Isogenies Between K3 Surfaces Over

Abstract

We generalize Mukai and Shafarevich’s definitions of isogenies between K3 surfaces over ${\mathbb{C}}$ to an arbitrary perfect field and describe how to construct isogenous K3 surfaces over $\bar{{\mathbb{F}}}_p$ 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 $p$.

Twisted Derived Equivalences and Isogenies between K3 Surfaces in Positive Characteristic

“…Our main result is a characterization of K3 surfaces isogenous to a given K3 surface X in terms of certain integral sublattices of the second rational ℓ-adic and crystalline cohomology groups of X. This is a positive characteristic analog of a result of Huybrechts [15], and extends results of [52]. We give applications to the reduction types of K3 surfaces and to the surjectivity of the period morphism.…”

“…1.2, we first reduce to the case when ι p (Λ ⊗ Z p ) = H 2 ét (X, Z p ) and (ι p ) K sends the slope 1 part, i.e., D(D * ), isomorphically onto itself. By Lubin-Tate theory, for some finite flat extension V of W , there exists a lift G V of Br X to V such that h lifts to End(G V )[1/p] ( [52,Lem. 4.8]).…”

“…where Ω is the period domain parametrizing Hodge structures of K3 type on L d ([35, §3.1, 3.2], [52,Def. 3.1]).…”

The Tate Conjecture for Motivic Endomorphisms of K3 Surfaces over Finite Fields

“…For reader's convenience we sketch the argument: First, note that ψ induces an isomorphism φ ∈ O(H 2 cris (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 X V of X such that Pic (X V ) → Pic (X) is an isomorphism, and the automorphism of H 2 dR (X V [1/p]) induced by φ via the Berthelot-Ogus isomorphism preserves the Hodge filtration.…”

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

“…Remark 1.2. After we completed the first draft of this article, the authors learned that Yang also proved the above theorem under the additional conditions that ≥ 5 and admits a quasi-polarisation whose degree is not divisible by ; see [74,Theorem 1.6]. Under these assumptions, our method (or a simplified version presented in Subsection 1.5) and Yang's method share several ingredients, but there is one difference; Yang used Kisin's result [41,Theorem 0.4] on the CM liftings, up to isogeny, of closed points of the special fibre of the integral canonical model of a Shimura variety of Hodge type, whereas we give a refinement of Kisin's result (or argument) itself; see Theorem 1.7 for details.…”