Finite and p-adic polylogarithms

Amnon Besser (BGU)

Abstract

The finite polylogarithms were introduced by Elbaz Vincent and Gangl following Kontsevich (who defined the finite (1)-logarithm) as the function from Z/p to itself given by the polynomial li_n(z) = Sum_{k = 1}^p-1 z^k/kⁿ. Kontsevich noticed that the finite logarithm satisfied a 4-term functional equation, known as the fundamental equation of information theory, and that this equation is also satisfied by the "derivative" of the complex dilogarithm. It was later checked by Elbaz-Vincent and Gangl that a similar relation exists between the finite 2-logarithm and the derivative of the complex 3-logarithm.

Kontsevich made the conjecture that (an appropriate version of) the p-adic n+1-logarithm would have the property that its derivative would have a reduction modulo p that would be exactly the finite n-logarithm and hoped that this conjecture explains the relation between the two functional equations. Recently we have proved this conjecture.

In the talk we will explain the complex p-adic and finite polylogarithms and the known functional equations. we will give the precise version of the theorem we proved and disucuss the proof (which is very simple).

File translated from T_EX by T_TH, version 2.60.
On 16 Feb 2000, 11:43.