Multiple polylogarithms, cyclotomy and modular complexes

this paper we investigate them at N-th roots of unity: x 1 = ::: = x m = 1. Notice that Li 1 (x) = log(1 x), so if N is a primitive N-th root of 1, then Li 1 ( N ) is a logarithm of a cyclotomic unit in Z[N ; N ]. In general the supply of numbers we get coincides with the linear combinations of multiple Dirichlet L-values L( 1 ; :::; m ; n 1 ; :::; nm ) := 1 (k 1 )::: m (k m ) They are periods of mixed Tate motives over the scheme SN := SpecZ[N ][ (see s. 11 of [G2] and [G4]). To study these numbers we introduce some tools from homological algebra (cyclotomic and dihedral Lie algebras, modular complex for GLm (Z)) and geometry (a realization of the modular complex in the symmetric space SLm (R)=SOm ). To motivate them we start from a conjecture