The Tate conjecture for ordinary K 3 surfaces over finite fields
type: Item
instance of: scholarly article
title: The Tate conjecture for ordinary K 3 surfaces over finite fields
zbMATH Open document ID: 0557.14002
DOI: 10.1007/BF01394314
published in: Inventiones Mathematicae
publication date: 1983-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143070
review text: Let \(X_ 0\) be an ordinary K 3 surface over a finite field \(k={\mathbb{F}}_ q\), \(q=p^ a\). The Galois group acts on the étale cohomology groups \(H^ 2_{et}(X_ 0\otimes \bar k,{\mathbb{Q}}_{\ell}(1)),\) \(\ell \neq p.\) According to the Tate conjecture on algebraic cycles, the subspace fixed under the Galois action is spanned by cohomology classes of algebraic cycles on \(X_ 0\). This conjecture is proved in the paper. Half of the paper is devoted to the study of a canonical lifting \(X_{can}\) of \(X_ 0\) over the Witt ring \(W(k)\). \(X_ 0\) is ordinary implies that via the Serre-Tate isomorphism one can get a structure of a formal group on the universal formal deformation space S of \(X_ 0\) over the Witt ring \(W(k)\). Let \({\mathcal X}_{can}\) be the formal lifting obtained by pulling back the universal formal family \({\mathcal X}/S\) along the 0-section \(Spf W(k)\to S.\) \(X_{can}\) is the algebraization of \({\mathcal X}_{can}\). It is proved that \(X_{can}\) is the same as the canonical lifting of Deligne and Illusie ({\S} 1). - If \(A_{{\mathbb{C}}}\) is the abelian variety associated to \(X_{{\mathbb{C}}}=X_{can}\otimes {\mathbb{C}}\) by Kuga-Satake-Deligne, then theorem 2.7 says that the Galois representation \(H^ 1_{et}(A_{{\mathbb{C}}},{\mathbb{Q}}_ p)\) is semi-simple. This leads to the proof that the reduction \(A_ 0\) of \(A_{{\mathbb{C}}}\) is an ordinary abelian variety when \(X_ 0\) is ordinary ({\S} 2.8, 2.5). The geometric Frobenius of \(\bar A_ 0\) lifts to an element \(\sigma_ A\in End(A_{{\mathbb{C}}})\times {\mathbb{Q}}\) such that the action of \(\sigma_ A\) on \(H^ 1_{et}(A_{{\mathbb{C}}},{\mathbb{Q}}_{\ell})\) coincides with the action of geometric Frobenius and \(\sigma_ A\) induces an automorphism of the rational Hodge structure \(H^ 1(A_{{\mathbb{C}}},{\mathbb{Q}})\). Also it is proved ({\S} 3.1, 3.2) that the geometric Frobenius on \(P^ 2_{et}(X_{{\mathbb{C}}},{\mathbb{Q}}_{\ell}(1))\) descends to an autmorphism \(\sigma_ X\) of…
Mathematics Subject Classification ID: 14C99
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14K15
zbMATH DE Number: 3887831
zbMATH Keywords: ordinary K 3 surface over a finite field
zbMATH Keywords: Tate conjecture on algebraic cycles
zbMATH Keywords: canonical lifting
zbMATH Keywords: abelian variety
zbMATH Keywords: Galois representation
zbMATH Keywords: rational Hodge structure
zbMATH Keywords: Hodge structures
Wikidata QID: Q122980867
author: Niels O. Nygaard
MaRDI profile type: Publication
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OpenAlex ID: W2094920452
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description: scientific article; zbMATH DE number 3887831