Classical Motivic Polylogarithm According to Beilinson and Deligne

The main purpose of this paper is the construction in motivic cohomology of the {\it cyclotomic}, or {\it classical polylogarithm} on the projective line minus three points, and the identification of its image under the regulator to absolute (Deligne or $l$-adic) cohomology. By specialization to roots of unity, one obtains a compatibility statement on {\it cyclotomic elements} in motivic and absolute cohomology of abelian number fields. As shown in \cite{BK}, this compatibility completes the proof of the Tamagawa number conjecture on special values of the Riemann zeta function.

The main constructions and ideas are contained in Beilinson's and Deligne's unpublished preprint ``Motivic Polylogarithm and Zagier Conjecture'' (\cite{BD}). We work out the details of the proof, setting up the foundational material which was missing from the original source: the paper contains an appendix on absolute Hodge cohomology with coefficients, and its interpretation in terms of Saito's Hodge modules. The second appendix treats $K$-theory and regulators for simplicial schemes.

1991 Mathematics Subject Classification: Primary 19F27; Secondary11R18, 11R34, 11R42, 14D07, 14F99.

Keywords: Polylogarithm, motivic and absolute cohomology, regulators, cyclotomic elements.

Correction, this volume: 297-299.

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