Isogenies between K3 Surfaces over $\bar{\mathbb{F}}_p$

Abstract

We generalize Mukai and Shafarevich's definition 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.Surjectivity of the period map gives us a complex analytic K3 X ′ which corresponds to the induced Hodge structure on Λ. Then one can use [3, IV Theorem 6.2] to see that X ′ has to be algebraic.We give an analogue of Theorem 1.2 for quasi-polarized K3 surfaces over Fp , using K3 crystals and Z p -lattices to replace Z-lattices. K3 crystals were originally defined by Ogus to characterize the F-crystals that arise from K3 surfaces (c.f. [34, Definition 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 free W (k)-module D equipped with a σ-linear injection ϕ : D → D and a perfect symmetric bilinear pairing −, − such that ϕ(x), ϕ(y) = p 2 σ( x, y ), p 2 D ⊂ ϕ(D), and rank ϕ⊗k = 1.Recall that a quasi-polarized K3 surface is a pair (X, ξ) where X is a K3 surface and ξ is a primitive big and nef line bundle on X. We call the self-intersection number of ξ the degree of (X, ξ). Customarily, we write P 2 * (X) := ch * (ξ) ⊥ ⊂ H 2 * (X) for the primitive cohomology of (X, ξ) ( * = dR, cris, ét, etc). Our main theorem states:

On Some Generalized Rapoport–Zink Spaces

“…Recently there has been a definition of isogeny between two K3 surfaces in characteristic p, cf. [77]. One can cheek that the locus J φ parametrizes an isogeny class of polarized K3 surfaces.…”

CM liftings of K3 surfaces over finite fields and their applications to the Tate conjecture

“…Remark 1.2. After we completed the first draft of this paper, the authors learned Yang also proved the above theorem under the additional conditions that p ≥ 5 and X admits a quasi-polarization whose degree is not divisible by p; see [73,Theorem 1.6]. Under these assumptions, our method (or a simplified version presented in Section 1.5) and Yang's method share several ingredients but there is one difference; Yang used Kisin's result [40,Theorem 0.4] on the CM liftings, up to isogeny, of closed points of the special fiber of the integral canonical model of a Shimura variety of Hodge type, while we give a refinement of Kisin's result (or argument) itself; see Theorem 1.7 for details.…”