On the Tate Conjecture for Squares of K3 Surfaces Over Finite Fields

On the Tate conjecture for Squares of K3 Surfaces over Finite Fields

Ziquan Yang

Abstract The Tate conjecture for squares of K3 surfaces over finite fields was recently proved by Ito–Ito– Koshikawa. We give a more geometric proof when the characteristic is at least 5. The main idea is to use twisted derived equivalences between K3 surfaces to link the Tate conjecture to finiteness results over finite fields, in the spirit of Tate.

1 Introduction

In this note, we give a proof of the Tate conjecture for squares of K3 surfaces over finite fields. We will phrase and study the problem from the perspective of motives, so we first introduce some notation. Let ¯ k be a perfect field of characteristic p with a chosen algebraic closure k, MChow(k) be the category of Chow motives, which comes equipped with fiber functors ω` (resp. ωcris) given by `-adic cohomology for every prime ` 6= p (resp. crystalline cohomology with W[1/p] coefficients). Let Mhom(k) be the category MChow(k) but with Hom groups taken modulo the kernel of ω := ω` × ωcris. Let h(−) denote the usual functor from the category of varieties over k to Mhom(k) given by X 7 (X, id, 0).

Theorem 1.1. Let X be a K3 surface over a finite field Fq of charactersitic p ≥ 5. Then natural morphism → ∗ ω : End(h(X))⊗ End (H (X¯ , )) ` Q` F ét Fq Q`

∗ is an isomorphism for ` 6= p, where F denotes the Frobenius action on H (X¯ , ). → ´et Fq Q` When X is a K3 surface over k, h(X) is known to admit a Chow-Künneth decomposition h(X) = h0(X) ⊕ h2(X) ⊕ h4(X). For two K3 surfaces X and X 0 over k, an isogeny from X 0 to X is an isomorphism f : h2(X) ∼ h2(X 0) which preserves the Poincaré pairing. We consider the isogeny category of K3 surfaces IK3(k), whose objects are K3 surfaces over k and whose morphisms are isogenies. For a give K3 surface X,→ we deﬁne the automorphism group scheme of X in IK3 to the the algebraic group IX over Q whose functor of points is given by

2 × IX(R) := {f ∈ (End(h (X))⊗R) : f preserves the Poincaré pairing} (1.2) for every Q-algebra R. Then we have a group-theoretic version of Thm 1.1:

Theorem 1.3. Let X be as in Thm 1.1. Assume that is big enough so that all line bundles on X¯ are Fq Fq 2 defined over and F ∈ SO(H (X¯ , (1))). Let I denote the centralizer of the geometric Frobenius Fq ét Fq Q` X,` 2 action in SO(H (X¯ , (1))). Then ω induces an isomorphism between I ⊗ =∼ I . ét Fq Q` ` X Q` X,`

We now explain the main ideas of proof. First, let X and Fq be as in the above theorem and t be the transcendental part of h(X), i.e., the orthogonal complement to the span of submotives. Similarly, we 2 consider It(X), which is deﬁned just as in (1.2) but with h (X) replaced by t and the centralizer It,` of the geometric Frobenius in SO(ω`(t(1))). It suﬃces to show that ω` : End(t(1))⊗Q` EndF(ω`(t(1))) and ω` : It⊗Q` It,` are isomorphisms. Next, we geometrically construct lots of elements in End(t) using twisted derived equivalences→ . A twisted K3 surface→ is a pair (X, α) where X is a K3 surface and α is a Brauer class. The order of (X, α) is the

1 order of α. Up to choices of B-fields, a derived equivalence D(X, α) =∼ D(X 0, α0) between two twisted K3 surfaces of ` -order gives rise to an isomorphism f : t ∼ t 0, where t0 is the transcendental part of h(X 0). We der,` call compositions∞ of such isomorphisms derived transcendental `-isogenies. Let St ⊂ End(t) denote the subgroup of derived transcendental `-isogenies from X→to itself. After developing the structural theorems for derived transcendental `-isogenies, we use the key fact that there are only finitely many K3 surfaces up der,` to isomorphism over Fq to show the quotient St \It(X),` is compact (Thm 3.7). der,` Note that this compactness statement already implies that St is big. Our next step is to find a 0 0 der,` 0 reductive Q-group I which is equipped with a morphism ν : I It(X),` and maps St I (Q) End(t) compatible with ν in the obvious sense. The compactness result and a theorem of Borel and Tis implies 0 0 that v(I ) ⊆ It,` is a parabolic subgroup (see Thm 3.6). However, v(I ) is reductive and It,` is connected, Q` → Q` → → 0 0 0 0 so v(I ) = It,`. Then using the density of I ( ) in I ( ¯ `) we check that the image of I ( ) in End(t) spans Q` Q Q Q EndFω`(t). This implies both Thm 1.1 and 1.3. It remains to explain how to construct I 0. Ideally, if X × X satisfies some of the standard conjectures, 0 then we can directly verify that It defines a reductive Q-group and simply take I = It. In fact, a little der,` refinement of the argument shows that EndFω`(t) is spanned by St . We will carry out this strategy out for K3 surfaces which arise from the Kummer construction (Ex. 3.10) and a slight variant of this strategy to ordinary K3 surfaces (Ex. 3.11). To treat the general case, we choose a polarization ξ on X and consider the Kuga-Satake abelian variety A attached to (X, ξ). The abelian variety A comes with a “CSpin-structure”. We consider the group scheme eI which parametrizes the automorphisms of A which preserve this CSpin- 0 structure and define I := eI/Gm. The main point of considering eI is that it fixes a polarization on A up to scalar, so that I 0 is reductive by the positivity of Rosati involution. The general mechinery of the 0 der,` 0 Kuga-Satake construction gives us a morphism I It,` and we construct maps St I (Q) End(t) by lifting-reduction arguments. → → → Remark 1.4. We remark that Thm 1.1 (including the case p = 2, 3) was previously proved in [10] by Ito–Ito–Koshikawa, whose method relies more heavily on the Kuga-Satake construction. The main idea there is to develop a refined CM lifting theory for K3 surfaces over finite fields, and then deduce Thm 1.1 using Kisin’s version of Tate’s theorem on the endomorphisms of abelian varieties, and the fact that squares of K3 surfaces with CM satisfies the Hodge conjecture, proved by Buskin. In comparison, our method relies more on the geometry of K3 surfaces themselves and Kuga-Satake abelian varieties are only used to bypass the standard conjectures. In particular, we do not make use of any CM liftings or Tate’s theorem1. There is a long standing connection between finiteness results and the divisorial Tate conjecture, as shown in the work of Tate [17] for abelian varieties, and works of Artin–Swinnerton-Dyer [2], Lieblich–Maulik–Snowden [12], and Charles [7] for K3 surfaces. Our main purpose of giving a new proof is to provide an example of such a connection for codimension 2 cycles, using theory of gerbes and twisted sheaves.

2 Derived `-Isogenies for K3 Surfaces

Let k be an algebraically closed ﬁeld. Let Y be a K3 surface or a product of two K3 surface over k and let α be a Brauer class in Br(Y)[` ]. The α-twisted coherent sheaves form an abelian category, so we can talk about its bounded derived category∞ D(Y, α). To take Chern characters of α-twisted sheaves, we need the notion of B-ﬁeld lifts:

2 Definition 2.1. A B-field lift of α is an element in Hét(Y, Q`(1)) with the following property: Sup- m m 2 pose ord(α) = ` and B = β/` . Then β lies in Hét(Y, Z`(1)) and maps to α under the composition 2 2 m Hét(Y, Z`(1)) Hét(Y, µ`m ) Br(Y), where the first map is the reduction map modulo ` . 1We do need to use the semisimplicity of Frobenius and the divisorial Tate conjecture for individual K3’s, whose known proofs indeed depend→ on Tate’s theorem→ via the Kuga-Satake construction. We will treat these inputs as given.

2 Given a complex of α-twisted sheaf E on Y, and a B-field lift B of α as above, we can define its twisted Chern character chB(E), which depends on the choice of B. The basic idea is that, given a choice of a m gerbe X representing β, E⊗` becomes an untwisted sheaf on X, so we can take the usual Chern character m ch(E⊗` ) and define chB(E) as `mpch(E⊗`m ), which depends only on B and not on X. For details, see [4, App. A] and [5, §2]. The twisted Mukai vector of E is defined by vB(E) := chB(E)ptd(Y). Now suppose we are given given two twisted K3 surfaces (X, α), (X 0, α0) and a derived equialence ∼ 0 0 0 −1 0 Φ : D(X, α) D(X , α ). By Orlov’s theorem, Φ is induced by a complex E ∈ D(X × X , α α ), which is unique up to quasi-isomorphism. To emphasize this dependence we denote Φ by ΦE. Given B-field lifts 0 0 0 −BB ∗ 0 B, B of α, α→, we have a twisted Mukai vector v (E). As an element of Ch (X × X )Q, it induces a 0 ∼ −BB ∗ ∗ 0 map Φ (E): H (X)Q H (X )Q. 0 ∼ −BB ∗ Theorem 2.2. The map Φ (E) as above restricts to a Z` 0 -integral isomorphism Hét(X, Z` 0 ) ∗ 0 → 0 −B B 0 H (X , Z` 0 ) for every prime ` 6∈ {`, char k}. If char k = p, then Φ (E) in addition restricts to a ∗ ∗ 0 W-integral isomorphism Hcris(X/W) Hcris(X /W). →

0 ∼ In general, Φ−BB (E): H∗(X) H∗(X 0) does not need to preserve the codimension filtration, or Q → Q restricts to an isogeny h2(X) ∼ h2(X 0). Nonetheless, it always restricts to an isomorphism t(X) ∼ t(X 0) which preserves the Poincaré pairing.→ One argues by Witt’s cancellation theorem that NS(X)Q is isomorphic 0 ∼ 0 to NS(X )Q as quadratic forms,→ so that given such t(X) t(X ) we can always complete to an→ isogeny h2(X) ∼ h2(X 0) by adding an isomorphism a2(X) ∼ a2(X 0) which preserves the Poincaré pairing. → Definition 2.3. (a)A primitive derived transcendental `-isogeny is an isomorphism t(X) ∼ t(X 0) → −B B 0 → 0 0 ∼ 0 0 is given by v (E) for some choices of B, B , α, α and ΦE : D(X, α) D(X , α ). → (b)A primitive derived `-isogeny f : h2(X) ∼ h2(X 0) is an isogeny whose restriction to the transcen- → dental parts is a primitive derived transcendental `-isogeny. → (c) A (transcendental) derived `-isogeny is a finite composition of the primitive ones.

Remark 2.4. We remark that isomorphisms between K3 surfaces and “reﬂection in (−2)-curves” on a single K3 (see [18, §6]) give rise to primitive derived `-isogenies with B, B0, α, α 0 all being zero.

2.1 Existence Theorems

2 Theorem 2.5. Let X be a K3 surface over k. Denote by Λ` the quadratic lattice given by Hét(X, Z`). For 2 0 every isometric embedding ι : Λ` , Hét(X, Q`), there exists another K3 surface X together with a derived 2 0 2 2 `-isogeny f : h (X ) h (X) such that f(Hét(X, Z`)) = ι(Λ`). → Proof. This is part of [5, Thm 1.3]. We explain the main steps for readers’ convenience: First, by Cartan- → Dieudonné theorem, every element of O(Λ`⊗Q`) can be written as a composition of relfections, so it suffices 2 2 to treat the case when ι(Λ`) = sb(Hét(X, Z`)), where sb is the reflection in b ∈ Hét(X, Q`). We may assume 2 b is a primitive element in Hét(X, Z`). Now set n = b2/2 and B := b/n. Let α be the Brauer class defined by B. Let X 0 be the moduli space of 0 0 ∼ stable α-twisted sheaves with Mukai vector (n, 0, 0). Then there is an equivalence ΦE : D(X , α ) D(X, α) 0 for some α 0 ∈ Br(X 0). For a suitable B-field lift B 0 of α 0, v−B B(E) restricts to the desired isogeny f. For details of this argument, see [5, Lem. 6.2(1)]. →

When k = C, this gives an adelic proof of [9, Thm 0.1], which reﬁnes [6, Thm 1.1]:

2 ∼ 2 Theorem 2.6. Let X, X0 be algebraic K3 surfaces over C. Every Hodge isometry g : H (X 0, Q) H (X, Q) is given by a derived isogeny. →

3 Proof. By applying Thm 2.5 for different `’s, we find a derived isogeny f : h2(X 00) ∼ h2(X) such that 2 2 f∗(H (X 00, Z)) = g(H (X 0, Z)) for some other K3 surface X 00. The composition g ◦ f∗ is an integral Hodge 2 ∼ 2 isometry H (X 00, Z) H (X 0, Z). By [18, Lem. 6.2], up to precomposing ±f by a composition→ of reflections in (−2)-curves, we may assume g ◦ f∗ preserves the ample cones. Therefore, g ◦ f∗ is induced by an isomorphism by the→ global Torelli theorem.

2.2 Torelli Theorem If a lifting XW of X over W carries a lifting of all line bundles on X, i.e., the natural map Pic (XW ) Pic (X) is an isomorphism, then we say that XW is a perfect lifting. By [11, Prop. 4.10], perfect liftings always exist for non-supersingular K3 surfaces. → ∼ Theorem 2.7. Let f : h2(X) h2(X 0) be a derived `-isogeny between non-supersingular K3 surfaces X, X0. 0 For every perfect lifting XW of X over W, there exists a perfect lifting XW such that f lifts to an isogeny 0 between their generic fibers of→XW and XW . Proof. It suffices to treat the case when f is primitive, so that the restriction of f to the transcendental ∼ 0 0 parts is given by a twisted Fourier-Mukai equivalence ΦE : D(X, α) D(X , α ) for some Brauer classes α, α0 of `-power order. Moreover, since we are considering perfect liftings, which in particular carry all liftings of (−2)-curves, using [18, Lem. 6.2] we reduce to the case when→ f is polarizable. Any lifting XW carries a unique lift αW of the Brauer class α, so we may now conclude by [5, Thm 4.6], which states that 0 0 0 for some lifting XW of X, E lifts up to quasi-isomorphism to a complex in D(XW × XW , −αW αW ), where 0 0 0 0 αW is the lift of α on XW . Moreover, if X is ordinary and XW is the canonical lifting, then X is also 0 0 ordinary and XW is the canonical lifting of X . 2 ∼ 2 0 0 For the second statement, note that since Hcris(X/W) = Hcris(X /W), X and X have the same height. 2 ∼ 2 0 Moreover, Hét(XK¯ , Zp) = Hét(XK¯ , Zp) as GalK-modules. It follows from [16, Thm C] that if XW is the 0 canonical lifting, then so is XW . Theorem 2.8. Let f : h2(X) ∼ h2(X 0) be derived `-isogeny. If f is polarizable and ω(f) restricts to an isomorphism H2(X) ∼ H2(X 0), then f is induced by a unique isomorphism X 0 ∼ X. → Remark 2.9. Thm 2.8 is proved in [5] in more generality, which also deals with derived equivlences → → between twisted K3 surfaces whose Brauer classes are not of prime-to-p order. The proof there also becomes significantly more involved because when the order of α ∈ Br(X) is not prime-to-p, it is not true that every lifting XW of X over W carries a lifting of α, so the proof of Thm 2.7 breaks down.

2.3 Galois Descent Using Thm 2.8 we give a Galois descent theorem for derived `-isogenies:

Theorem 2.10. Let k be a perfect field with algebraic closure k¯. Let f : X¯ 0 ∼ X¯ be a derived `-isogeny ¯ ¯ 0 ¯ ¯ ∗ 2 ¯ 0 bewteen two K3 surface X, X over k. Suppose X admits a model X over k and the Z`-lattice f (Hét(X , Z`)) 2 ¯ ¯ 0 0 in Hét(X, Z`) is invariant under the induced Galk-action, then X admits a model→ X over k such that f decends to a derived `-isogeny h2(X 0) ∼ h2(X).

Proof. For each σ ∈ Galk, we use supscript σ to denote base change via σ : k¯ k¯. Let us write bσ : ¯ X 0 0 σ 0 → ¯σ ∼ ¯ and bσ :(X ) X for the canonical base change morphisms. Suppose {ϕσ : X X}σ∈Galk is the ¯ 0 descent datum for X deﬁned by the model X. Our goal is to transport {ϕσ} to a→ descent datum for X .→ Let 0 2 0 σ 2 0 ϕσ : h ((X ) ) →h (X ) be the deﬁned by the composition →

σ −1 h2((X¯ 0)σ) f h2(X¯σ) ϕσ h2(X¯) f h2(X¯ 0).

0 It is clear that ϕσ is a derived `-isogeny. → → → 0 2 ¯ 0 σ ∼ 2 ¯ We claim that ω(ϕσ) restricts to an isomorphism Hét((X ) , Z`) Hét(X, Z`). Since f is an `- 0 0 isogeny, it suffices to check that ϕσ is `-integral. Recall that bσ and bσ induce canonical identifications → 4 2 ¯σ 2 ¯ 2 0 σ 2 ¯ 0 2 ¯ Hét(X , Z`) = Hét(X, Z`) and Hét((X ) , Z`) = Hét(X , Z`), the action of σ on Hét(X, Z`) comes exactly ∗ 2 ¯ 0 from ϕσ. Therefore, the claim follows from the assumption that f (Hét(X , Z`)) is Galk-stable. 0 0 By Thm 2.8, ϕσ is induced by an isomorphism, which we still denote by ϕσ. Since ϕσ satisfies the cocycle condition, one easily checks using the cohomological rigidity of K3 surfaces ([15, Prop. 3.4.2]) that 0 0 0 {ϕσ} also satisfies the cocyle condition. Therefore, {ϕσ} defines a descent datum for X , which gives rise to a 0 ∗ 2 ¯ 0 2 ¯ model Y over k. By construction, the map f : Hét(X , Q`) Hét(X, Q`) is Galk-equivariant for the Galk- actions induced by X, X0. By a standard averaging argument, f descends to an isogeny h2(X 0) ∼ h2(X). → 3 Relations between cycle conjectures →

3.1 Linear Algebra Let V` be a Q` vector space of dimension 2r equipped with a non-degenerate ∨ bilinear pairing q : V` × V` Q`. Note that q provides an identification of V` with V` , through which ⊗(1,1) ∼ induces a natural pairing q on End(V`) = V`⊗V` given by the formula → q⊗(1,1)(α, β) = tr(αβ†) (3.1) where β† denotes the adjoint of β under q. × Let U(E) denote the group scheme defined by units of E: For each Q-algebra R, set U(E) := (E⊗QR) . In particular, if E is a field extension of Q, then U(E) is nothing but the usual Weil restriction Res E/QGm.

Lemma 3.2. If the natural map E⊗Q` End(V`) is injecture, or equivalently, dimQ E = dimQ` Q`hEi, then the natural map U(E)⊗Q` GL(V`) is injective. → Proof. When evaluated on a -algebra R, U(E)⊗ GL(V ) becomes the map (E⊗ R)× GL(V ⊗R). Q` → Q` ` Q ` This map is a restriction of the R-linear map E⊗QR End(V`⊗R) which is obtained from E⊗Q` End(V`) via base change. Then we use the fact that algebras over→ a ﬁeld are ﬂat. → → → For a quick non-example, note that if dim V` = 1, E = Q(α) for some α ∈ Q` which is a root to some irreducible polynomial over Q of degree ≥ 2, then U(E)⊗Q` cannot inject into GL(V`) as dim U(E) ≥ 2.

Lemma 3.3. Let W` be any finite dimensional Q` vector space equipped with a non-degenerate pairing W` × W` Q`. Let W be a Q-vector space embedded in W` such that the natural map j` : W⊗Q` W` is surjective. If the pairing on W` restricts to a pairing W × W Q on W, then j` must also be injective. → → Proof. Choose a section of the projection W W/(W ∩ ker(j )). Then we obtain a finite dimensional →` subspace N ⊂ W such that the restriction of j` to N⊗Q` is an isomorphism. In other words, N provides a Q-linear structure on W. Note that the pairing→ on N which is inherited from W is necessarily non- ∨ ∨ ∨ degenerate. Since j` is surjective, the map W W` = (N⊗Q`) is injective. As this map factors as ∨ ∨ ∨ ∨ ∨ W N (N⊗Q`) , W N must also be injective, so that N = W. → Set-up 3.4. Let F ∈ SO(V`) be a semisimple element such that the set of eigenvalues of F is of the form ± →± → ± → ¯ {α1 , α2 , ··· , αr } for distinct αi’s in Q` and none of αi’s is a root of unity. Let E be a vector space over ⊗(1,1) Q embedded in End(V`) such that E is closed under multiplication and q restricts to a Q-bilinear pairing E × E Q on E. Let I` be the centralizer of F in the reductive group SO(V`) over Q`. ± ± We first make some elementary observations: Let Wi be the eigenspace of the eigenvalue αi . Then ¯ ¯ → r + − − V` := V`⊗Q` admits a decomposition V` = ⊕i=1(Wi ⊕ Wi ). In particular, every Wi (resp. Wi ) is − isotropic and only pairs nontrivially with Wi (resp. Wi). More precisely, q restricts to a perfect pairing + − ¯ + − ∨ Wi × Wi Q` with which we can identify Wi with (Wi ) . There is a natural isomorphism

+ ∼ + − + − → GL(Wi ) = SOF(Wi ⊕ Wi ) ⊂ End(Wi ) × End(Wi ) (3.5)

5 given by g 7 (g, (g∨)−1).

Theorem 3.6. Suppose I is a group over together with a morphism i : I( ) E of sets and a morphism → Q Q i` : I⊗Q` I` of algebraic groups such that the diagram → I( ) I( ) → Q Q` i i`(Q`)

E EndV` commutes. If i(I(Q))\I(Q`) is compact, then the following are equivalent: (a) I is reductive. (b) The image of i` is I`. (c) The image of I(Q) in EndV` span EndF(V`).

Proof. We first remark that by [3, Prop. 9.3], the compactness of I(Q)\I(Q`) ensures that I⊗Q` is a parabolic subgroup of I`. (a) (b) The quotient Q := I⊗Q`\I` is proper. Since I⊗Q` is reductive, Q must also be affine. The connectedness of I` implies that Q is just a point. ¯ (b) ⇒ (c): Since I is reductive, I(Q) is Zariski dense in I`. Therefore, it suffices to show that I(Q`) spans ¯ r + EndF(V`). This follows easily from the observation that I`⊗Q` = i=1 GL(Wi ) under (3.5). Indeed, it suffices⇒ to check that for any vector space W over any field, the image of GL(W) in End(W) × End(W) Q under the embedding g 7 (g, g−1) spans the entire space. ¯ r + ¯ r (c) (a): Under the isomorphism I`⊗Q` = i=1 GL(Wi ), I⊗Q` takes the form i=1 Gi, where Gi + is a parabolic subgroup→ of GL(Wi ). Suppose for some i, Gi is a proper subgroup of GL(Wi). Then Gi Q¯ Q has to⇒ fix some nontrivial flag in Wi, so that Gi(Q`) ⊂ End(Wi) cannot possibly span End(Wi). This contradicts (c). Therefore, we must have (b), which implies (a).

3.2 Finiteness and Tate Conjecture Let X be a polarized K3 surface over Fq. Let F be the Frobenius 2 2 2 action on H` (X)(1) and assume that F ∈ SO(H` (X)Q(1)), and the F-invariants of H` (X)Q(1) are the same as F2-invariants. Let a and t denote the algebraic part and the transcendental part of h2(X) respectively. 2 F Recall that by the Tate conjecture for K3 surfaces ([13, Thm 1]), H` (X)Q(1) can be identiﬁed with NS(X)⊗Q`. Let T`(X) denote the orthogonal complemenet of NS(X)⊗Q`. Let I` (resp. It,`) denote the 2 centralizer of F in SO(H` (X)Q(1)) (resp. SO(T`(X)Q)). Note that I` canonically decomposes as Ia,` × It,`, 2 F where Ia,` = SO(H` (X)Q(1) ). Deﬁne It,` to be the functor which sends every Q-algebra R to the subgroup of (End(t)⊗R)× which preserves the Poincaré pairing t(1)⊗t(1) 1. Let Sder,` ⊂ End(h2(X)) (resp. der,` St ⊂ End(t)) be the subgroup of derived (resp. derived transcendental) `-isogenies.

der,` → Theorem 3.7. The quotient St \It,`(Q`) is compact.

Proof. Note that I` = Ia,` × It,`. Let pa(K) and pt(K) be the projections of the Ia,`(Q`) × It,`(Q`) repsectively. Then K ⊆ pa(K) × pt(K) is a subgroup of finite index. Therefore, it suffices to show that the der,` double quotient S \I`(Q`)/K is finite. By combining Thm 2.5 and Thm 2.10, one quickly sees that the elements in the double quotient der,` 0 S \I`(Q`)/K corresponds bijectively to the set of isomorphism class of K3 surfaces X over Fq which admits a derived `-isogeny f : h2(X 0) h2(X). Thanks to [13, Cor. 2], which is a corollary to [12, Main Theorem] and the Tate conjecture for K3’s [13, Thm 1], there are only finitely isomorphism classes of K3 der,` surfaces over Fq. Therefore, S \I`(→Q`)/K is finite.

Corollary 3.8. Suppose there is a reductive group I over Q which is equipped with (set-theoretic) maps der,` j i St I(Q) End(t) and a morphism i` : I It,` such that i` ◦ j = ω` and ω` ◦ i = i`. Then

EndFT`(X)Q is spanned by ω`(i(I(Q)), and ω` : It⊗Q` It,` is an isomorphism. In particular, Thm 1.1 and 1.3→ hold for→X. → → 6 Proof. The ﬁrst statement is a direct consequence of Thm 3.6 and Thm 3.7. For the second, we ﬁrst apply

Lem. 3.3 to W` = EndFT`(X)Q and W = E to obtain the ﬁnite-dimensionality of End(t) and the injectivity of E⊗Q` EndFT`(X)Q. Then we apply Lem. 3.2 to V` = T`(X)Q and E = End(t) to conclude that the morphism It⊗Q` It,` is injective.

→ ,` Remark 3.9. If we know that (A) the span QhSder i in E is ﬁnite dimensional, and (B) the Poincaré →der,` pairing on pt(QhS i) is negative deﬁnite, as predicted by the standard conjectures, then we can simply 0 der,` 0 take I to be the subgroup It := U(QhS i)∩It of It. Indeed, using (3.1) one quickly checks that It embeds der,` 0 into O(pt(QhS i)) via right multiplication. This implies that It(R) is compact, so that It is reductive. der,` In fact, in this case, we know that EndFT`(X)Q is spanned by St . Example 3.10. Suppose X comes from the Kummer construction, i.e., X is the desingularization of the quotient A/A[2] for some abelian surface A. Set Ae := X ×A/A[2] A. Then we have a diagram

Ae .

X A

This correspondence induces an isomorphism Ψ : h2(A) ⊕ (L⊕16) =∼ h2(X), where each L corresponds to an 2 ⊗2 2 ⊗2 exceptional divisor, which has self-intersection number −2. If qX : h (X)(1) 1 and qA : h (A)(1) ⊗2 1 denote the Poincaré pairings, then qX ◦ Ψ = 2qA. It is easy to see, using [1, Thm 1.1] and [8, Thm 1], that X satisfies (A) and (B) in Rmk 3.9. → → Example 3.11. Suppose X is an ordinary K3 surface. Then by Thm 2.7 elements in Sder,` all lift to the canonical lifting of X. Therefore, it follows from Hodge theory that X satisfies (A) and (B) in Rmk 3.9. We remark that the Tate conjecture for any power of an ordinary K3 surface was proved by Zarhin in [19]. The main point is that the transcendental Tate classes are all spanned by graphs of powers of the Frobenius −1 endomorphism. The q -multiple of the graph of the Fq-linear Frobenius endomorphism is indeed an isogeny. However, it cannot be a derived `-isogeny as long as X is not supersingular. Therefore, for squares of ordinary K3’s, our proof yields a different explanation for algebraic cycles spanning the transcendental classes such that the method of constructing these cycles works for arbitrary fields.

4 Uncondtional Proof

◦ 4.1 Kuga-Satake Construction Let M2d denote the moduli stack of polarized K3 surface of degree 2d over Z(p). Let Ld denote the lattice deﬁned in [13, (3.10)]. Let S(Ld) (resp. Se(Ld)) denote the canonical integral model of the associated orthogonal (resp. spinor) Shimura variety. There are sheaves

LB, LdR, L`, Lp, Lcris deﬁned on S(Ld)(C), S(Ld), S(Ld), S(Ld)Q, S(Ld)Fp respectively, with respect to the understood Grothendieck topology ([13, §4.1]).

◦ Theorem 4.1. There exists an étale morphism ρ : Me 2d S(Ld) such that for every algebraically closed ◦ field κ, u ∈ Me 2d(κ) and t := ρ(u), the following holds: If (X, ξ) is the fiber of the universal family over u, ∼ 2 then there are canonical isometry α∗ : L∗,t(−1) P∗(X) →for ∗ = B, dR, `, p, cris whenever the cohomology theory ∗ is applicable. → If p | d, then Ld is not self-dual at p. Nonetheless, by [14, §6.8] we can find a quadratic lattice Le which is self-dual at p and of signature (r, 2) for some r > 19, together with an isometric embedding Ld , Le onto ⊥ an orthogonal direct summand. This gives rise to maps S(Ld) S(Le). Set N := (Ld) ⊂ Le. If p - d, then we just set Le = Ld. We additionally consider Se(Le), which is equipped with a polarized abelian scheme→ Ae, and sheaves LeB, LedR, Le`, Lep, Lecris. To simplify notation, we view→ each scheme T over S(Ld) simultaneously

7 as a scheme over S(Le). Then there is a canonical embedding N ⊂ LEnd(AeT ) together with compatible ∼ ⊥ isometries L∗,T N ⊂ Le∗,T (∗ = B, `, dR, p, cris whenever applicable) [13, Prop. 4.9].

CSpin-structures→ Let He denote Cl(Le), viewed as a Cl(Le)-bimodule. Equip End(He Q) with a pairing r+2 by (α, β) 7 2 tr(α ◦ β). Then the map LeQ End(He Q) given by left multiplication is an isometric embedding. Denote by π 0 the idempotent projector from End(H 0 ) onto L . The group CSpin(L ) is by Q eQ eQ 0 definition→ the stabilizer in GL(He Q) of the Z/2Z-grading,→ right Cl(Le)-action, and the tensor π . We can endow a nondegenerate symplectic pairing Ψ : He × He Z such that the morphism CSpin(LeQ) GL(He Q) factors through GSp(He Q,Ψ). This gives an embedding of Se(Ld) into a Siegel Shimura variety, from which it inherits a weakly polarized universal abelian scheme→ (Ae, λ0). Moreover, Ae is equipped with→ a “CSpin- structure”: a Z/2Z-grading, a left Cl(Le)-action and various realizations of π. More precisely, for each algebraically closed fieldκ, and κ-point t on S(Le) with lift s on Se(Le), there exists an idempotent projector 1 πe∗,s on End(H∗(Aes)) whose image is canonically identified with Le∗,t for ∗ = B, dR, `, p, cris whenever the cohomology ∗ is applicable. These tensors are compatible in a natural way: If char κ = 0, the tuple

(πdR,s, π ) is absolute Hodge. In particular, if κ = , (πdR,s, π ) comes from the Betti tensor πB,s. If e eZb,s C e eZb,s char κ = p, K is a ﬁnite extension of W(κ)[1/p], and sK, tK are K-points which specialize to s, t respectively,

π p and π p are compatible via the smooth and proper base change theorem and πcris,s and πdR,sK eZb ,sK¯ eZb ,s e e 1 ∼ 1 are compatible via the Berthelot-Ogus isomorphism Hcris(AesK )⊗K = HdR(AesK ). ◦ ρ 0 Denote the composition Me 2d S(Ld) S(Le) by ρ . Lemma 4.2. Let κ be a ﬁeld. Let t ∈ Se(Le)(k) be a point and s ∈ S(Le)(k) be a lift of the image of t. → → 1 1 1 (a) Fil HdR(As) = ker(x) for any generator x of the one-dimensional subspace Fil LedR,s. Under the 1 1 inclusion LdR ⊂ LedR we have that Fil LdR = Fil LedR.

1 (b) Let R be any Q`-algebra. If ψ` ∈ GL(H` (Aes)⊗R) is an element which preserves the Z/2Z-grading, 1 Cl(Le)-action and πe`, then ψ` preserves the symplectic pairing on H` (Aes)⊗R induced by the weak 0 × polarization λs, which is well deﬁned up to a multiple of R . Proof. (a) The ﬁrst statement follows from [14, Prop. 4.7(v)]. The second statement follows from the 0 1 fact that N ⊆ Fil LedR is perpendicular to Fil LedR. (b) By [14, Prop. 3.14], there exists an isomorphism η : H 0 ∼ H1(A ) which preserves the /2 -grading, Cl(L)-action and sends π to π such that η sends ` Q ` s Q Z Z e e` ` 0 Ψ to the pairing induced by the weak polarization λs. Then we use the fact that the stablizer of the /2 -grading, Cl(L)-action and sends π in GL(H 0 ) is CSpin(L ), which lies in GSp(H 0 ,Ψ). Z Z → e Q` eQ` Q`

◦ 4.2 Unconditional Proof Let k be a finite field over Fp. Let u ∈ Me 2d(k) be a point given by the ∼ 2 tuple (X, ξ, ) for some choice of : det(Ld⊗Z2) det(P2(X)). Set t = ρ(u) and choose a lift s ∈ Se(Le)(k) of t. Up to replacing k by a finite extension, we assume that all line bundles on Xk¯ and all endomorphisms der,` 2 der,` of Aes,k¯ are defined over k. Let S0 be the stabilizer→ of ξ and det(P2(X)) in S . Since rank Pic (X) ≥ 2, der,` der,` the restriction of pt to S0 has the same image as S . Suppose that X has finite height.

der,` Lemma 4.3. For every f ∈ S0 , there exists a CSpin-isogeny ψ : Aes Aes which fixes Ns and induces p × the same action on L∗,t for ∗ = cris, Zb as f via α∗. Moreover, ψ is unique up to a Q -multiple. 0 → 2 Proof. By Thm 2.7, we can find perfect liftings XW and XW of X such that f lifts to an isogeny fK : h (XK) 2 0 ¯ ∼ 0 h (XK) between the generic fibers. Choose an isomorphism K = C. Let tW , tW be the liftings of t given by 0 0 0 ∼ 0 XW ,XW and tC, t be tW ⊗C, tW ⊗C. Then fK⊗C gives a rational Hodge isometry LB,t ⊗Q = LB,t ⊗Q→. C ∼ C C Recall that Nt is canonically identified with Nt 0 , so that can extend LB,t ⊗Q = LB,t 0 ⊗Q to a Hodge C C C C ∼ 0 isometry LeB,t ⊗Q = LeB,t 0 ⊗Q. Since the map Se(Le) S(L) is étale, the liftings tW , tW of t can be lifted C C to liftings s , s0 of s over W. Set s , s0 to be the -fibers of s , s0 . By [5, Lem. 5.2], the rational W W C C C W W → 8 ∼ Hodge isometry LeB,t ⊗Q = LeB,t 0 ⊗Q can be lifted to a CSpin-isogeny Aes Aes 0 . We get the desired ψ C C C C by specialzation. → Lemma 4.4. For each CSpin-isogeny ψ : Aes Aes which preserves Ns ⊂ LEnd(Aes) there exists an 2 2 0 isogeny f : h (X) h (X ) which preserves ξ and such that ψ and f induce the same actions on L∗,t⊗Q p for ∗ = cris, Zb . → → Proof. This follows from a slight refinement of the argument as [18, Prop. 5.2]2. For reader’s convenience we 2 sketch the argument: First, note that ψ induces an isomorphism φ ∈ O(Hcris(X/W)[1/p]) which preserves the class of ξ. By [18, Lem. 4.5], there exists a finite flat extension V of W, and a lifting XV of X such 2 that Pic (XV ) Pic (X) is an isomorphism, and the automorphism of HdR(XV [1/p]) induced by φ via the Berthelot-Ogus isomorphism preserves the Hodge filtration. This XV gives rise to a V-valued point uV 0 which lifts u. Set→ tV = ρ (uV ). Since the map Se(Ld) S(Ld) are étale, uV induces liftings tV and sV of t 1 1 and s over V. Next, set K := V[1/p] and recall that Fil HdR(AsK ) = ker(x) for any generator x of the one- 1 1 dimensional subspace Fil LedR,tK . By Hodge theory, the→ cycle classes of N in LedR,tK cannot lie in Fil LedR,tK , 1 1 so we have Fil LdR,tK = Fil LedR,tK under the inclusion LdR,tK ⊆ LedR,tK . By our construction of sV , ψ 1 1 preserves Fil HdR(AsK ) via the Berthelot-Ogus isomorphism. This implies that, up to replacing ψ by a p-power multiple, ψ lifts to a CSpin-isogeny ψK : AsK AsK , which necessarily preserves N ⊂ LEnd(AsK ). ¯ ∼ 2 Now choose an isomorphism K = C. Then ψK⊗C induces an isometry in O(H (XsK (C), Q)) which preserves ξK(C) and the Hodge structure. Finally, apply Thm→ 2.6 to obtain an auto-isogeny of XK⊗C. Specialize this isogeny to X and we are done.

Proofs of Thm 1.1 and 1.3. Consider group scheme eI deﬁned by

× p eI(R) := {ψ ∈ (End(Aes)⊗R) : ψ respects the Z/2Z-grading Cl(Le)-action and π∗⊗ZR for ∗ = cris, Zb }

0 for each Q-algebra R. Set I := eI/Gm. It follows from Lem. 4.2(b) and the positivity of Rosati involution 0 0 0 that I (R) is compact, so that I is reductive. Note that I ⊗Q` admits a natural morphism into the α` 2 centralizer of F in SO(Ls,`¯ ) = SO(P´et(X, Q`(1))). Let t denote t(X) and let It be as deﬁned in § 3.2. Since 0 der,` 0 I (Q) = eI(Q)/Gm(Q) by Hilbert theorem 90, Lem. 4.3 and 4.4 give us maps S0 I (Q) It(Q) which der,` der,` are compatible with the morphisms to It(Q`). It is easy to check that pt(S0 ) = St is surjective, so der,` der,` there exists at least a set-theoretic section St S0 . Then we conclude by Cor.→ 3.8. →

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