Geometry of configurations, polylogarithms, and motivic cohomology
type: Item
instance of: scholarly article
title: Geometry of configurations, polylogarithms, and motivic cohomology
zbMATH Open document ID: 0863.19004
published in: Advances in Mathematics
publication date: 1996-10-23T00:00:00Z
review text: This paper gives a full proof account of the state of the art of the theory of polylogarithms from a geometric point of view as advocated by the author. The case of the dilogarithm being known since some time by work of Gabrielov et al., Bloch, Wigner, Zagier,\( \dots\), the underlying paper focusses on the trilogarithm. Classically the \(p\)-th polylogarithm \(\text{Li}_p (z)\) is defined as the analytic continuation of the expression \(\text{Li}_p(z)=\sum_{n=1}^\infty {z^n \over n^p}\), \(|z|\leq 1\). With \(\text{Li}_1(z)= - \log(1-z)\), one has the inductive formula \(\text{Li}_p(z)=\int^z_0 \text{Li}_{p-1} (z) {dt \over t}\) with its (multivalued) continuation to \(\mathbb{P}^1_\mathbb{C} \backslash \{0,1, \infty\}\). It turns out to be advantageous to consider modified functions \({\mathcal L}_p (z)={\mathcal R}_p \left( \sum^p_{j=0} {2^j B_j \over j!} (\log |z |)^j \cdot \text{Li}_{p-j} (z) \right)\), where the \(B_j\) are the Bernoulli numbers, and \({\mathcal R}_m\) denotes the real part for odd \(m\) and the imaginary part for even \(m\), \(\text{Li}_0 (z):= -{1 \over 2}\). Thus one has \[ {\mathcal L}_2 (z)={\mathfrak I} \bigl( \text{Li}_2 (z) \bigr) + \arg (1-z) \cdot \log |z |, \] the Bloch-Wigner function. For the present paper the most interesting function becomes \[ {\mathcal L}_3 (z)={\mathfrak R} \Bigl( \text{Li}_3 (z)-\log |z |\cdot \text{Li}_2 (z)+ \textstyle {{1\over 3}} \log^2 |z|\cdot \text{Li}_1(z) \Bigr). \] The functions \({\mathcal L}_p(z)\) are single-valued, real analytic on \(\mathbb{P}^1_\mathbb{C} \backslash \{0,1, \infty\}\) and continuous at \(0,1,\infty\). In particular, \({\mathcal L}_3 (0)= {\mathcal L}_3 (\infty) =0\) and \({\mathcal L}_3 (1)= \zeta_\mathbb{Q} (3)\), the Riemann zeta-function at \(s=3\). The functions \({\mathcal L}_p (z)\) admit a Hodge theoretic interpretation. As a first goal one tries to find the generic functional equation for the (modified) trilogarithm. Motivated by results on the Bloch-Wigner function wh…
Mathematics Subject Classification ID: 19F27
Mathematics Subject Classification ID: 14C35
Mathematics Subject Classification ID: 14F20
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 11R70
Mathematics Subject Classification ID: 33E20
zbMATH DE Number: 814488
zbMATH Keywords: algebraic \(K\)-theory
zbMATH Keywords: values of \(L\)-functions
zbMATH Keywords: Beilinson conjecture
zbMATH Keywords: finite dimensional representations of graded pro-Lie algebra
zbMATH Keywords: polylogarithms
zbMATH Keywords: Bernoulli numbers
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: generic functional equation
zbMATH Keywords: trilogarithm
zbMATH Keywords: Bloch-Wigner function
zbMATH Keywords: Spence-Kummer relation
zbMATH Keywords: motivic cohomology
zbMATH Keywords: characteristic classes
zbMATH Keywords: Dedekind zeta function
zbMATH Keywords: number field
zbMATH Keywords: dilogarithm
zbMATH Keywords: regulator map
zbMATH Keywords: Deligne-Beilinson cohomology
zbMATH Keywords: mixed Tate category
zbMATH Keywords: Tannakian categories
zbMATH Keywords: duality of configurations
zbMATH Keywords: Grassmannian trilogarithm
author: A. B. Goncharov
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
OpenAlex ID: W2034590487
DOI: 10.1006/AIMA.1995.1045
Recommended article: Motives and homotopy theory in logarithmic geometry
Recommended article: Q2988254
Recommended article: Motivic polylogarithm and related classes
Recommended article: Q4819930
Recommended article: Logarithmic geometry, minimal free resolutions and toric algebraic stacks
Recommended article: Q4650879
Recommended article: Q5436873
Recommended article: Geometric motives and the \(h\)-topology
Recommended article: The algebraic and anabelian geometry of configuration spaces
Recommended article: Q4524273
description: scientific article; zbMATH DE number 814488