The dihedral Lie algebras and Galois symmetries of \(\pi_1^{(l)}(\mathbb P^1-(\{0,\infty\}\cup\mu-\{N\}))\).
type: Item
instance of: scholarly article
title: The dihedral Lie algebras and Galois symmetries of \(\pi_1^{(l)}(\mathbb P^1-(\{0,\infty\}\cup\mu-\{N\}))\).
zbMATH Open document ID: 1113.14020
DOI: 10.1215/S0012-7094-01-11031-4
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: For a natural number \(N\), denote by \(\mu_N\) the group of complex \(N\)-th roots of unity and consider the variety \[ X_N:= \mathbb{P}_\mathbb{C}^1\setminus\bigl\{\{0,\infty\} \cup\mu_N\bigr\}=\mathbb{C}^* \setminus\mu_N. \] In the study of higher cyclotomy, understood as the motivic theory of multiple polylogarithms at roots of unity, it has turned out that there should be some fundamental link between the structure of the motivic fundamental group of the variety \(X_N\), on the one hand, and the geometry of certain modular varieties for the congruence subgroups \(\Gamma_1(m:N)\) of \(\text{GL}_m(\mathbb{Z})\), \(m\geq 1\), on the other. In the case of \(N=1\), this mysterious link had been sketchily foretold by A. Grothendieck in the early 1980s, and was then completely established by the work of P. Deligne (l987), Y.Ihara (1986--1999), V. G. Drinfeld (1991), and others in the sequel. The paper under review is devoted to an extensive study of this link for \(\dot N>1\) from a particular viewpoint. Namely, according to P. Deligne's motivic philosophy, the motivic fundamental group of \(X_N\) can be interpreted as a Lie algebra object in the category of mixed motives, appearing there as a pro-nilpotent completion of the topological fundamental group of \(X_N\) equipped with various additional structures of analytic, geometric, and arithmetic nature. Moreover, the \(\ell\)-adic realization of the motivic fundamental group is obtained from the action of the absolute Galois group \(\text{Gal}(\overline \mathbb{Q}/\mathbb{Q})\) on the pro-\(\ell\) completion \(\pi_1^{(\ell)}(X_N)\) of \(\pi_1(X)\), and that is why a detailed study of this Galois action constitutes one main part of the present paper. The author's approach is based on a fine analysis of certain Lie algebras constructed from the \(\ell\)-adic pro-Lie algebra \(\mathbb{L}_N^{(\ell)}\) corresponding via Deligne-Maltsev theory to the group \(\pi_1^{(\ell)} (X_N)\). The central object in the author's buildi…
Mathematics Subject Classification ID: 14G32
Mathematics Subject Classification ID: 11F67
Mathematics Subject Classification ID: 11G55
Mathematics Subject Classification ID: 14H30
zbMATH DE Number: 1820861
zbMATH Keywords: fundamental groups
zbMATH Keywords: modular varieties
zbMATH Keywords: Galois groups
zbMATH Keywords: motives
zbMATH Keywords: Lie algebras
zbMATH Keywords: complexes
zbMATH Keywords: polylogarithms
zbMATH Keywords: cohomology of arithmetic groups
author: A. B. Goncharov
reviewed by: Werner Kleinert
MaRDI profile type: Publication
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full work available at URL: s0012-7094-01-11031-4
OpenAlex ID: W2043511149
description: scientific article; zbMATH DE number 1820861