Abstract
AbstractWe 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$3$.
Highlights
The integral canonical models of orthogonal Shimura varieties and the Kuga-Satake morphism have applications to the arithmetic of 3 surfaces over finite fields
We shall attach an algebraic group over Q to each polarised 3 surface (, L) of finite height over F, which is an analogue of the algebraic group attached by Kisin to each mod point on the integral canonical model of a Shimura variety of Downloaded from https://www.cambridge.org/core
In the course of writing this article, we found an error in the proof of the étaleness of the Kuga-Satake morphism in characteristic 2, which was used in the proof of the Tate conjecture for 3 surfaces in characteristic 2 [38]
Summary
The integral canonical models of orthogonal Shimura varieties and the Kuga-Satake morphism have applications to the arithmetic of 3 surfaces over finite fields. Combined with the results of Mukai and Buskin on the Hodge conjecture for products of 3 surfaces, we shall prove that the action of every element of (Q) is induced by an algebraic cycle of codimension 2 on × Applying this result for several maximal tori ⊂ , the result (2) follows. In the course of writing this article, we found an error in the proof of the étaleness of the Kuga-Satake morphism in characteristic 2, which was used in the proof of the Tate conjecture for 3 surfaces in characteristic 2 [38] We correct it using our results on -crystals on orthogonal Shimura varieties, which depend on the integral comparison theorem of Bhatt-Morrow-Scholze [6]; see Remark 6.9 for details. We explain our results on characteristic 0 liftings (see Theorem 1.7) and how to obtain (1) and (2) from them
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