Finite and p-adic polylogarithms, by Amnon Besser

The finite n-th polylogarithm lin(z) in Z/p[z] is defined as the sum on k from 1 to p-1 of zk/kn. We state and prove the following theorem. Let Lik:Cp to Cp be the p-adic polylogarithms defined by Coleman. Then a certain linear combination Fn of products of polylogarithms and logarithms, with coefficients which are independent of p, has the property that p1-n DFn(zp) reduces modulo p > n+1 to lin-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.

Amnon Besser <bessera@math.bgu.ac.il>