## Abstract

The finite nth polylogarithm li_{n}(z) ε ℤ/p[z] is defined as ∑_{k=1}^{p-1}z^{k}/k^{n}. We state and prove the following theorem. Let Li_{k}: ℂ_{p} → 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) 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 language | English |
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Pages (from-to) | 215-223 |

Number of pages | 9 |

Journal | Compositio Mathematica |

Volume | 130 |

Issue number | 2 |

DOIs | |

State | Published - 1 Dec 2002 |

## Keywords

- Functional equations
- Polylogarithms
- p-adic integration

## ASJC Scopus subject areas

- Algebra and Number Theory