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

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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 !` (resp. !cris) 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 ! := !` × !cris. 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 ! ∗ ! : End(h(X))⊗ End (H (X¯ ; )) ` Q` F ´et 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) := ff 2 (End(h (X))⊗R) : f preserves the Poincaré pairingg (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 2 SO(H (X¯ ; (1))). Let I denote the centralizer of the geometric Frobenius Fq ´et Fq Q` X;` 2 action in SO(H (X¯ ; (1))). Then ! induces an isomorphism between I ⊗ =∼ I . ´et 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(!`(t(1))). It suffices to show that !` : End(t(1))⊗Q` EndF(!`(t(1))) and !` : 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 compositions1 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 nIt(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 EndF!`(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 EndF!`(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 category1 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 H´et(Y; Q`(1)) with the following property: Sup- m m 2 pose ord(α) = ` and B = β/` . Then β lies in H´et(Y; Z`(1)) and maps to α under the composition 2 2 m H´et(Y; Z`(1)) H´et(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 2 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).
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