Higher regulators and values of \(L\)-functions
type: Item
instance of: scholarly article
title: Higher regulators and values of \(L\)-functions
zbMATH Open document ID: 0588.14013
DOI: 10.1007/BF02105861
published in: Journal of Soviet Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Let \(X\) be a complex algebraic variety, \(K_j(X)\) its algebraic \(K\)-groups. More subtle analytic invariants of elements of \(K_j(X)\) than usual Chern classes are constructed. In the case of the Chow groups they are the well-known Abel-Jacobi-Griffiths periods of an algebraic cycle. The invariants seem to be closely related with values of \(L\)-functions at integral points; conjectures and supporting computations (for curves uniformized by modular functions) are given. The main tool is a cohomology theory related with the Hodge filtration and satisfying the Poincaré duality. The characteristic classes on \(K_j(X)\) valued in those cohomology groups are defined. In terms of these classes, higher regulators are defined which reduce to the Borel regulators in the case when \(X\) is a point. Here are some topics mentioned in the paper: Hodge conjecture, motives, Riemann-Roch theorem, multidimensional analog of Arakelov's construction of Néron-Tate height on curves, deformation of Chern classes, Tsygan-Feigin's theorem on the stable cohomology of current algebras. After this paper was written, \textit{C. Soulé} [Sémin. Bourbaki, 37e année 1984/85, Exp. No. 644, Astérisque 133/134, 237--254 (1986; Zbl 0617.14008) and more recent papers] and R. Ramakrishnan obtained results related with those in the paper.
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 14G10
zbMATH DE Number: 3944014
zbMATH Keywords: periods of an algebraic cycle
zbMATH Keywords: values of L-functions
zbMATH Keywords: higher regulators
zbMATH Keywords: Hodge conjecture
zbMATH Keywords: motives
zbMATH Keywords: Riemann-Roch theorem
zbMATH Keywords: multidimensional analog of Arakelov's construction of Néron-Tate height on curves
zbMATH Keywords: deformation of Chern classes
zbMATH Keywords: stable cohomology of current algebras
author: Alexander A. Beilinson
reviewed by: L. N. Vaserstein
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3944014