Geodesic
CM liftings of $K3$ surfaces over finite fields and their applications to the Tate conjecture
Forum of Mathematics, Sigma, Tome 9 (2021)

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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 $3$. 
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     author = {Kazuhiro Ito and Tetsushi Ito and Teruhisa Koshikawa},
     title = {CM liftings of $K3$ surfaces over finite fields and their applications to the {Tate} conjecture},
     journal = {Forum of Mathematics, Sigma},
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     year = {2021},
     doi = {10.1017/fms.2021.24},
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Kazuhiro Ito; Tetsushi Ito; Teruhisa Koshikawa. CM liftings of $K3$ surfaces over finite fields and their applications to the Tate conjecture. Forum of Mathematics, Sigma, Tome 9 (2021). doi: 10.1017/fms.2021.24

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