Tate's conjecture for \(K3\) surfaces of finite height
type: Item
instance of: scholarly article
title: Tate's conjecture for \(K3\) surfaces of finite height
zbMATH Open document ID: 0591.14005
DOI: 10.2307/1971327
author: Arthur Ogus
author: Niels O. Nygaard
published in: Annals of Mathematics. Second Series
publication date: 1985-01-01T00:00:00Z
review text: Let \(X/\mathbb{F}_q\) be a \(K3\) surface over a finite field \(\mathbb{F}_q\), \(q=p^d\) and \(h\) be the height of the formal Brauer group \(^{\wedge} X\). It is proved that Tate's conjecture on algebraic cycles and order of pole of zeta function is true for \(X\) provided that \(h<\infty\) and \(p\geq 5\). The result is deduced from the theorem that if \(V\) is a finite extension of \(W_q\), \(Y/V\) is a quasicanonical lifting of \(X\) then \(X\) satisfies the Tate conjecture. When \(p\geq 13\) the quasicanonical lifting is constructed by standard crystalline deformation theory. For \(p=5,7,11\), this depends on a comparison of the deformations of \(K3\) surfaces, Dieudonné modules of the dual of the enlarged formal Brauer groups and of abstract \(K3\) crystals. The paper contains the theory of quasi-canonical varieties over \(p\)-adic fields and of equicharacteristic deformations of abstract \(F\)-crystals.
Mathematics Subject Classification ID: 14C99
Mathematics Subject Classification ID: 14F30
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 14J20
Mathematics Subject Classification ID: 32J25
zbMATH DE Number: 3948456
zbMATH Keywords: K3 surface over a finite field
zbMATH Keywords: Tate's conjecture on algebraic cycles
zbMATH Keywords: order of pole of zeta function
zbMATH Keywords: crystalline deformation theory
zbMATH Keywords: quasi-canonical varieties over p-adic fields
zbMATH Keywords: equicharacteristic deformations of abstract F-crystals
Wikidata QID: Q29012170
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971327
OpenAlex ID: W2018742429
description: scientific article; zbMATH DE number 3948456