Classical Motivic Polylogarithm According to Beilinson and Deligne

Abstract

The main purpose of this paper is the construction in motivic cohomology of the cyclotomic, or classical polylogarithm on the projective line minus three points, and the identi cation 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 cyclotomic elements in motivic and absolute cohomology of abelian number fields. As shown in [BlK], 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The main constructions and ideas are contained in Beilinson's and Deligne's unpublished preprint “Motivic Polylogarithm and Zagier Conjecture” ([BD1]). 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 coe cients, and its interpretation in terms of Saito's Hodge modules.

A correction to this paper is available.