2018/09/25 by Kazuhiro Ito, Ito, Kazuhiro, Tetsushi Ito +3 · 1 citation
Mathematics · #14G35 #14J28 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Primary 11G18 #Secondary 11G15
paper · pdf · doi:10.48550/arxiv.1809.09604
openalex publication_date 2018/09/25 · openalex created_date 2018/10/05 · openalex updated_date 2026/07/28
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 every K3 surface of finite height over a finite field admits a characteristic 0 lifting whose generic fiber 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.