Zagier’s conjecture and wedge complexes in algebraic \(K\)-theory. (English) Zbl 0868.19002

De Jeu, Rob

Compos. Math. 96, No. 2, 197-247 (1995).

In “Polylogarithms, Dedekind zetafunctions, and the algebraic \(K\)-theory of fields” [in: Arithmetic algebraic geometry, Prog. Math. 89, 391-430 (1991; Zbl 0728.11062)], D. Zagier formulated a conjecture about the higher odd \(K\)-theory groups \(K_{2n-1}(F)\) of a number field \(F\) based on Bloch’s earlier work in the case \(n=2\). Beilinson and Deligne reformulated the conjecture as follows:
Let \({\mathcal L}=F^*\otimes \mathbb{Q}\) and let \(\{x\}_1= 1-x\) for \(x\in F\setminus \{0,1\}\). There should exist abelian groups \({\mathcal L}_n\) for \(n\geq 1\) generated by symbols \(\{x\}_n\), \(x\in F\setminus \{0,1\}\) together with a map \(d_n:{\mathcal L}\to\bigwedge^2 (\bigoplus^{n-1}_{l=1}{\mathcal L}_l)\) given by \(\{x\}_n\mapsto x\wedge\{ x\}_{n-1}\). There should exist an isomorphism \(\varphi_n:\ker d_n\to K^{(n)}_{2n-1}(F)\) and the composite of \(\varphi_n\) and the Borel regulator map should be given explicitly in terms of the \(n\)-th polylogarithm.
Using a formalism of multi-relative \(K\)-theory the author constructs for each \(n\geq 1\) complexes \(\widetilde{\mathcal M}_{(n)}\) of \(\mathbb{Q}\)-vector spaces, whose \(p\)-th cohomology \(H^p(\widetilde{\mathcal M}_{(n)})\) maps to \(K^{(n)}_{2n-p}(F)\). This leads to a proof of part of the Zagier conjecture for all \(n\geq 2\), since for \(p=1\) this map is an injection. Moreover, using work of Suslin, Goncharov and Zagier, the author shows that for \(n=2,3\) surjectivity holds as well, i.e., the maps \(H^1(\widetilde{\mathcal M}_{(n)})\to K^{(n)}_{2n-1}(F)\) are isomorphisms for \(n=2,3.\) The same result holds for all \(n\geq 2\) provided \(F=\mathbb{Q}(\zeta_N)\) is a cyclotomic field.

Reviewer: M.Kolster (Hamilton/Ontario)

MSC:

19F15	Symbols and arithmetic (\(K\)-theoretic aspects)
11R70	\(K\)-theory of global fields
14C35	Applications of methods of algebraic \(K\)-theory in algebraic geometry
11R42	Zeta functions and \(L\)-functions of number fields

Keywords:

wedge complex; Borel regulator; polylogarithm; multi-relative \(K\)-theory; Zagier conjecture

Citations:

Zbl 0728.11062

References:

[1]	Beilinson, S. : Higher regulators and values of L-functions , J. Sov. Math. 30 (1985), 2036-2070. · Zbl 0588.14013
[2]	Beilinson, S. and Deligne, P. : Polylogarithms and regulators , Preprint, 1990. · Zbl 0799.19004
[3]	Beilinson, S. and Deligne, P. : Motivic polylogarithm and Zagier’s conjecture , Preprint, 1992.
[4]	Bloch, S. : Higher Regulators, Algebraic K-Theory and Zeta Functions of Elliptic Curves , Lecture Notes U.C., Irvine, 1978.
[5]	Bloch, S. and Kriz, I. : Mixed Tate Motives , to appear in Ann. Math. · Zbl 0935.14014
[6]	Borel, A. : Stable real cohomology of arithmetic groups , Ann. Sci. ENS 4 (1974), 235-272. · Zbl 0316.57026
[7]	Borel, A. : Cohomology de SLn et valeurs de fonctions zêta aux points entires , Ann. Sci. Norm. Sup. Pisa 4 (1977), 613-636. · Zbl 0382.57027
[8]	Bousfield, A.K. and Kan, D.M. : Homotopy Limits, Completions and Localizations , Lecture Notes in Math. 304, Springer-Verlag, Berlin, 1972. · Zbl 0259.55004
[9]	Deligne, P. : Théorie de Hodge, II , Publ. Math. IHES 40 (1972), 5-57. · Zbl 0219.14007
[10]	Deligne, P. : Interprétation motivique de la conjecture de Zagier reliant polylogarithms et régulateurs . Preprint, 1991. · Zbl 0799.19004
[11]	Dold, A. and Puppe, D. : Homologie nicht-additiver Funktoren. Anwendungen , Ann. Inst. Fourier 11 (1961), 201-312. · Zbl 0098.36005
[12]	Esnault, H. and Viehweg, E. : Deligne-Beilinson cohomology, in Beilinson’s Conjectures on Special Values of L-functions , Academic Press, New York, 1988, pp. 43-91. · Zbl 0656.14012
[13]	Gillet, H. : Riemann-Roch theorems for higher algebraic K-theory , Adv. in Math. 40 (1981), 203-288. · Zbl 0478.14010
[14]	Gillet, H. and Soulé, C. : Filtrations on higher algebraic K-theory, unpublished . · Zbl 0951.19003
[15]	Goncharov, A.B. : Geometry of configurations, polylogarithms and motivic cohomology , to appear in Adv. Math. · Zbl 0863.19004
[16]	Goncharov, A.B. : Polylogarithms and motivic Galois groups . 1992, Preprint. · Zbl 0842.11043
[17]	Hartshorne, R. : Residues and Duality , Lecture Notes in Math. 20, Springer-Verlag, Berlin, 1966. · Zbl 0212.26101
[18]	Jannsen, U. : Deligne homology, Hodge-D-conjecture, and motives , in Beilinson’s Conjectures on Special Values of L-functions , Academic Press, New York, 1988, pp. 305-372. · Zbl 0701.14019
[19]	Joyal, A. : Letter to A. Grothendieck , 1984.
[20]	Kratzer, Ch. : \lambda -structure en K-théorie algébrique , Comm. Math. Helv. 55 (1980), 233-254. · Zbl 0444.18008
[21]	Levine, M. : On the derived motivic category , Preprint.
[22]	Mac Lane, S. : Homology , Springer-Verlag, Berlin, 1963.
[23]	Milnor, J. : Introduction to Algebraic K-theory , Princeton University Press, Princeton, 1971. · Zbl 0237.18005
[24]	Quillen, D. : Homotopical Algebra , Lecture Notes in Math. 43, Springer-Verlag, Berlin, 1967. · Zbl 0168.20903
[25]	Quillen, D. : Higher algebraic K-theory I, in Algebraic K-theory 1 , Lecture Notes in Math., 341, Springer-Verlag, Berlin, 1973, pp. 85-147. · Zbl 0292.18004
[26]	Soulé, C. : K-théorie des anneaux d’entiers de corps de nombres et cohomologie étale , Inventiones Math. 55 (1979), 251-295. · Zbl 0437.12008
[27]	Soulé, C. : Opérations en K-théorie algébrique , Can. J. Math. 37(3) (1985), 488-550. · Zbl 0575.14015
[28]	Suslin, A.A. : K3 of a field and the Bloch group , Trudy Math. Inst. Steklov 189 (1990), 180-199 (in Russian). English translation: Proc. Steklov Inst. Math., 183-184 (1991), 217-239. · Zbl 0741.19005
[29]	Tamme, G. : The theorem of Riemann-Roch, in Beilinson’s Conjectures on Special Values of L-functions , Academic Press, New York, 1988, pp. 103-168. · Zbl 0664.14004
[30]	Voevodsky, V. : Homology of schemes and covariant motives , Preprint, 1992.
[31]	Zagier, D. : Polylogarithms, Dedekind Zetafunctions, and the Algebraic K-Theory of Fields , in Arithmetic Algebraic Geometry , Birkhäuser, 1991, pp. 391-430. · Zbl 0728.11062