Finite and p-adic polylogarithms

Abstract

The finite n-th polylogarithm li_n(z) in Z/p[z] is defined as the sum on k from 1 to p-1 of z^k/k^n. We state and prove the following theorem. Let Li_k:C_p to 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-n} DF_n(z^p) reduces modulo p>n+1 to li_{n-1}(z) where D is the Cathelineau operator z(1-z) d/dz. 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.