Finite and p-adic polylogarithms

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13 Scopus citations


The finite nth polylogarithm lin(z) ε ℤ/p[z] is defined as ∑k=1p-1zk/kn. We state and prove the following theorem. Let Lik: ℂp → 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-nDFn(z) reduces modulo p > n + 1 to li n-1(σ(z)), where D is the Cathelineau operator z(1-z)d/dz and σ is the inverse of the p-power map. 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.

Original languageEnglish
Pages (from-to)215-223
Number of pages9
JournalCompositio Mathematica
Issue number2
StatePublished - 1 Dec 2002


  • Functional equations
  • Polylogarithms
  • p-adic integration

ASJC Scopus subject areas

  • Algebra and Number Theory


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