) = chˆp (ξS), (∧ η)(δ⊗1) = (β`,S)}. Z Z ,S Z

p,ad It comes equipped with a natural action by the constant sheaf of KL . For every compact 0 ad ad 0p 0p ad p 0 open subgroup K ⊂ KL of the form Kp K for some K ⊂ G (Af ), we let Me 2d,K ,Z(p) denote the relative moduli problem over Me 2d,Z(p) which attaches to each morphism S → 0 p 0p 0 p 0p 0 Me 2d,Z(p) the set H (S, I /K ). A section [η] ∈ H (S, I /K ) is called a K -level structure on S. Denote by M 0 the corresponding pullback of M . Denote by M ad e 2d,K ,Z(p) e 2d,Z(p) e 2d,Kp ,Z(p) 0p ad p the limit lim Me 2d,KadK0p, as K varies over the compact open subgroups of G ( ). ←−K0p p Z(p) Af Remark 3.8. There is a minor issue in Rizov’s construction of the period map in [41, 3.9], see [43, Rmk 5.12].3 Our definition of level structures is a slight generalization of Def. 5.11 in [43].

0 ad ad 0p 0p ad p 3.3.4 Let K ⊂ KL be a compact open subgroup of the form Kp K with K ⊂ G (Af ) ad 0 0 0 sufficiently small. The map ρQ lifts naturally to a map ρK ,Q : Me 2d,K ,Q → ShK (G , Ω), and the extension property of the canonical integral model allows us then to extend the map

0 ρK ,Q to (cf. [28, Prop. 5.7])

ad 0 0 0 ρK ,Z(p) : Me 2d,K ,Z(p) → SK (G , Ω).

In turn, by taking quotients, we obtain a morphism of Deligne-Mumford stacks ρZ(p) : ad M2d, → S ad (G , Ω). It follows from local Torelli theorem for K3 surfaces that ρ , and e Z(p) KL C hence ρQ, is étale. Madapusi-Pera showed that ρZ(p) is still étale [28, Thm 5.8].

A major part of [28] is to show that ρZ(p) still records the cohomological data of K3 sur- ad faces, just like ρ . This is done by comparing sheaves on M2d, with those on S ad (G , Ω) C e Z(p) KL through ρZ(p) . For simplicity, we write ρZ(p) simply as ρ. By construction, we have isome- ∗ ∼ 2 ∗ ∼ 2 tries αB : ρ LB → PB(1) and αdR : ρ LdR → PdR(1) over Me 2d,C. We summarize some properties which we shall need in the following proposition:

∗ ∼ Proposition 3.9. (a) αB extends to an isometry of Z` étale local systems α` : ρ L`(−1) → 2 P` over Me 2d,Z(p) for every ` 6= p and an isometry of Zp étale local systems αp : ∗ ∼ 2 ρ Lp(−1) → Pp over Me 2d,Q.

(b) αdR,C extends to an isometry of filtered vector bundles with flat connection αdR : ∗ ∼ 2 2 1 1 2 ρ LdR(−1) → PdR which sends Fil LdR = Fil LdR(−1) to Fil PdR over Me 2d,Z(p) .

(c) Let T → Me 2d,Fp be an étale map. αdR induces a canonical isomorphism αcris,T : ∗ ∼ 2 ∗ ρ Lcris(−1)T → Pcris,T , where ρ Lcris(−1)T is the crystal on Cris(T/Zp) defined by 3The author is grateful to an anonymous referee for point out the issue with orientations to him.

14 ρ ad pulling back Lcris(−1) via the composite morphism T → M2d, → S ad (G , Ω) . e Fp KL Fp αcris,T is a morphism of F-crystals.

Proof. These results are proved in [28, Sect. 5].

Technically, the universal family over Me 2d is étale locally an algebraic space, whereas the references for de Rham or crystalline comparison isomorphisms used in [28, Sect. 5] often only consider schemes. This issue is addressed in [20, Sect. 11]. Here we give a brief sketch: Let k be a perfect field of characteristic p, V be the ring of integers of a finite totally ramified extension K of W (k)[1/p], and YV be an algebraic space over V . Assume that the generic fiber YK and the special fiber Yk are both schemes. This assumption is always satis-

fied, for example, if YV is a K3 space, because smooth proper algebraic spaces over a field of dimension ≤ 2 are schemes. It has been explained in [20, Sect. 11.2] that under this assump- tion, [6, Thm 14.6] and GAGA results for étale cohomology such as [?, Thm 3.7.2] give rise ∗ ∼ ∗ ¯ to an isomorphism Hcris(Yk/W (k))⊗W (k)Bcris → Hét(YK ⊗K K, Zp)⊗Zp Bcris. Moreover, it is ∗ ∼ compatible via the Berthelot-Ogus isomorphism with the isomorphism HdR(YK /K)⊗BdR → ∗ ¯ Hét(YK ⊗K, Zp)⊗Zp BdR provided by [?, Thm 8.4] and GAGA results. [20, Sect. 11.3] ex- plained that Blasius’ results in [7] hold with these comparison isomorphisms and the X in [7, Thm 3.1] is allowed to be an algebraic space. These observations fill in the details in the proofs of [28, Prop. 5.3, 5.6(4)].

p p 3.4 Special Endomorphisms Let K ⊂ KL be a sufficiently small compact open sub- p ad group and set K = K K ⊂ G( ). We write ρ ad : M ad → ad (G , Ω) simply p Af K ,Z(p) e 2d,K ,Z(p) SK as ρ.

Definition 3.10. (cf. [29, 5.22]) Let k be a perfect field with algebraic closure k¯, s ∈ 4 SK(G, Ω)(k) and f ∈ End(As) defined over k. If char k = 0, we say f is a special endo- morphism if its `-adic realization lies in L`,s⊗k¯⊗Q` ⊂ End(H`,s⊗k¯⊗Q`) for every prime `. If char k = p, we say f is a special endomorphism if its `-adic realization lies in L`,s⊗k¯⊗Q` ⊂ End(H`,s⊗k¯⊗Q`) and its crystalline realization lies in Lcris,s[1/p] ⊂ End(Hcris,s[1/p]).

In either case, the special endomorphisms form a subspace L(As) ⊂ End(As). If f ∈

L(As), then f ◦ f is a scalar. Therefore, L(As) has the structure of a quadratic lattice given by f 7→ f ◦ f. Now we relate special endomorphisms to line bundles on K3 surfaces.

Proposition 3.11. Let t ∈ M ad (¯ ) be a point and (X , ξ ) be the associated quasi- e 2d,K ,Z(p) Fp t t polarized K3 surface. Let s ∈ (G, Ω)(¯ ) be a lift of ρ(t). Let hξ i⊥ denote the orthogonal SK Fp t Q complement of ξt in Pic (Xt)Q. Then there is an isomorphism of quadratic lattices over Q

L(A ) =∼ hξ i⊥ s Q t Q

∼ whose `-adic and crystalline realizations agree with the isomorphisms L`,s = L`,ρ(t) → 2 ∼ 2 Pét(Xt, Z`) and Lcris,s = Lcris,ρ(t) → Pcris(X/W ) given by α` and αcris in Proposition 3.9.

4 p SK(G, Ω) := SKp (G, Ω)/K is a Deligne-Mumford stack which carries a descent of the universal abelian scheme A on SKp (G, Ω).

15 Note that we have identifications L`,s = L`,ρ(t) and Lcris,s = Lcris,ρ(t) because L` and ad Lcris on (the appropriate fibers of) SKad (G , Ω) are descent of the corresponding sheaves on SK(G, Ω).

Proof. This is a coarser form of [28, Thm 5.17(4)], except that we allow quasi-polarized K3 surfaces as well. We slightly adapt the arguments in loc. cit. Let αcris,t be the isomorphism ⊥ provided by Prop. 3.9(c) for T = t. It suffices to construct a map L(As) → hξti ⊂ Pic (Xt) such that the following diagram commutes,

⊥ L(As) hξti

F =p αcris,t 2 F =p Lcris,s(−1) (Pcris(Xt/W )) because the left vertical arrow is known to be an isomorphism after the domain is tensored with Qp by [28, Theorem 6.4(2)]. Let f ∈ L(As) be a special endomorphism. By [29, Prop. 5.21], there exists a charac- teristic 0 field F and an F -point sF on SK(G, Ω) which specializes to s such that f lifts an element fF ∈ L(AsF ). By the étaleness of the period map, there exists an F -point tF ad on M ad which specializes to t and whose image in ad (G , Ω) lifts to s . Choose e 2d,K ,Z(p) SK F an embedding F,→ C and let tC, sC denote the base change of tF , sF to C. Since we may 2 identify LB,s with P (Xt , (1)), the element fF ⊗ ∈ L(As ) produces a Hodge class in C C Z C C 2 P (Xt , (1)), which by Lefschetz (1, 1)-theorem comes from a line bundle ζt ∈ Pic (Xt ). C Z C C Specialize ζt to an element ζ ∈ Pic (Xt). Note that ζ does not depend on the choice of sF C 2 ⊥ because c1(ζ) ∈ Pcris(Xt/W ) depends only on f. The desired map L(As) → hξti ⊂ Pic (Xt) is given by sending f to ζ. ¯ Let V be a finite flat extension of W (Fp). Denote the fraction field of V by K and fix an ¯ ∼ ¯ isomorphism K = C. Let XV over V be a lift of a K3 surface X over Fp such that XV ⊗C has CM. One can infer from [19, Theorem 1.1] that if X has finite height, then the specialization map Pic (XV ⊗C)Q → Pic (X)Q is an isomorphism. In view of Proposition 3.11, there should be a corresponding statement for special endomorphisms: ¯ Proposition 3.12. Let s ∈ SK(G, Ω)(Fp) be a point such that As is non-supersingular.

Suppose there exists a lift sV of s over V such that sC := sV ⊗C is a special point on ShK(G, Ω). Then the specialization map

L(As ) → L(As) C Q Q is an isomorphism.

Here we are implicitly using [29, Lem. 5.13], which implies that the specalization map of endomorphisms End(As ) → End(As) restricts to a map of special endomorphisms C L(As ) → L(As). C Proof. This can be seen as a reinterpretation of [19, Thm 1.1]. To explain the idea, we first assume that there exists t ∈ M ad (V ) such that ρ ad (t ) is the image of s . Set V e 2d,K ,Z(p) K ,Z(p) V V ¯ t = tV ⊗Fp. Since AsV ⊗C has CM, so does XtV ⊗C (cf. Lem. 2.8). Then we can conclude

16 by Prop. 3.11. Now we sketch how to interpret the computation in [19, Section 4] without appealing to K3 surfaces.

(0,0) ⊥ Step 1: Recall that LB,s carries a Hodge structure of K3 type. Set T (LB,s ) := (LB,s ) ⊂ C C C LB,s ⊗ . By Thm 2.2 and Lem. 2.8, EndHdgT (LB,s ) = E for some CM field E such that C Q C T (LB,s ) is one-dimensional over E. Let E0 be the maximal totally real subfield of E. C Thm 2.2(b) tells us that

MT(LB,s ) = MT(T (LB,s )) = ker(Nm : Res m → Res m) C C E/QG E0/QG

Denote this group by G0.

Step 2: By extending V if necessary, we can find a number field F with E ⊂ F ⊂ K = V [1/p] such that sK arises from a F -valued point sF on ShK (G, Ω). Let v be the finite place above p given by the inclusion F ⊂ K and let q be the place of E below v. Let Fv be the completion of F at v and OFv be the ring of integers of Fv with residue field k(v). Then sFv := sF ⊗Fv extends to a O -valued point s on (G, Ω) such that s := s ⊗k(v) gives the Fv OFv SK k(v) OFv ¯ point s when base changed to Fp. To sum up, we have a commutative diagram s

¯ sV Spec Fp Spec V SK(G, Ω) sO Fv

Spec k(v) Spec OFv Spec Fv

The descent of s to sk(v) clearly endows L`,s with an action by the geometric Frobenius ¯ Fr ∈ Gal(Fp/k(v)).

Step 3: Choose algebraic closures F¯ and F¯v for F and Fv and embeddings F¯ ⊂ F¯v ⊂ K¯ which are compatible with F ⊂ Fv ⊂ K. Let sF be the geometric point over s given by

¯ ∼ p ∼ p F . Then we have natural isomorphisms L(AsF ) = L(As ) and L ,s = L ,s . Therefore, C Af C Af F if we define T (LAf ,sF ) to be the orthogonal complement of the image of L(AsF ) in LAf ,sF , ∼ then LB,s ⊗ f = L ,s restricts to an isomorphism C A Af F

∼ T (LB,s )⊗ f = T (L ,s ). C A Af F

¯ The Gal(F /F )-action on LAf ,sF restricts to one on T (LAf ,sF ). The theory of canonical mod- els tells us that the induced action of Gal(F¯ /F ) on T (LB,s )⊗ f is given by a homorphism C A ¯ γ : Gal(F /F ) → G0(Af ). Moreover, we obtain the following commutative diagram Nm × AF /AE × proj × AF AE AE,f

artF y7→c(y)y−1

ab γ Gal(F /F ) G0(Af ) G0(Af )/G0(Q) where F ab denotes the maximal abelian extension of F , c the complex conjugation on E ⊂ C,

17 5 and artF the Artin reciprocity map (cf. [19, Thm 2.1], [41, Cor. 3.9.2], [44, Thm 12]). Here artF is normalized so that the images of the uniformizers under the local Artin reciprocity maps act as lifts of geometric Frobenii on unramified extensions.

× Step 4: Let πv be a uniformizer of Fv and x ∈ AF be the element with πv at v and 1 at other places. Let σe := artF (x). We use the diagram in step 3 to analyze the character- istic polynomial fσ of σ acting on T (LB,s )⊗ ` for ` 6= p. Let z be the image of x under e e C Q the composition

−1 × Nm × pr × y7→c(y)y × AF → AE → AE,f → G0(Af ) ⊂ (E⊗Af ) .

The commutativity of the diagram in step 3 tells us that there exists a ∈ G0(Q) such that γ(σe) = za. Then we have that f = f [E:Q(q)] σe a × where fa is the minimal polynomial of a ∈ G0(Q) ⊂ E .

Step 5: We claim that q splits in E. By way of contradiction, suppose that q does not split in E. Then by [19, Lemma 4.1] a ∈ E× is a root of unity. This implies that a power 1 of Fr acts trivially on L`,s ⊂ End(H (As, Q`)). By the proof of [11, Proposition 21], As must be supersingular. We repeat the argument in loc. cit. using our notation for conve- nience of the reader: By construction of the étale local system L`, there is a Fr-equivariant isomorphism

∼ 1 Cl(L`,s) = EndCl(Ld)(Hét(As, Z`)). (3.13)

1 Here Cl(L) acts on Hét(As, Z`) from the right. Recall that As carries a Z/2Z-grading and a (left) Cl(L)-action. Let Cl+(L) (resp. Cl−(L)) be the even (resp. odd) part of Cl(L). + − Similarly, let As (resp. As ) be the even (resp. odd) part of As. Any invertible element + − + − of odd degree in the algebra Cl(L) induces a quasi-isogeny As → As , so As and As are + + isogenous. Therefore, it suffices to show that As is supersingular. For As , (3.13) restricts to an isomorphism

+ 1 + ∼ + Cl (L`,s) = EndCl (Ld)(Hét(As , Z`)). (3.14)

m 1 + m + If for some m, Fr acts trivially on EndCl (Ld)(Hét(As , Z`)), then Fr acts through the + + m center of Cl (L`,s). However, Cl (L`,s) is a central simple algebra, so Fr must be a ho- mothety. This forces As to be supersingular, which contradicts our assumption. Therefore, q splits in E. By [19, Lemma 4.1] again, none of the roots of f can be a root of unity. σe

Step 6: Now embed L(As ) into L`,s⊗ ` via the composition of maps C Q

L(As ) → L(As) ,→ L`,s⊗ ` C Q 5In these references, the above diagram is stated in terms of K3 surfaces. However, Rizov’s theorem [41,

Cor. 3.9.2] is a formal consequence of the rationality of the period map ρK,C. The diagram itself comes from ad ad the canonical model ShKad (G , Ω) of ShKad (G , Ω)C.

18 where the first map is given by specialization. By the smooth and proper base change theo- rem, f is equal to the characteristic polynomial of Fr acting on the orthogonal complement σe of L(As ) in L`,s⊗ `. Since none of the roots of fσ is a root of unity, the composition C Q e

Frm=1 L(As )⊗ ` → L(As)⊗ ` → lim(L`,s⊗ `) C Q Q −→ Q m is an isomorphism. Therefore, L(As ) → L(As) is an isomorphism. C Q Q

4 Formal Groups and Cohomology

In this section, we review Nygaard-Ogus theory and prove an important lifting lemma which we shall use in Section 5.

4.1 Formal Brauer Groups An abelian variety’s crystalline cohomology can be recov- ered from its p-divisible group via Dieudonné modules. There is a similar story for K3 surfaces of finite height, for which the role of p-torsion is taken by the formal Brauer group. Let k be an algebraically closed field of characteristic p > 2.

4.1.1 Let X be a K3 surface over R, where R is an Artinian local ring over Wn(k) := n W (k)/(p ) for some n. We denote the formal Brauer group of X by Brˆ X . Assume that ˆ BrX⊗Rk has finite height. Define a functor ΨX from the category ArtR of Artinian local R-algebras with residue field k to the category Ab of abelian groups by setting

2 ΨX (A) = Hfl(X⊗RA, µp∞ )

2 2 where Hfl denotes the second flat cohomology. Set D = Hfl(X⊗Rk, µp∞ ). By [35, Lem. 3.1], D is an abstract p-divisible group. We will view it as an étale p-divisible group over k.

Proposition 4.1. ΨX defines a p-divisible group over Spec R with connected component 0 ˆ ét ΨX = BrX and étale part ΨX = DR, where DR denotes the unique p-divisible group over Spec R lifting D/Spec k.

Proof. This is a corollary to [3, IV Prop. 1.8]. See also [35, Prop. 3.2].

Recall that Tate established an equivalence between the category of divisible connected formal Lie groups and that of connected p-divisible groups over a complete Noetherian local ring with residue field of characteristic p (cf. [45, Prop. 1]). We will freely make use of this equivalence of categories. There is a canonical sequence of F-crystals on Cris(R/W (k)) (cf. [35, Thm 3.20]):

∗ % 2 θ ˆ 0 → D(ΨX ) → Hcris(X) → D(BrX )(−1) → 0 (4.2)

For any object S in the nilpotent crystalline site of R/W (k), the above sequence induces 2 a short exact sequence of (Zarsiki) sheaves over S. Here Hcris(X) denotes the crystal 2 R fcris∗OX/W (k), where f : X → Spec R is the structure morphism.

19 The readers are refered to [35, Sect. 3] for the construction of %. Here we explain that ˆ ∗ ∗ θ is constructed out of % by duality: Let ι : D(BrX ) → D(ΨX ) be the map induced by the ˆ ˆ ∗ ∨ ∼ ˆ inclusion BrX → ΨX and let κ : D(BrX ) → D(BrX )(1) be the isomorphism induced by the ˆ ∗ ˆ 2 canonical pairing D(BrX ) × D(BrX ) → OR/W (k)(−1). Poincaré duality on Hcris(X) induces ∨ 2 ∗ ∨ a map % : Hcris(X) → D(ΨX ) (−2). The map θ is defined as the composition

∨ ∨ 2 % ∗ ∨ ι (−2) ˆ ∗ ∨ κ(−2) ˆ Hcris(X) → D(ΨX ) (−2) → D(BrX ) (−2) → D(BrX )(−1). (4.3)

4.2 Constructing Liftings For the rest of section 4, assume that p ≥ 5. Let X be a K3 surface over k with Brˆ X of finite height. There is a unique splitting of (4.2) for X, so that there is a canonical identification (cf. the proof of [35, Prop. 5.4])

2 ˆ ∗ ˆ ∗ ∗ ˆ Hcris(X/W (k)) = K(BrX ) := D(BrX ) ⊕ D(D ) ⊕ D(BrX )(−1). (4.4)

ˆ 2 2 The construction of K(BrX ) explains the slope filtration on Hcris(X/W (k)): Hcris(X/W (k))<1 = ˆ ∗ 2 ∗ 2 ˆ D(BrX ),Hcris(X/W (k))=1 = D(D ) and Hcris(X/W (k))>1 = D(BrX )(−1). Let K be a finite extension of K0 := W (k)[1/p] and let V = OK be the ring of integers ˆ of K. Suppose GV is a lifting of BrX over V . Then GV determines a filtration FilGV ⊂ (Brˆ )⊗ K. We can use Fil to define on (Brˆ )⊗ K a two-step filtration Fil• : D X W (k) GV K X W (k) GV

Fil2 ⊂ Fil1 ⊂ Fil0 = (Brˆ )⊗ K GV GV GV K X W (k) by setting Fil2 := Fil (−1) ⊂ (Brˆ )(−1)⊗ K and Fil1 := (Fil2 )⊥ with respect GV GV D X W (k) GV GV ˆ 2 to the bilinear pairing on K(BrX )⊗W K = Hcris(X/W (k))⊗W (k)K. The following result of Nygaard-Ogus allows us to choose a lift a K3 surface by choosing one for its formal Brauer group. ˆ Proposition 4.5. For each GV lifting BrX as above, there exists a K3 surface XV over ∼ V lifting X with the following properties: (a) ΨXV = GV ⊕ DV , where DV is the unique p- divisible group over Spec V lifting D. (b) The natural map Pic (XV ) → Pic (X) is an isomor- 2 ∼ 2 phism. (c) The Berthelot-Ogus isomorphism σcris : HdR(XV /V )⊗K → Hcris(X/W (k))⊗K = (Brˆ )⊗K respects the pairings and takes the Hodge filtration Fil• to the filtration Fil• . K X Hdg GV

Proof. This is [35, Prop. 5.5], except that we do not tensor σcris with K¯0 and let the crystalline-Weil group Wcris(K¯0) (cf. [5, Def. 4.1]) act on both sides. Here K¯0 is an al- gebraic closure of K0 containing K as a subfield. Indeed, [35, Prop. 5.5] tells us that there is lifting XV such that the isomorphism

2 ¯ ∼ 2 ¯ ˆ ¯ Hcris(XV /V )⊗V K0 → Hcris(X/W (k))⊗W (k)K0 = K(BrX )⊗W (k)K0 (4.6) takes Fil• to Fil• and respects the pairings and the action of the crystalline-Weil group Hdg GV ¯ Wcris(K0). The above isomorphism is constructed as σcris⊗V idK¯ 0 (cf. [5, Thm 4.2] and its proof). As explained in loc. cit., this isomorphism is independent of the choice of K in 0 0 the following sense: Suppose that K ⊂ K¯0 is a finite extension of K and V is the ring 0 0 of integers of K . By [5, Prop. 2.7], if we form the Berthelot-Ogus isomorphism σcris : 2 0 0 0 ∼ 2 0 0 0 ¯ HdR(XV ⊗V V /V )⊗V 0 K → Hcris(X/W (k))⊗W (k)K for XV ⊗V V , then σcris⊗V 0 K0 =

20 ¯ 2 0 2 0 0 0 σcris⊗V K0 when we identify HdR(XV /V )⊗V K with HdR(XV ⊗V V /V )⊗V 0 K . The proof of [35, Prop. 5.5] mentioned that the arguments for [36, Prop. 1.8] show that the natural map Pic (XV ) → Pic (XV ⊗V/(p)) is an isomorphism. In fact, the arguments for [36, Prop. 1.8] imply the stronger statement (b).6 Here we present an alternative proof in order to be a little more self-contained. To begin, we give some detail on the construction of XV . Let t be a uniformizer of V and let e be the ramification degree of V over W (k). For n+1 each number n = 0, 1, ··· , e − 1, set An := V/(t ), Gn := GV ⊗An and Dn := DV ⊗An. ∗ ∗ Set K(Gn) := D(Gn ) ⊕ D(Dn) ⊕ D(Gn)(−1). The paragraph above [35, Prop. 5.4] has explained how to put a K3 crystal structure (cf. [35, Def. 5.1]) on K(Gn). Since Gn is a deformation of G0 = BrX , [35, Prop. 5.4] gives us a deformation Xn of X0 = X to An, ∼ ∼ together with isomorphisms hn : BrXn = Gn, jn :ΨXn = Gn ⊕ Dn and an isomorphism of K3 2 ∼ crystals in : Hcris(Xn) = K(Gn). One can check from the proofs of [35, Thm. 5.3, Prop. 5.4] that each Xn+1 is constructed as a deformation of Xn, and (hn+1, jn+1, in+1) restricts to

(hn, jn, in). Finally, note that (V, (p)) has a natural PD structure. XV is constructed as the 2 ∼ 2 deformation of Xe−1 with the desired Hodge filtration on HdR(XV /V ) = Hcris(Xe−1)V . In particular, XV ⊗V/(p) = Xe−1.

We first show that for n ≤ e − 2, every line bundle ζ ∈ Pic (Xn) lifts to Xn+1. This implies that Pic (Xe−1) → Pic (X) is an isomorphism. Note that there is a canonical iso- 2 ∼ 2 morphism Hcris(Xn)An+1 = HdR(Xn+1/An+1). By [37, Prop. 1.12], it suffices to check that 1 2 2 2 ⊥ c1,cris(ζ)An+1 lies in Fil HdR(Xn+1/An+1), which is equal to [Fil HdR(Xn+1/An+1)] in our ∗ case. It follows from the K3 crystal structure on K(Gn+1) that D(Dn+1) is orthogonal to 2 2 D(Gn+1)(−1) and Fil HdR(Xn+1/An+1) lies (via in+1) in the D(Gn+1)(−1)An+1 component. ∗ Therefore, it suffices to show that c1,cris(ζ)An+1 ∈ D(Dn+1)An+1 . Let R a finite flat extension of W (k) with ramification degree n + 2 and fix an iso- ∼ morphism R/(p) = An+1. Let L be the fraction field of R. Let π be a uniformizer of R ∼ which is sent to the image of t ∈ V in An+1. Then R/(p) = An+1 descends to an iso- n+1 ∼ n+1 morphism R/(π ) = An. By [5, Lem. 3.9], the ideal (π ) ⊂ R has a canonical PD structure. By applying [5, Cor. 2.2] to Xn and X⊗An, we obtain a canonical isomorphism 2 ∼ 2 Hcris(Xn)R⊗RL = Hcris(X/W )⊗W L. By [5, Cor. 3.6], this isomorphism is compatible with the Chern class maps, so that c1,cris(ζ)R⊗1 is sent to c1,cris(ζ0)⊗1, where ζ0 is the restric- tion of ζ to X0 = X. Similarly, one slightly adapts the proof of [5, Prop. 3.14] to obtain ∼ ˆ a natural isomorphism K(Gn)R⊗RL = K(BrX )W ⊗W L which respects their three compo- 2 ∼ 2 nents. This isomorphism is compatible with Hcris(Xn)R⊗RL = Hcris(X/W )⊗W L. Finally, 2 ∗ ∗ since c1,cris(ζ0) ∈ Hcris(X/W ) lies in the D(D )W component, c1,cris(ζ)R lies in the D(Dn)R component. Therefore, c1,cris(ζ)An+1 , which is the mod p reduction of c1,cris(ζ)R, lies in ∗ ∗ D(Dn)An+1 = D(Dn+1)An+1 . A little extension of the above argument shows that Pic (XV ) → Pic (Xe−1) is an isomorphism. Let ζe−1 be an element in Pic (Xe−1). Recall the natural isomorphism 2 ∼ 2 Hcris(Xe−1)V = HdR(XV /V ). By the above argument, we know that c1,cris(ζe−1)V lies in ∗ 2 2 D(De−1)V . By the construction of XV , Fil HdR(XV ) lies in the D(Ge−1)(−1)V compo- 2 2 2 nent of Hcris(Xe−1)V . Hence c1,cris(ζe−1)V is orthogonal to Fil HdR(XV /V ) and lies in 1 2 Fil HdR(XV /V ). By [37, Prop. 1.12] again, ζe−1 lifts to XV .

6We thank an anonymous referee for pointing this out.

21 Remark 4.7. Let X and XV be as above. It follows from [23, Thm 1.2.17] that the ample cone of Pic (X) lies inside the ample cone of Pic (XK ). By a theorem of Kleiman

(cf. [23, Thm 1.4.23]), the nef cone of X or XK is the closure of the ample cone, so the nef cone of Pic (X) also lies inside that of Pic (XK ). Finally, a nef line bundle is big if and only if its top self-intersection is strictly positive (cf. [23, Thm 2.2.16]), so the lift of a big and nef divisor on X to XK is also big and nef. ¯ Lemma 4.8. Let G be a one dimensional formal group of height h over k = Fp and α : G → G be an isogeny. There exists a finite flat extension V over W of ramification index

≤ h and a lift GV of G to V such that α lifts to an isogeny αV : GV → GV .

Proof. View G as a Zp-module (for this terminology, see [46, 1.1]). Set M = Qp(α) and let ˆur OM be its ring of integers. Let V := OM be the completion of the maximal unramified exten- sion of OM . Let e := [M : Qp]. Then as an OM -module G has height h/e. The universal de- ˆur formation space of an OM -module of height h/e is isomorphic to OM [[t1, ··· , th/e−1]]. Hence ˆur we may obtain a lift GV of OM -module G to OM simply by setting t1 = ··· = th/e−1 = 0. Clearly GV carries an action of α and its special fiber is isomorphic to G by construction.

Now we can prove a lifting lemma which we shall use in Section 5.

0 ¯ Lemma 4.9. Let X and X be two K3 surfaces of finite height over k = Fp. For any isometry of F-isocrystals

2 ∼ 2 0 φ : Hcris(X/W )⊗K0 → Hcris(X /W )⊗K0

0 we can find a finite flat extension V of W and lifts XV ,XV of X,X over V with the following properties:

0 0 (a) The natural maps Pic (XV ) → Pic (X) and Pic (XV ) → Pic (X ) are isomorphisms.

2 ∼ 2 0 (b) The map HdR(XV /V )⊗K → HdR(XV /W )⊗K induced by φ via the Berthelot-Ogus isomorphism preserves the Hodge filtrations, where K = V [1/p].

0 0 When X = X , XV and XV can be taken to be the same lifting.

Proof. Since φ respects the Frobenius action, under the identification (4.4) φ restricts to ˆ ∼ ˆ ˆ an isomorphism of isocrystals D(BrX )(−1)⊗K0 → D(BrX0 )(−1)⊗K0. Therefore, BrX0 and Brˆ X must have the same height, and hence they are abstractly isomorphic. Let φBr be the quasi-isogeny BrX0 → BrX induced by φ. 0 ˆ By Lem. 4.8, we can find a finite flat extension V of W and lifts GV and GV of BrX ˆ ˆ ˆ 0 and BrX0 such that φBr : BrX0 → BrX lifts to an quasi-isogeny φBr,V : GV → GV . The 0 desired XV and XV can be taken to be lifts given by Prop. 4.5 such that the Berthelot-Ogus 2 ∼ ˆ 2 0 ∼ ˆ isomorphisms HdR(XV /V )⊗V K → K(BrX )⊗W K and HdR(XV /V )⊗V K → K(BrX0 )⊗W K 2 2 0 • • send the Hodge filtrations on H (XV /V )⊗K and H (X /V )⊗K to Fil and Fil 0 dR dR V GV GV respectively. 0 ˆ ˆ When X = X , we identify BrX and BrX0 and lift φ to a quasi-isogeny GV → GV . Then 0 of course XV and XV can be taken to be the same lifting.

22 Remark 4.10. Not surprisingly, if X and X0 are ordinary, we can always take V to be 0 W and XV ,XV to be the canonical liftings. The reason is that the canonical lifting of an ordinary K3 surface induces a filtration which coincides with the slope filtration ( [47, Lem. 1.9]).

5 Proofs of Theorems

Convention The universal family over M ad is denoted by (X , ξ). For any scheme e 2d,Kp ,Z(p) ad T , we denote the image of a point t ∈ M ad (T ) in ad (G , Ω)(T ) still by t. Similarly, e 2d,Kp ,Z(p) SKp we will also make use of the isomorphisms in Prop. 3.9 with the symbol for period morphism suppressed. Denote the fiber of (X , ξ) over t by (Xt, ξt). We first give a simple lemma on correspondences:

0 Lemma 5.1. Let Y,Y be two algebraic surfaces over a field. Every morphism ψ : NS(Y )Q → 0 0 NS(Y )Q is induced by a correspondence on Y × Y . Proof. Recall that by the Hodge index theorem, the intersection pairing on NS(Y ) is non- ∗ ∗ degenerate. Let e1, ··· , en be a basis for NS(Y )Q and let e1, ··· , en ∈ NS(Y )Q be a dual ∗ ∗ basis, i.e., under the intersection pairing ei ·ei = 1 and ei ·ej = 0 for i 6= j. Let f1, ··· , fn0 be 0 ∗ a basis for NS(Y ) . For each i, let Ei,Ei ,Fi be formal Q-linear combinations of curves on Y Q 0 0 ∗ 0 Pn or Y representing the classes ei, ei , fi. If ψ : NS(Y )Q → NS(Y )Q sends ei to j=1 aijfj for 0 Pn Pn ∗ 0 aij ∈ Q, then the desired correspondence is given by i=1 j=1 aijEi × Fj on Y × Y .

5.1 Proof of Proposition 1.5 Proposition 1.5 is a direct consequence of the following more precise statement:

0 Proposition 5.2. Assume p d and p ≥ 5. Let t, t ∈ M ad (¯ ) be two points. Let - e 2d,Kp ,Z(p) Fp 0 ad W := W (¯ ) and K := W [1/p]. Suppose the images of t, t in ad (G , Ω)(¯ ) lift to Fp 0 SKp Fp 0 ¯ 0 points s, s ∈ SKp (G, Ω)(Fp) respectively. For each CSpin-isogeny ψ : As → As , there exists an isogeny φ : Xt Xt0 which sends ch∗(ξt0 ) to ch∗(ξt) for ∗ = cris, ét such that the following diagrams commute

∗ ∗ conj. by ψ conj. by ψét cris p p 0 L 0 L Lcris,t ⊗K0 Lcris,t⊗W K0 Af ,t Af ,t

φ∗ ∗ 2 cris 2 2 p φét 2 p Pcris(Xt0 /W )⊗K0(1) Pcris(Xt/W )⊗K0(1). Pét(Xt0 , Af )(1) Pét(Xt, Af )(1)

Proof. We treat the supersingular case and the finite height case separately. Supersingular case: The map ψ, as a CSpin-isogeny, clearly preserves special endo- ∼ 0 morphisms, so it induces an isometry L(As )Q → L(As)Q. By Thm 3.11, we obtain ⊥ ⊥ ∼ ⊥ ⊥ ⊥ an isometry i : hξt0 i → hξti . Here the orthogonal complements hξti and hξt0 i ⊥ 0 are taken inside NS(Xt)Q and NS(Xt ) respectively. We may extend i to an isometry ∼ 0 i : NS(Xt )Q → NS(Xt)Q by sending ξt0 to ξt. By Lemma 5.1, i is given by a correspondence ⊥ ⊥ 2 p 2 p φ : Xt Xt0 . On the other hand, hξti and hξt0 i span all of Pét(Xt, Af ) and Pét(Xt0 , Af ), ∗ 2 p 2 p the induced map φét : Pét(Xt0 , Af ) → Pét(Xt, Af ) is completely determined and has to agree

23 with the map induced by ψ. The argument for crystalline cohomology is the same.

Finite height case: By Lem. 4.9 and Rmk 4.7, for some finite flat extension V of W 0 0 and K := V [1/p], we may choose quasi-polarized K3 surfaces (XV , ξV ) and (XV , ξV ) over

V which lift (Xt, ξt) and (Xt0 , ξt0 ) such that

∗ 2 2 conj. by ψcris : Pcris(Xt0 /W )⊗W K → Pcris(Xt/W )⊗W K (5.3)

0 0 preserves the Hodge filtrations induced by (XV , ξV ) and (XV , ξV ). Note that the Berthelot- 2 ∼ 2 Ogus isomorphism Hcris(Xt/W )⊗W K = HdR(XV /V )⊗V K restricts to an isomorphism

2 ∼ 2 Pcris(Xt/W )⊗W K = PdR(XV /V )⊗V K

0 0 and the same holds for X and X 0 . We can then choose lifts t , t : Spec V → M ad V t V V e 2d,Kp ,Z(p) 0 of t, t such that (Xt , ξ ) and (Xt0 , ξ 0 ) can be respectively identified with (XV , ξV ) and V tV V tV 0 0 (XV , ξV ). ad Since the map (G, Ω) → ad (G , Ω) is pro-étale, we can lift the V -valued points SKp SKp 0 0 0 0 tV , tV to sV , sV on S (G, Ω) such that s, s are special points of sV , sV . Note that under the 2 1 1 inclusion PdR(XtV (1)) ⊂ End(HdR(AsV )), the Hodge filtration on HdR(AsV ) is given by

1 1 1 2 Fil HdR(AsV ) = ker Fil PdR(XtV )(1).

∗ 1 ∼ 1 Therefore, ψcris⊗K : Hcris(As0 /W )⊗W K → Hcris(As/W )⊗W K preserves the Hodge filtrations induced by A and A 0 . By [5, Thm 3.15], ψ lifts to a quasi-isogeny ψ : sV sV V A → A 0 . Choose an isomorphism K¯ =∼ , which induces an embedding V ⊂ . We sV sV C C claim that ψ ⊗ respects the /2 -grading, Cl(L)-action and sends π to π 0 . V C Z Z B,sV ⊗C B,sV ⊗C In particular, ψV ⊗C is a CSpin-isogeny. Indeed, we can check this by looking at the maps induced by ψV ⊗C on `-adic cohomology for any ` 6= p, so that the conclusion follows from the smooth and proper base change theorem and the assumption that ψ is a CSpin-isogeny. Now ψV ⊗C induces by conjugation a Hodge isometry

2 ∼ 2 P (X 0 , ) → P (X , ). tV ⊗C Q tV ⊗C Q

Extend the above map to full Hodge structures by sending the class of ξt0 to that of ξt. Buskin’s result tells us that this Hodge isometry is given by an isogeny X X 0 . tV ⊗C tV ⊗C Now we can complete the proof by specializing this correspondence to φ : Xt Xt0 . By the compatibility between specialization and cycle class maps, the diagrams in the statement of the proposition commute.

5.2 Proof of Theorem 1.4 Lemma 5.4. Let k be a field, M be a vector space over k and G ⊂ GL(M) be a closed 0 0 reductive subgroup. Let µ, µ : Gm → G be two cocharacters. If µ and µ induce the same filtration on V , then they are conjugate under G(k). Proof. This can be extracted from the proof of [22, Lem. 1.1.9]. We present the argument for readers’ convenience. Let Fil•M be the filtration on M induced by µ, µ0 and let P ⊂ G

24 denote the parabolic subgroup which respects Fil•. Let U ⊂ P be the subgroup which acts trivially on associated graded vector space gr•M, so that U is the unipotent radical of P . Since µ, µ0 induce the same filtration, they induce the same grading on gr•M, which means 0 µ,µ 0 0 that their compositions Gm → P → P/U are equal. Let L, L be the centralizers of µ, µ in G. Then L, L0 are Levi subgroups of P , so L = uL0u−1 for some u ∈ U(k). The cocharacter 00 0 −1 ∼ µ := uµ u : Gm → L and µ induce the same cocharacter under the projection L → P/U. Hence µ00 = µ.

Now let k be a perfect field of characteristic p and let (D, ϕ, h−, −i) be a K3 crystal over • k. Recall that the Frobenius action ϕ gives an abstract Hodge filtration Filϕ on D/pD:

i −1 i Filϕ(D/pD) := ϕ (p D) mod p

2 By Mazur-Ogus inequality (cf. [5, Thm 8.26]), if (D, ϕ) is given by Hcris(X/W (k)) for • some K3 surface X over k, then Filϕ agrees with the Hodge filtration on

2 2 ∼ 2 Hcris(X/W (k))/pHcris(X/W (k)) = HdR(X/k).

Let (D, ϕ) be a K3 crystal. Assume that k is algebraically closed. Set K0 := W (k)[1/p].

Suppose that the quadratic lattice D is given by N⊗Zp W (k) for some self-dual quadratic ¯ ¯ lattice N over Zp and ϕ is given by pbσ for some b ∈ SO(N⊗Zp K0). For the following two lemmas, we write SO for the group scheme SO(N) over Zp. The lemma below is an analogue of [22, Lem. 1.1.12].

Lemma 5.5. (a) There exists a cocharacter µ : Gm → SOW (k) such that

¯b ∈ SO(W (k))µ(p)SO(W (k))

−1 • and σ (µ) gives the filtration Filϕ(1).

2 (b) Filϕ is an isotropic direct summand of D/pD of dimension 1.

0 Proof. (a) By the Cartan decomposition, there exists a SOW (k)-valued cocharacter µ and ¯ 0 −1 0 h1, h2 ∈ SOW (k) such that b = h1µ (p)h2 = h1h2µ(p) for µ = h2 µ h2. We check that −1 • 2 σ (µ) gives the filtration Filϕ(1). The condition that p D ⊂ ϕ(D) tells us that the de- −1 composition defined by σ (µ) takes the form D = D1 ⊕ D0 ⊕ D−1 where Di = {d ∈ −1 i 2 D : σ (µ)(z) · d = z d}. Now we verify that D1/pD1 = Filϕ(D/pD), or equivalently, −1 2 D1 and ϕ (p D) have the same image modulo p. One can easily check by definitions −1 2 D1 is a sub W (k)-module of ϕ (p D). Conversely, we need to show for every d ∈ D 2 such that ϕ(d) ∈ p D, d is congruent to an element of D1/pD2. Let d = d1 + d0 + d−1 ¯ 2 be the decomposition such that di ∈ Di. Since ϕ(d) = pbσ(d) = ph1h2µ(p)σ(d) ∈ p D, 1 d0 ≡ d−1 ≡ 0 mod p. Therefore, (d mod p) ∈ D1/pD1. We can check that Filϕ(D/pD) is equal to (D1 ⊕ D0) modulo p similarly.

(b) The condition that rank ϕ⊗W (k)k = 1 implies that rank D−1 = 1. Since D1 = ∨ (D−1) , rank D1 = 1.

25 Lemma 5.6. Suppose (D0, ϕ0) is another K3 crystal with D0 =∼ N⊗W (k) and there is an 0 0 embedding ι : D ,→ D⊗K0 which respects the Frobenius action by ϕ and ϕ⊗1. If g ∈ 0 −1¯ SO(K0) is an element such that g(D) = ι(D ), then g bσ(g) ∈ SO(W (k))µ(p)SO(W (k)).

Proof. We may assume that D0 = N⊗W (k) and ι is given by restricting the domain of ∼ 0 0 ¯0 g : N⊗K0 → N⊗K0 to N⊗W (k). The action of ϕ ⊗1 on D ⊗K0 is then given by pb σ, where ¯b0 = g−1¯bσ(g). Now we use the assumption that (D0, ϕ0) is a K3 crystal to deduce that g−1¯bσ(g) ∈ 0 ¯0 SO(W (k))µ(p)SO(W (k)). Let µ be a cocharacter of SOW (k) such that b belongs to 0 −1 0 • 0 SO(W (k))µ (p)SO(W (k)) and σ (µ ) gives the filtration Filϕ0 (1) on D ⊗k. Clearly it suffices to show that µ and µ0 are conjugate by an element in SO(W (k)). Note that the filtrations on N⊗k induced by σ−1(µ) and σ−1(µ0) are both of the form 0 = Fil−2 ⊂ Fil1 ⊂ 0 −1 0 1 ⊥ 1 Fil ⊂ Fil = N⊗k with Fil = (Fil ) and dimk Fil = 1. This implies that reductions of σ−1(µ) and σ−1(µ0) modulo p are conjugate by an element in SO(k). By [14, IX 3.3], σ−1(µ) and σ−1(µ0), and hence µ and µ0, are conjugate by element of SO(W (k)).

Proof of Theorem 1.4. Let (X, ξ) be the quasi-polarized K3 surface of Thm 1.4. Clearly it suffices to prove the case when ξ is primitive. Let t ∈ M ad (¯ ) be a point such e 2d,Kp ,Z(p) Fp that (Xt, ξt) can be identified with (X, ξ). Let L = Ld and set up period morphisms as ad in section 3.3. Let s ∈ (G, Ω)(¯ ) be a lift of the image of t in ad (G , Ω)(¯ ). SKp Fp SKp Fp ∼ 1 As in section 3.2, fix an isomorphism H⊗W = Hcris(As/W ) of Z/2Z-graded right Cl(L)- p ¯ modules which send π to πcris,s and construct maps ιs : Xp × X → SKp (G, Ω)(Fp) and ad ad ad,p ad ι : X × X → ad (G , Ω)(¯ ). s p SKp Fp ¯ 2 Set W = W (Fp) and K0 = W [1/p]. Recall that we can identify Pcris(X/W )(1) with ∼ 1 Lcris,t by Prop. 3.9 and the isomorphism H⊗W = Hcris(As/W ) induces an isomorphism ∼ 2 Lcris,t = L⊗W . Let (Mp, ϕ) ⊂ Pcris(X/W )⊗K0 be the embedding in the statement of the ad ad ad 2 2 theorem. There exists gp ∈ G (K0) such that gp · Pcris(X/W ) = Mp ⊂ Pcris(X/W )⊗K0. ad ad By Lem. 5.6, gp ∈ Xp . p 2 p p ∼ 2 ˆp Similarly, for every isometric embedding M ⊂ Pét(X, Af ) with M = Pét(X, Z ), we can ad,p ad,p ad,p 2 ˆp p pick g ∈ X such that g · Pét(X, Z ) = M . Under the assumption p > 18d + 4, ad the period map M ad → ad (G , Ω) is known to be surjective on ¯ -points ( [30, e 2d,Kp ,Z(p) SKp Fp Thm 4.1]). It follows from Prop. 3.4 and Prop. 5.2 that any point in the preimage of ad ad ad,p 0 0 ιs ((gp , g )) under the period map will give us the desired (X , ξ ).

5.3 Proof of Theorem 1.6

Proof. Let (X, ξ) be the quasi-polarized surface of Thm 1.6. Clearly we can assume that

ξ is primitive. Let t ∈ M ad (¯ ) be a point such that (X , ξ ) can be identified with e 2d,Kp ,Z(p) Fp t t ¯ (X, ξ). Let s ∈ SKp (G, Ω)(Fp) be a lift of t. By [22, Thm (0.4)], the isogeny class of s 0 contains a point s which lifts to a CM point on ShKp (G, Ω). More precisely, there exists 0 ¯ 0 0 a finite flat extension V of W := W (Fp) and a V -point sV 0 lifting s such that for some ¯ 0 ∼ 0 0 0 (and hence any) isomorphism K = , where K := V [1/p], the complex point s := s 0 ⊗ C C V C is a special point. Choose an K¯ 0 =∼ and let t0 and t0 denote the images of s0 and s0 on C C C ad ad (G , Ω) respectively. SKp

26 By the surjectivity of the period map over , we can choose a preimage of t0 in C C 0 M ad ( ), which we still denote by t . The choice of the preimage will not be im- e 2d,Kp ,Z(p) C C portant. Let (Xt0 , ξt0 ) be the associated quasi-polarized complex K3 surface. By Thm 2.2, C C 0 E := EndHdgT (Xt0 ) is a CM field generated by some Hodge isometry τ as a Q-algebra, C Q 0 0 2 0 and we have dimE T (Xt0 ) = 1. We extend τ to τ ∈ EndHdg(P (Xt0 , Q)) such that τ fixes C Q e C e the Hodge classes. 0 0 By lemma 2.9, there exists a CSpin-isogeny ψ : As0 → As0 which induces τ via the C C C e identifications

2 ∼ 1 P (Xt0 , Q(1)) → LB,t0 ⊗Q = LB,s0 ⊗Q ,→ End(HB,s0 )⊗Q = End(H (As0 , Q)). C C C C C

0 Now specialize ψ to a quasi-isogeny ψ 0 : A 0 → A 0 . One easily checks that ψ 0 is a CSpin- C τ s s τ isogeny using the smooth and proper base change theorem and Rmk 3.2. Let ψ : As → As0 0 −1 be a CSpin-isogeny connecting s and s . Set ψτ := ψ ◦ ψτ 0 ◦ ψ. ψτ induces an isometry 2 2 of F-isocrystals τcris : Pcris(Xt/W )[1/p] → Pcris(Xt/W )[1/p]. By Lem. 4.9, we can find a lift tV of t for some finite flat extension V over W with fraction field K = V [1/p] such that the 2 filtration on Pcris(Xt/W )⊗K induced by (XtV , ξtV ) is respected by τcris⊗K. We will show that XtV ⊗V C has CM for any V,→ C. ¯ Let sV ∈ SKp (G, Ω)(V ) be a lift of tV such that s = sV ⊗Fp. As in the proof of Thm 1.5, 1 ψτ respects the Hodge filtration induced on Hcris(As/W )⊗K by AsV . By [5, Thm 3.15] again, we may lift ψτ to a quasi-isogeny ψτ,V : AsV → AsV . For any embedding V,→ C, the CSpin-isogeny ψ ⊗ : A → A induces a Hodge isometry τ : P 2(X , ) → τ,V C sV ⊗C sV ⊗C e tV ⊗C Q P 2(X , ). One may easily check from the construction that τ⊗ is sent precisely to tV ⊗C Q e Q` 0 τe ⊗Q` via the isomorphisms

∗ −1 2 2 conj. by (ψét) 2 ∼ 0 ∼ 0 P (XtV ⊗ , Q)⊗Q` = Pét(Xt, Q`) = L`,t → L`,t = P (Xt , Q)⊗Q`. (5.7) C C

∼ 0 Therefore, we have an isomorphism of Q-algebras Q(τe) = Q(τe ). Moreover, we have a com- mutative diagram

⊥ ⊥ ⊥ 0 hξtV ⊗ i hξti hξt i C C

conj. by ψ L(As ) L(As) L(As0 ) L(As0 ) C C

⊥ ⊥ ⊥ Here the orthogonal complements hξ i , hξ i , and hξ 0 i are taken inside Pic (X ), tV ⊗ t t tV ⊗C C C Pic (Xt), and Pic (Xt0 ) respectively. All arrows in the diagram are isomorphisms of quadratic C lattices: The top horizontal arrow is an isomorphism because the construction in Lem. 4.9 lifts the entire Picard group, the vertical arrows are isomorphisms given by Prop. 3.11, and the specialization map L(As0 ) → L(As0 ) is an isomorphism by Prop. 3.12. The diagram C ⊥ ∼ ⊥ 0 gives us an isometry hξtV ⊗ i → hξt i which is clearly compatible with (5.7). Therefore, C C (5.7) restricts to an isometry

∼ 0 T (XtV ⊗ )⊗Q` = T (Xt )⊗Q`. C C

Let τ be the restriction of τ to T (X , ). Then the above isomorphism tells us that e tV ⊗C Q

27 (τ) ∼ (τ 0) restricts to (τ) ∼ (τ 0). Since τ acts as an Hodge endomorphism on T (X ) Q e = Q e Q = Q tV ⊗C and dimQ(τ) T (XtV ⊗C)Q = 1, by Thm 2.2 XtV ⊗C has CM.

Although we introduced the K3 surface Xt0 to make the argument more symmetric, one C can also argue purely in terms of Hodge structures of K3 type.

6 Further Remarks on Isogenies

Motivated by global Torelli theorems, we make the following definitions.

Definition 6.1. Let k be a perfect field with algebraic closure k¯. Let X,X0 be K3 surfaces 0 over k and let f : X X be an isogeny over k. • We say that f is polarizable (resp. quasi-polarizable) if there exists an ample (resp. 0 ∗ big and nef) class ξ ∈ Pic (X )Q such that f (ξ) is still ample (resp. big and nef).

2 0 ∼ 2 • If char k = 0, we say f is Z-integral if the induced isometries Hét(Xk¯, Q`) → Hét(Xk¯, Q`) preserves the Z`-integral structures for every prime `.

2 0 ∼ 2 • If char k = p > 0, we say f is Z-integral if the induced map Hét(Xk¯, Q`) → Hét(Xk¯, Q`) 2 0 ∼ preserves the Z`-integral structures for every prime ` 6= p and Hcris(X /W (k))[1/p] → 2 Hcris(X/W (k))[1/p] preserves the W (k)-integral structures. From now on, we let k denote some algebraically closed field of characteristic p 6= 2. 0 If an isogeny f : X X is equivalent (cf. Def. 1.1) to the graph of an isomorphism ι : X →∼ X0, then we simply say that f is induced by the isomorphism ι. The classical Torelli 0 theorem implies that when k = C, an isogeny f : X X is induced by an isomorphism if and only if f is polarizable and Z-integral. In fact, in view of Buskin’s theorem, the classical Torelli theorem is equivalent to this statement about isogenies. We will explain that Ogus’ crystalline Torelli theorem [38, Thm II] can be formulated in terms of isogenies just as the classical Torelli theorem (cf. Thm 6.5 below). For abelian varieties there is no distinction between isogenies and polarizable isogenies, because the pullback of an ample divisor along an isogeny of abelian varieties is clearly always ample. For K3 surfaces however, there is a special class of isogenies customarily called “reflection in (-2)-curves", which are never polarizable: Let X be a K3 surface over k and let β be a line bundle on X with β2 = −2. Then up to replacing β by β∨, β is effective (cf. [17, 1.1.4]). Let C be a curve representing β and OC be the structure sheaf of

C, which we view as a coherent sheaf on X. Then OC is a spherical object in the bounded derived category of X, which we denote by D(X). As a spherical object, −⊗OC induces a ∼ Fourier-Mukai auto-equivalence TC : D(X) → D(X). By Orlov’s theorem [39, 2.2], RC is induced by a Fourier-Mukai kernel PC , which is a perfect complex on X × X. The Chow realization of the PC (cf. [25, 2.9]) induces an action on the second cohomology which is nothing but reflection in β, i.e., x 7→ x + hx, βiβ, for every cohomology theory.7 We can view the Chow realization of PC as an isogeny RC : X X. One quickly checks from the formula x 7→ x + hx, βiβ that RC is Z-integral. 7 The action on the entire Mukai lattice by PC is reflection in the Mukai vector h0, β, 1i (cf. the proof of [25, Prop. 6.2]). This of course restricts to reflection in β on second cohomology. One can also use OC (−1) instead as in [16, 10.3(iii)] to obtain a reflection in h0, β, 0i on the Mukai lattice.

28 It turns out that every isogeny differs from a polarizable isogeny by a composition of reflections in (−2)-curves up to a sign. To make this statement precise, view RC as an element of the Q-algebra of correspodences of degree 0 from X to X, where two correspon- dences are viewed as equivalent if their crystalline and `-adic realizations all agree. Let R be the group of auto-isogenies X X generated by {RC : C is a (−2)-curve on X} in this correspondence algebra. Let ±R be the group generated by R and −1. Then we have the following:

Lemma 6.2. Let X,X0 be K3 surfaces over k.

(a) For every class ξ ∈ NS(X) with ξ2 > 0, there exists an element α ∈ ±R such that α∗(ξ) is big and nef.

0 (b) For every isogeny f : X X , there exists an element α ∈ ±R such that f ◦ α is a polarizable isogeny.

Proof. (a) is a restatement of [37, Lem. 7.9] in terms of isogenies. To prove (b), let a0 ∈ 0 ∗ 0 NS(X ) be an ample class and set a := f (a ) ∈ NS(X). Take an open neighborhood Da0 of 0 0 0 ∗ 2 a in NS(X )R which is contained in the ample cone of X . Since f is an isometry, a > 0. By (a) there exists an element α ∈ ±R such that α∗(a) is big and nef. Since the nef cone is ∗ ∗ the closure of the ample cone ( [23, 1.4 C, 2.2 B]), α (f (Da0 )) intersects the ample cone of X.

Remark 6.3. The above result and arguments are well known to experts and are usually used as a reduction step (e.g., [10, Prop. 6.2], [25, Lem. 6.2]).

Lemma 6.4. Let X be a supersingular K3 surface over k. Then the maps

2 (a) c1 : NS(X)⊗Z` → Hét(X, Z`) for every prime ` 6= p;

2 F =p (b) c1 : NS(X)⊗Zp → Hcris(X/W (k)) are isomorphisms.

2 Proof. Let ` be any prime not equal to p. First, we show that NS(X)⊗Q` → Hét(X, Q`) 2 F =p ¯ and NS(X)⊗Qp → Hcris(X/W (k)) [1/p] are isomorphisms. If k = Fp, this follows from the Tate conjecture for K3 surfaces [28, Thm 1]. One can then show this for a general algebraically closed k by a theorem of Artin [2, Thm 1.1] and a standard spreading out argument. 2 Now it suffices to show that the maps NS(X)⊗Z` → Hét(X, Z`) and NS(X)⊗Zp → 2 F =p Hcris(X/W (k)) have torsion-free cokernels. For the `-adic Chern class map, this follows from a Brauer group argument (cf. [24, Lem. 2.2.2] and its proof). For the cristalline Chern class map, this follows from [13, Rmk 3.5] (see also [24, Lem. 2.2.4]).

0 0 Theorem 6.5. An isogeny f : X X between two supersingular K3 surfaces X,X over k is induced by an isomorphism if and only if f is polarizable and Z-integral.

29 Proof. Clearly we only need to show the “if” direction. By Lem. 6.4, f ∗ induces an isomor- phism NS(X0) → NS(X) of quadratic lattices over Z. By [38, Thm II, II00], it suffices to show that f ∗ maps the ample cone of NS(X0) to that of NS(X). Define

2 2 VX = {x ∈ NS(X)R : x > 0 and hx, βi= 6 0 for all β = −2}

∗ 0 and define VX0 verbatim. Since f (VX0 ) = VX , and the ample cones of X,X are connected 0 components of VX ,VX0 , it suffices to show that the preimage of the ample cone of X under f ∗ intersects the ample cone X. This follows from our assumption that f is polarizable.

By Lem. 5.1 and Lem. 6.4, every isometry NS(X0) → NS(X) comes from an isogeny. Therefore, the above theorem is a reformulation of [38, Thm II].

Remark 6.6. We conjecture that Theorem 6.5 holds for non-supersingular K3 surfaces as well.

Now we turn our attention to the formulation of the main theorem. An exact analogue of Theorem 1.2 in positive characteristic will be:

p Conjecture 6.7. Let X be a K3 surface over k. Let Λp (resp. Λ ) be a quadratic lattice ˆp 2 2 ˆp over W (k) (resp. Z ) which is abstractly isomorphic to Hcris(X/W (k)) (resp. Hét(X, Z )). Equip Λp with a Frobenius action ϕ such that (Λp, ϕ) has the structure of a K3 crystal. 2 p Then for each pair of isometric embeddings (Λp, ϕ) ⊂ (Hcris(X/W (k))[1/p],F ) and Λ ⊂ 2 p 0 0 Hét(X, Af ), there exists another K3 surface X together with an isogeny f : X X such ∗ 2 0 ∗ 2 0 ˆp p that f Hcris(X /W (k)) = Λp and f Hét(X , Z ) = Λ .

2 By “(Λp, ϕ) ⊂ (Hcris(X/W (k))[1/p],F )” we just mean an isometric embedding Λp ⊂ 2 Hcris(X/W (k))[1/p] such that ϕ agrees with F . Below we state the quasi-polarized version of Conjecture 6.7 in terms of pointed lattices. A pointed lattice is a pair (M, m) where M is a quadratic lattice and m ∈ M is a distinguished element. Morphisms between pointed lattices are defined in the obvious way.

Conjecture 6.8. Let (X, ξ) be a quasi-polarized K3 surface over k. Let (Λp, λp) (resp. p p ˆp p (Λ , λ )) be a pointed lattice over W (k) (resp. Z ) such that Λp (resp. Λ ) is abstractly 2 2 ˆp isomorphic to Hcris(X/W (k)) (resp. Hét(X, Z ). Equip Λp with a Frobenius action ϕ such that (Λp, ϕ) has the structure of a K3 crystal. Then for each pair of isometric embeddings 2 p p 2 p ((Λp, λp), ϕ) ⊂ ((Hcris(X/W (k))[1/p], c1(ξ)),F ) and (Λ , λ ) ⊂ (Hét(X, Af ), c1(ξ)), there 0 0 0 exists another quasi-polarized K3 surface (X , ξ ) together with an isogeny f : X X such ∗ 2 0 0 ∗ 2 0 ˆp 0 p p that f (Hcris(X /W (k)), c1(ξ )) = (Λp, λp) and f (Hét(X , Z ), c1(ξ )) = (Λ , λ ). Proposition 6.9. Conjecture 6.7 and Conjecture 6.8 are equivalent.

Proof. We first show that Conjecture 6.8 implies Conjecture 6.7. Suppose that (X, Λp ⊂ 2 p 2 ˆp Hcris(X/W (k))[1/p], Λ ⊂ Hét(X, Z )) is a tuple which satisfies the hypothesis of Conjec- ϕ=p 2 F =p ture 6.7. Note that Λp is a Zp-lattice in the Qp-vector space (Hcris(X/W (k)) )⊗Qp. Let ξ ∈ NS(X) be the class of any quasi-polarization. Up to replacing ξ by a Z-multiple, ϕ=p we can assume that its crystalline Chern class is λp for some λp ∈ Λp and its étale Chern p p p 0 0 class is some λ for some λ ∈ Λ . We get the desired X and f : X X by applying p p Conjecture 6.8 to (X, (Λp, λp), (Λ , λ )).

30 Conversely, Conjecture 6.7 implies Conjecture 6.8 by Lemma 6.2(a).

2 ¯ Remark 6.10. Let 2d = ξ . Under the assumptions k = Fp and p > 18d + 4, Theorem 1.4 p proves the p-part and the prime-to-2d part of Conjecture 6.8, i.e., Conjecture 6.8 with Af replaced by the restricted product

Y Y (Q` : Z`) = {(a`) ∈ Q` : a` ∈ Z` for all but finitely many ` - 2dp}. `-2dp `-2dp

2 2 The reason is that when ` - 2dp, Hét(X, Z`) = Pét(X, Z`) ⊕ c1(ξ) as quadratic lattices. If 2 2 ` | 2d and ` 6= p, then Pét(X, Z`) ⊕ c1(ξ) embeds as a finite index sublattice of Hét(X, Z`). Therefore, even if it were not for the restriction p > 18d + 4, Theorem 1.4 is slightly weaker 2 than Conjecture 6.8. We also remark that if p | 2d, Pcris(X/W ) is not a K3 crystal because it is no longer self-dual. Therefore, giving an anologue of Theorem 1.4 for p | d will entail a different formulation. However, our main interest is in Conjecture 6.8, whose formulation works whether or not p | 2d.

Remark 6.11. We do not expect Conjecture 6.8 to hold if quasi-polarizations are replaced by polarizations. The reason is that f does not preserve the integral structures of Néron-

Severi groups in general, so it does not have to pull back VX0 to VX as in the proof of

Thm 6.5. Nonetheless, f certainly does pull back the closure of VX0 to that of VX . As we have discussed in the proof of Lem. 6.2, for any smooth projective variety, a big and nef class is a class which has positive self-intersection number and lies in the closure of the ample cone. This partially explains the usefulness of considering big and nef classes.

Finally, we explain that the surjectivity statement for Ogus’ crystalline period map [38, Prop. 1.16] and Lem. 6.4 yield a complete proof of Conjecture 6.7 in the supersingular case:

Proposition 6.12. Conjecture 6.7 holds for X supersingular.

p p ϕ=p ˆ p Proof. Let N := Λ , Np := Λp , and N := N × Np. By Lem. 6.4, the embedding 2 (Λp, ϕ) ⊂ (Hcris(X/W (k))[1/p],F ) restricts to an embedding Np ⊂ NS(X)⊗Qp. Similarly c p p 2 p ∼1 ˆp the embedding Λ = N ⊂ Hét(X, Af ) = NS(X)⊗Z . Together we obtain an embedding ˆ ˆ N ⊂ NS(X)⊗Af . Set N = N ∩ NS(X)Q. Then N is quadratic lattice over Z such that the ˆp p natural maps N⊗Zp → Np, N⊗Z → N and N⊗Q → NS(X)Q induced by inclusions are isomorphisms of quadratic lattices. We check that N is a K3 lattice in the sense of [38, Def. 1.6]. First, N is even, as ∼ ∼ N⊗Z2 = NS(X)⊗Z2. Since N⊗Q = NS(X)⊗Q, (a) and (b) in [38, Def. 1.6] are clearly satisfied. We only need to check that the cokernel of the embedding N → N ∨ induced by the symmetric bilinear form on N is annihilated by p. Note that (Λp, ϕ) is a supersingular ∨ K3 crystal and N⊗Zp = Np is the Tate module of (Λp, ϕ) (cf. [37, 3.1, 3.2]). That N /N is annihilated by p follows from [37, 3.13, 3.14]. ∨ Let MN be the k-scheme parametrizing the characteristic subspaces of (N /N)⊗Fp k (see [37, Sect. 4]). It is the moduli space of N-rigidified K3 crystals.8 By [38, Prop. 1.16], the natural period map from the moduli space of N-marked supersingular K3 surfaces

8The reader can look at [27, 4.4, 5.2] for an exposition of the results in [37, Sect. 4] and the period map.

31 0 (cf. [38, Thm 2.7]) to MN is surjective. Therefore, there exists another K3 surface X together with an isomorphism N →∼ NS(X0) which fits into a commutative diagram

N i NS(X0)

c1 .

∼ 2 0 (Λp, ϕ) (Hcris(X /W (k)),F ) Note that the bottom arrow is completely determined by the top arrow, as the maps 0 c1⊗1 2 0 N⊗ZK0 → Λp[1/p] and NS(X )⊗ZW (k)[1/p] → Hcris(X /W (k))[1/p] are isomorphisms.

By composing the identification N⊗Q = NS(X)Q given by the inclusion N ⊂ NS(X)Q with −1 0 ∼ 0 i ⊗Q we get an isometry f : NS(X )Q → NS(X)Q such that NS(X ) is sent to N and 2 0 0 0 Hcris(X /W (k)) is sent to Λp. By Lem. 5.1, f is induced by an isogeny X X . X together with this isogeny is what we seek.

Remark 6.13. In [38], Ogus gave a slightly different definition of K3 crystals. In partic- ular, he asked the crystalline discriminant to be −1 (cf. [38, Def. 1.4]). This additional requirement does not affect the formulation of Conjecture 6.7 and 6.8, because if a K3 crys- 2 tal (Λp, ϕ) embeds isometrically into (Hcris(X/W (k))[1/p],F ) for any K3 surface X, then (Λp, ϕ) necessarily has crystalline discriminant −1.

Remark 6.14. We remark that in fact any two supersingular K3 surfaces X,X0 over k ∼ 0 are isogenous by Lem. 5.1 and Lem. 6.4, as NS(X)Q = NS(X )Q as Q-quadratic lattices. By ∼ works of Artin and Rudakov-Shafarevich (see [17, Prop. 17.2.19, 17.2.20]), NS(X) = Np,σ for some 1 ≤ σ ≤ 10, where Np,σ is the unique even, non-degenerate Z-lattice with signature 2σ ∼ (1, 21) and discriminant group (Z/pZ) . One may check that Np,σ⊗Q = Np,σ0 ⊗Q for any 0 ∼ σ, σ by the Hasse principle: Np,σ⊗R = Np,σ0 ⊗R as a real quadratic form is determined by its signature. For any ` 6= p, we see that NS(X)⊗Q` = Λ⊗Q` by a lifting argument. Finally, ∼ Np,σ⊗Qp = Np,σ0 ⊗Qp as they have the same discriminant and Hasse invariant.

Acknowledgments First and foremost I need to thank my advisor Mark Kisin for sug- gesting this problem to me and supporting me through the project. It is also a pleasure to thank F. Charles, L. Chen, J. Lam, Q. Li, T. Nie, L. Mocz, A. Petrov and A. Shankar for helpful conversations. I thank K. Madapusi Pera for clarifying several details about his work, C. Schoen for reading a previous version of this paper and L. Mocz for giving many helpful comments. I am deeply grateful for the two anonymous referees who pointed out many inaccuracies in an earlier version of the paper and offered suggestions which greatly improved the text.

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