Finite and p-adic polylogarithms

The finite nth polylogarithm li_n(z) ε ℤ/p[z] is defined as ∑_k=1^p-1z^k/kⁿ. We state and prove the following theorem. Let Li_k: ℂ_p → ℂ_p be the p-adic polylogarithms defined by Coleman. Then a certain linear combination F_n of products of polylogarithms and logarithms, with coefficients which are independent of p, has the property that p^1-nDF_n(z) reduces modulo p > n + 1 to li _n-1(σ(z)), where D is the Cathelineau operator z(1-z)d/dz and σ is the inverse of the p-power map. A slightly modified version of this theorem was conjectured by Kontsevich. This theorem is used by Elbaz-Vincent and Gangl to deduce functional equations of finite polylogarithms from those of complex polylogarithms.