Period relations and the Tate conjecture for Hilbert modular surfaces
type: Item
instance of: scholarly article
title: Period relations and the Tate conjecture for Hilbert modular surfaces
zbMATH Open document ID: 0631.14005
DOI: 10.1007/BF01389081
author: V. Kumar Murty
author: Dinakar Ramakrishnan
published in: Inventiones Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143483
review text: Harder, Langlands and Rapoport had proved the Tate conjecture for algebraic cycles for the non-CM-submotives of \(H^ 2\) of Hilbert modular surfaces [\textit{G. Harder}, \textit{R. P. Langlands}, \textit{M. Rapoport}, J. Reine Angew. Math. 366, 53--120 (1986; Zbl 0575.14004)]. The authors consider Hilbert modular forms of CM-type and prove the Tate conjecture for the CM-submotives in \(H^ 2\). This gives the full Tate conjecture asserting the algebraicity of all Tate cycles on Hilbert modular surfaces over an arbitrary number field. The authors also relate, in the CM case, the number of independent divisor classes to the order of pole of the \(L\)-function of the surface at the edge of convergence. The results have interesting corollaries about the Picard group of the surface. There exist non-trivial divisors defined over metabelian fields which are not in the linear span of the Hirzebruch-Zagier cycles, canonical divisors and the cycles coming from the desingularization of the surface. The methods are quite different from the methods of Harder, Langlands and Rapoport.
Mathematics Subject Classification ID: 14C99
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 14A20
Mathematics Subject Classification ID: 14C22
zbMATH DE Number: 4025577
zbMATH Keywords: Hodge cycles
zbMATH Keywords: CM-motives
zbMATH Keywords: Tate conjecture for algebraic cycles
zbMATH Keywords: Hilbert modular surfaces
zbMATH Keywords: Hilbert modular forms of CM-type
zbMATH Keywords: Picard group
reviewed by: A. N. Parshin
Wikidata QID: Q102398538
MaRDI profile type: Publication
cites work: Cohomologie d'intersection et fonctions $L$ de certaines variétés de Shimura
cites work: Q3050551
cites work: Hodge cycles, motives, and Shimura varieties
cites work: Q4195061
cites work: Q3695374
cites work: Q3855317
cites work: On Abelian Varieties with Complex Multiplication as Factors of the Jacobians of Shimura Curves
cites work: Hilbert modular surfaces
cites work: Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus
cites work: Die Tate-Vermutungen für Hilbert-Blumenthal-Flächen. (The Tate conjectures for Hilbert-Blumenthal surfaces)
cites work: L-Indistinguishability For SL (2)
cites work: Algebraic cycles on compact Shimura surface
cites work: Q3939877
cites work: Periods of Hilbert modular surfaces
cites work: Q3030933
cites work: Q3714175
cites work: On Elliptic Curves with Complex Multiplication as Factors of the Jacobians of Modular Function Fields
cites work: On the factors of the Jacobian variety of a modular function field
cites work: Q3849064
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description: scientific article; zbMATH DE number 4025577