The dihedral Lie algebras and Galois symmetries of \(\pi_1^{(l)}(\mathbb P^1-(\{0,\infty\}\cup\mu-\{N\}))\).
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 building up is a Lie algebra bigraded by negative integers \(-w\) and \(-m\), called the weight and the depth indices, which is called the ``level \(N\) Galois Lie algebra associated to \(\pi_1^{(\ell)}(X_N)\)''. In this context, the author's main goal is prove the following general statement: Let \(V\) be the standard \(m\)-dimensional representation of the linear group \(\text{GL}_m\). Then the depth \(m\) and weight \(w\) part of the standard cochain complex of the level \(N\) Galois Lie algebra associated to \(\pi_1^{(\ell)}(X_N)\) is explicitely related to the geometry of the local system with fibre \(S^{w-m}V_m\) over the level \(N\) modular variety for the congruence subgroup \(\Gamma_1(m;N)\) of \(\text{GL}_m(\mathbb{Z})\). In particular, the depth \(m\) quotient of the level \(N\) Galois Lie algebra can be completely described by the geometry of that modular variety. In the present paper, this is made explicit (and then-proved) in the special cases \(m=2\) and \(m=3\). The case \(m=1\) can be deduced from the motivic theory of classical polylogarithms due to P. Deligne and A. Beilinson, whereas there are indications that the next complicated case \((m=4)\) could also be tackled by the author's methods developed in the present paper. As for the novel and subtle techniques established here, the author builds upon his previous related results obtained in a series of foregoing papers the author [Math. Res. Lett. 4, No. 5, 617--636 (1997; Zbl 0916.11034); ibid. 5, No. 4, 497--516 (1998; Zbl 0961.11040)], this time emphasizing the Galois-theoretic part of his global program. Finally, the (partial) results of the present paper lead the author to a number of conjectures, in this present paper lead the author to a number of conjectures, in this context, which are explained and illustrated in detail.
scientific article; zbMATH DE number 1820861
scientific article