### Finite and p-adic polylogarithms, by Amnon Besser

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.

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