scholarly journals A note on the Tate conjecture for $K3$ surfaces

1980 ◽  
Vol 56 (6) ◽  
pp. 296-300 ◽  
Author(s):  
Takayuki Oda
Keyword(s):  
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.

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 in odd characteristic

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

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 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 Finiteness of Brauer groups of K3 surfaces in characteristic 2

2018 ◽  
Vol 14 (06) ◽  
pp. 1813-1825
Author(s):  
Kazuhiro Ito
Keyword(s):  
K3 Surfaces ◽  
Torsion Subgroup ◽  
Brauer Group ◽  
Base Field ◽  
Brauer Groups ◽  
Finitely Generated ◽  
Tate Conjecture ◽  
Characteristic 2 ◽  
Primary Torsion

For a [Formula: see text] surface over a field of characteristic [Formula: see text] which is finitely generated over its prime subfield, we prove that the cokernel of the natural map from the Brauer group of the base field to that of the [Formula: see text] surface is finite modulo the [Formula: see text]-primary torsion subgroup. In characteristic different from [Formula: see text], such results were previously proved by Skorobogatov and Zarhin. We basically follow their methods with an extra care in the case of superspecial [Formula: see text] surfaces using the recent results of Kim and Madapusi Pera on the Kuga-Satake construction and the Tate conjecture for [Formula: see text] surfaces in characteristic [Formula: see text].

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