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

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)
Full Text: DOI
[1] S. Yu. Arakelov, ”Theory of intersections of divisors on an arithmetic surface,” Izv. Akad. Nauk SSSR, Ser. Mat.,38, No. 6, 1179–1192 (1974).
[2] M. F. Atiyah, K-Theory, W. A. Benjamin (1967).
[3] A. A. Beilinson, ”Higher regulators and values of the L-functions of curves,” Funkts. Anal. Prilozhen.,14, No. 2, 46–47 (1980). · Zbl 0453.46007 · doi:10.1007/BF01078419
[4] A. Weil, Elliptic Functions According to Eisenstein and Kronecker, Springer-Verlag (1976). · Zbl 0318.33004
[5] H. Jacquet and R. P. Langlands, Automorphic Forms on GL(2), Springer-Verlag (1970). · Zbl 0236.12010
[6] Yu. I. Manin, ”Correspondences, motifs, and monoidal transformations,” Mat. Sb.,77, No. 4, 475–507 (1968). · Zbl 0199.24803
[7] A. A. Suslin, ”Algebraic K-theory,” Itogi Nauki i Tekh. VINITI. Algebra. Topologiya. Geometriya, Vol. 20 (1982), pp. 71–152. · Zbl 0519.18011
[8] S. Bloch, ”Applications of the dilogarithm function in algebraic K-theory and algebraic geometry,” Proc. Int. Symp. on Alg. Geometry, Kyoto (1977), pp. 103–114. · Zbl 0416.18016
[9] S. Bloch, ”Higher regulators, algebraic K-theory, and zeta-functions of elliptic curves,” Irvine Univ. Preprint (1978).
[10] S. Bloch, ”Lectures on algebraic cycles,” Duke Univ. Math. Series, No. 4 (1980). · Zbl 0436.14003
[11] S. Bloch, ”The dilogarithm and extensions of Lie algebras,” Lect. Notes Math.,B54, 1–23 (1981). · Zbl 0469.14009 · doi:10.1007/BFb0089515
[12] A. Borel, ”Stable and real cohomology of arithmetic groups,” Ann. Sci. ENS,7, 235–272 (1974). · Zbl 0316.57026
[13] A. Borel, ”Cohomologie de SLn et valeurs de fonctions zeta aux points entiers,” Ann. Scu. Norm. Super. Pisa Cl. Sci.,4, No. 4, 613–636 (1977). · Zbl 0382.57027
[14] A. K. Bousfield and D. M. Kan, ”Homotopy limits, completions and localizations,” Lect. Notes Math.,304 (1972). · Zbl 0259.55004
[15] P. Deligne, ”Theorie de Hodge. II,” Publ. Math. Inst. Hautes Etudes Scient.,40, 5–58 (1971). · Zbl 0219.14007 · doi:10.1007/BF02684692
[16] P. Deligne, ”Theorie de Hodge. III,” Publ. Math. Inst. Hautes Etudes Scient.,44, 5–77 (1974). · Zbl 0237.14003 · doi:10.1007/BF02685881
[17] P. Deligne, ”Les constantes des equations fonctionelles des fonctions L,” Lect. Notes Math.,349, 501–597 (1973). · doi:10.1007/978-3-540-37855-6_7
[18] P. Deligne, ”Valeurs de fonctions L et periodes d’integrales,” Automorphic Forms, Representations, and L-Functions, Proc. Symp. Pure Math. Am. Math. Soc., Corvallis, Ore. (1977), Part 2, Providence, R.I. (1979), pp. 313–346.
[19] P. Deligne, ”Le symbole modere,” Manuscript (1979).
[20] P. Deligne and M. Rapoport, ”Les schemas de modules de courbes elliptiques,” Lect. Notes Math.,349, 143–316 (1973). · Zbl 0281.14010 · doi:10.1007/978-3-540-37855-6_4
[21] J. Dupont, ”Simplicial de Rham cohomology and characteristic classes of flat bundles,” Topology,15, No. 3, 233–245 (1976). · Zbl 0331.55012 · doi:10.1016/0040-9383(76)90038-0
[22] H. Gillet, ”Riemann-Roch theorems for higher algebraic K-theory,” Adv. Math.,40, 203–289 (1981). · Zbl 0478.14010 · doi:10.1016/S0001-8708(81)80006-0
[23] H. Gillet, ”Comparison of K-theory spectral sequences with applications,” Lect. Notes Math., No. 854, 141–167 (1981). · Zbl 0478.14011 · doi:10.1007/BFb0089520
[24] B. Gross, ”Higher regulators and values of Artin’s L-functions,” Preprint (1979).
[25] H. Jacquet, ”Automorphic forms on GL(2),” Lect. Notes Math., No. 78 (1971). · Zbl 0236.12010
[26] Ch. Kratzer, ”\(\lambda\)-structure en K-theorie algebrique,” Comment. Math. Helv.,55, No. 2, 233–254 (1980). · Zbl 0444.18008 · doi:10.1007/BF02566684
[27] J.-L. Loday, ”Symboles en K-theorie algebrique superiore,” C. R. Acad. Sci.,292, 863–867 (1981).
[28] D. Quillen, ”Higher algebraic K-theory. I,” Lect. Notes Math., No. 341 (1973). · Zbl 0292.18004
[29] B. Saint-Donat, ”Technique de descent cohomologique,” Lect. Notes Math.,270, 83–162 (1972). · doi:10.1007/BFb0061321
[30] Ch. Soulé, ”Operations en K-theorie algebrique,” Preprint (1980).
[31] A. Suslin, ”Homology of GLn, characteristic classes, and Milnor’s K-theory,” Preprint, LOMI (1982).
[32] J. Tate, ”Algebraic cycles and poles of zeta functions,” Arithmetical Algebraic Geometry, New York (1965), pp. 93–100.
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