The Tate conjecture for ordinary K3 surfaces over finite fields

1983 ◽  
Vol 74 (2) ◽  
pp. 213-237 ◽  
Author(s):  
N. O. Nygaard
Keyword(s):  
Finite Fields ◽  
K3 Surfaces ◽  
Tate Conjecture

scholarly journals The Tate conjecture for K3 surfaces over finite fields

2012 ◽  
Vol 194 (1) ◽  
pp. 119-145 ◽  
Author(s):  
François Charles
Keyword(s):  
Finite Fields ◽  
K3 Surfaces ◽  
Tate Conjecture

scholarly journals Integral canonical models for Spin Shimura varieties

2015 ◽  
Vol 152 (4) ◽  
pp. 769-824 ◽  
Author(s):  
Keerthi Madapusi Pera
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
K3 Surfaces ◽  
Shimura Varieties ◽  
Good Reduction ◽  
Intersection Numbers ◽  
Orthogonal Groups ◽  
Canonical Models ◽  
Tate Conjecture ◽  
Precise Sense

We construct regular integral canonical models for Shimura varieties attached to Spin and orthogonal groups at (possibly ramified) primes$p>2$where the level is not divisible by$p$. We exhibit these models as schemes of ‘relative PEL type’ over integral canonical models of larger Spin Shimura varieties with good reduction at$p$. Work of Vasiu–Zink then shows that the classical Kuga–Satake construction extends over the integral models and that the integral models we construct are canonical in a very precise sense. Our results have applications to the Tate conjecture for K3 surfaces, as well as to Kudla’s program of relating intersection numbers of special cycles on orthogonal Shimura varieties to Fourier coefficients of modular forms.

scholarly journals Point counting on K3 surfaces and an application concerning real and complex multiplication

2016 ◽  
Vol 19 (A) ◽  
pp. 12-28 ◽  
Author(s):  
Andreas-Stephan Elsenhans ◽  
Jörg Jahnel
Keyword(s):  
Finite Fields ◽  
Efficient Method ◽  
Complex Multiplication ◽  
K3 Surfaces ◽  
Point Counting

We report on our project to find explicit examples of K3 surfaces having real or complex multiplication. Our strategy is to search through the arithmetic consequences of RM and CM. In order to do this, an efficient method is needed for point counting on surfaces defined over finite fields. For this, we describe algorithms that are$p$-adic in nature.

Transcendental cycles on ordinary $K3$ surfaces over finite fields

1993 ◽  
Vol 72 (1) ◽  
pp. 65-83 ◽  
Author(s):  
Yuri G. Zarhin
Keyword(s):  
Finite Fields ◽  
K3 Surfaces

scholarly journals The Tate conjecture for K3 surfaces in odd characteristic

2014 ◽  
Vol 201 (2) ◽  
pp. 625-668 ◽  
Author(s):  
Keerthi Madapusi Pera
Keyword(s):  
K3 Surfaces ◽  
Tate Conjecture

scholarly journals One-cycles on rationally connected varieties

2014 ◽  
Vol 150 (3) ◽  
pp. 396-408 ◽  
Author(s):  
Zhiyu Tian ◽  
Hong R. Zong
Keyword(s):  
Complete Intersection ◽  
Finite Fields ◽  
Chow Group ◽  
Rational Curves ◽  
Positive Answer ◽  
Tate Conjecture ◽  
Rationally Connected ◽  
Rationally Connected Varieties

AbstractWe prove that every curve on a separably rationally connected variety is rationally equivalent to a (non-effective) integral sum of rational curves. That is, the Chow group of 1-cycles is generated by rational curves. Applying the same technique, we also show that the Chow group of 1-cycles on a separably rationally connected Fano complete intersection of index at least 2 is generated by lines. As a consequence, we give a positive answer to a question of Professor Totaro about integral Hodge classes on rationally connected 3-folds. And by a result of Professor Voisin, the general case is a consequence of the Tate conjecture for surfaces over finite fields.

scholarly journals Finiteness of K3 surfaces and the Tate conjecture

2014 ◽  
Vol 47 (2) ◽  
pp. 285-308 ◽  
Author(s):  
Max Lieblich ◽  
Davesh Maulik ◽  
Andrew Snowden
Keyword(s):  
K3 Surfaces ◽  
Tate Conjecture

scholarly journals CM liftings of surfaces over finite fields and their applications to the Tate conjecture

2021 ◽  
Vol 9 ◽  
Author(s):  
Kazuhiro Ito ◽  
Tetsushi Ito ◽  
Teruhisa Koshikawa
Keyword(s):  
Finite Field ◽  
Algebraic Group ◽  
Finite Fields ◽  
Complex Multiplication ◽  
Shimura Varieties ◽  
Canonical Models ◽  
Tate Conjecture ◽  
Finite Height ◽  
Generic Fibre

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 $3$ .

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