Finite and p-adic Polylogarithms

The finite nth polylogarithm li_n(z) /p(z) is defined as _k=1^p–1z^k/kⁿ. We state and prove the following theorem. Let Li_k: _{p 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((z)), where D is the Cathelineau operator z(1–z)d/dz and 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.