Abstract
Abstract
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
$3$
.
Publisher
Cambridge University Press (CUP)
Subject
Computational Mathematics,Discrete Mathematics and Combinatorics,Geometry and Topology,Mathematical Physics,Statistics and Probability,Algebra and Number Theory,Theoretical Computer Science,Analysis
Cited by
2 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献