The double logarithm and Manin’s complex for modular curves

Multiple polylogarithms are periods of mixed Tate motives (s.12 of [G2], [G3]). Let Γ1(N ;m) ⊂ GLm(Z) be the subgroup of the matrices whose bottom row is congruent to (0, ..., 0, 1) modulo N and Vm the standard m-dimensional representation of GLm(Z). In this note we begin to study a mysterious connection between the multiple polylogarithms of depth m at N -th roots of unity and cohomology of Γ1(N ;m) with coefficients in Sw−m(Vm) where m > 1. We will work out in details the simplest case: m = 2, n1 = n2 = 1, p is prime. In s.4 we touch the general situation and will return to it in [G1], see also [G4]. So we investigate the double logarithm function