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 <firstname.lastname@example.org>