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

Abstract

We give applications of integral canonical models of orthogonal Shimura varieties and the Kuga-Satake morphism to the arithmetic of $K3$ surfaces over finite fields. We prove that every $K3$ surface of finite height over a finite field admits a characteristic $0$ lifting whose generic fibre is a $K3$ surface with complex multiplication. Combined with the results of Mukai and Buskin, we prove the Tate conjecture for the square of a $K3$ surface over a finite field. To obtain these results, we construct an analogue of Kisin’s algebraic group for a $K3$ surface of finite height and construct characteristic $0$ liftings of the $K3$ surface preserving the action of tori in the algebraic group. We obtain these results for $K3$ surfaces over finite fields of any characteristics, including those of characteristic $2$ or $3$ .

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

“…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.…”

Construction of quasi-canonical liftings of K3 surfaces of finite height in odd characteristic

“…Note that Tate's conjecture for K3 surfaces is previously proved by Madapusi Pera [MP1, Theorem 1], [KM,Theorem A.1]. (See also [MP2], [IIK,Section 6.4]. )…”

“…Remark 1.4. If an extension of the base field k is allowed, Theorem 1.1 is previously obtained by Ito-Ito-Koshikawa [IIK,Corollary 9.11] using the étaleness of the Kuga-Satake morphism. However, it seems difficult to control the extension degree by the method of [IIK].…”

Unpolarized Shafarevich conjectures for hyper-Kähler varieties

“…Proof. A proof for the statement for K3 surfaces can be found in for example [23,Proposition 9.3], which works for general Hodge structures of K3-type. We sketch it here for the convenience of readers.…”