Article contents
CM liftings of
$K3$ surfaces over finite fields and their applications to the Tate conjecture
Published online by Cambridge University Press: 08 April 2021
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
$3$.
- Type
- Number Theory
- Information
- Creative Commons
- This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
- Copyright
- © The Author(s), 2021. Published by Cambridge University Press
References
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