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 $3$ .

中文翻译：

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有限域上表面的 CM 提升及其在 Tate 猜想中的应用

我们将正交 Shimura 簇的积分规范模型和 Kuga-Satake 态射应用于 $K3$ 有限域上的曲面。我们证明每 $K3$ 有限域上有限高度的表面承认一个特征 $0$ 提升其通用纤维是 $K3$ 具有复数乘法的曲面。结合 Mukai 和 Buskin 的结果，我们证明了 A 平方的 Tate 猜想 $K3$ 有限域上的曲面。为了获得这些结果，我们构造了一个 Kisin 代数群的类比 $K3$ 有限高度的表面和构造特征 $0$ 的起重 $K3$ 曲面在代数群中保留了 tori 的作用。我们获得这些结果 $K3$ 任何特征的有限域上的表面，包括那些特征 $2$ 要么 $3$ .