Finite and p-adic Polylogarithms

Abstract

The finite nth polylogarithm li_n(z) ∈ Z/p(z) is defined as [sum ]_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 p¹⁻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.