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CM liftings of $K3$ surfaces over finite fields and their applications to the Tate conjecture

Part of: Arithmetic algebraic geometry Surfaces and higher-dimensional varieties Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  08 April 2021

Kazuhiro Ito Open the ORCID record for Kazuhiro Ito [Opens in a new window] ,
Tetsushi Ito Open the ORCID record for Tetsushi Ito [Opens in a new window]  and
Teruhisa Koshikawa
Kazuhiro Ito
Affiliation:
Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, 91405, Orsay, France; E-mail: kazuhiro.ito@universite-paris-saclay.fr
Tetsushi Ito
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto606-8502, Japan; E-mail: tetsushi@math.kyoto-u.ac.jp
Teruhisa Koshikawa
Affiliation:
Research Institute for Mathematical Sciences, Kyoto University, Kyoto606-8502, Japan; E-mail: teruhisa@kurims.kyoto-u.ac.jp

Abstract

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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 $3$.

MSC classification

Primary: 11G18: Arithmetic aspects of modular and Shimura varieties
Secondary: 11G15: Complex multiplication and moduli of abelian varieties 14G35: Modular and Shimura varieties 14J28: $K3$ surfaces and Enriques surfaces
Type
Number Theory
Information
Forum of Mathematics, Sigma , Volume 9 , 2021 , e29
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Creative Commons
Creative Common License - CCCreative Common License - BY
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

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