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

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