Multiple polylogarithms, cyclotomy and modular complexes
The multiple polylogarithms are the functions \[ Li_{n_1, \dots,n_m} (x_1,\dots,x_m) =\sum_{0<k_1<k_2< \cdots<k_m}{x_1^{k_1} x_2^{k_2} \cdots x_m^{k_m} \over k_1^{n_1} k_2^{n_2} \cdots k_m^{n_m}}. \] Call \(w=n_1+ \cdots+n_m\) the weight and \(m\) the depth. When \(x_1=\cdots=x_m=1\), the functions reduce to Euler's multiple zeta function which has recently been rediscovered in the theory of quantum groups, knot invariants, mixed Tate motives and modular forms. The author is interested in the connections between the special values of the multiple polylogarithms at roots of unity and mixed motives, but this paper is `elementary'. Let \(\mu_N\) be the group of \(N\)-th roots of unity. Let \({\mathcal Z}_{\leq w}(N)\) be the \({\mathcal Q}\)-vector space spanned by the numbers (*) \((2\pi i)^{-w}Li_{n_1, \dots,n_m} (\zeta_N^{\alpha_1}, \dots, \zeta_N^{ \alpha_m})\), where \(\zeta_N\) is a primitive \(N\)-th root of unity, and \({\mathcal Z} (N)=\cup {\mathcal Z}_{\leq w}(N)\). Let \(Z_{*,*}(N)\) be the associated graded quotient with respect to the weight and depth filtrations of the algebra \({\mathcal Z}(N)\) and let \(\overline Z_{*,*} (N)={Z_{*,*}(N) \over (Z_{>0,>0} (N))^2}\). Construct a Lie coalgebra \({\mathcal D}_{*,*}(\mu_N)\) by taking as generators of the \({\mathcal Q}\)-vectors space \({\mathcal D}_{w,m} (\mu_N)\) the projections of numbers (*) to \(\overline Z_{w,m}(N)\). By exploiting functional relations for the multiple polylogarithms and properties of the Voronoi complex, the author arrives at estimates for the Euler characteristic of the complexes \((\Lambda^*{\mathcal D} (1))_{w,m}\) for \(m=2\) and 3. For example, if \(w\) is odd, \(\dim \overline Z_{w,2} (1) =0\) and, if \(w\) is even, \(\dim\overline Z_{w,2} (1)\leq[{w-2 \over 6}]\). Again, if \(w\) is even, \(\dim\overline Z_{w,3} (1)=0\) and, if \(w\) is odd, \(\dim \overline Z_{w,3} (1)\leq[{(w-3)^2 -1\over 48}]\).
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