Inventiones mathematicae

, Volume 155, Issue 2, pp 253–286

p-adic multiple zeta values I.

p-adic multiple polylogarithms and the p-adic KZ equation

DOI: 10.1007/s00222-003-0320-9

Cite this article as:
Furusho, H. Invent. math. (2004) 155: 253. doi:10.1007/s00222-003-0320-9


Our main aim in this paper is to give a foundation of the theory of p-adic multiple zeta values. We introduce (one variable) p-adic multiple polylogarithms by Coleman’s p-adic iterated integration theory. We define p-adic multiple zeta values to be special values of p-adic multiple polylogarithms. We consider the (formal) p-adic KZ equation and introduce the p-adic Drinfel’d associator by using certain two fundamental solutions of the p-adic KZ equation. We show that our p-adic multiple polylogarithms appear as coefficients of a certain fundamental solution of the p-adic KZ equation and our p-adic multiple zeta values appear as coefficients of the p-adic Drinfel’d associator. We show various properties of p-adic multiple zeta values, which are sometimes analogous to the complex case and are sometimes peculiar to the p-adic case, via the p-adic KZ equation.

Copyright information

© Springer-Verlag 2003

Authors and Affiliations

  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan