Higher regulators and values of \(L\)-functions. (English. Russian original) Zbl 0588.14013

Beilinson, A. A.

J. Sov. Math. 30, 2036-2070 (1985); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 24, 181-238 (1984).

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, 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.

Reviewer: L. N. Vaserstein (University Park)

MSC:

14C35	Applications of methods of algebraic \(K\)-theory in algebraic geometry
18F25	Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)

Keywords:

periods of an algebraic cycle; values of L-functions; higher regulators; Hodge conjecture; motives; Riemann-Roch theorem; multidimensional analog of Arakelov’s construction of Néron-Tate height on curves; deformation of Chern classes; stable cohomology of current algebras

Citations:

Zbl 0617.14008

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