p-adic multiple zeta values I -- p-adic multiple polylogarithms and the p-adic KZ equation

摘要

Our main aim in this paper is to give a foundation of the theory of $p$-adicmultiple zeta values. We introduce (one variable) $p$-adic multiplepolylogarithms by Coleman's $p$-adic iterated integration theory. We define$p$-adic multiple zeta values to be special values of $p$-adic multiplepolylogarithms. 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 appearas coefficients of a certain fundamental solution of the $p$-adic KZ equationand our $p$-adic multiple zeta values appear as coefficients of the $p$-adicDrinfel'd associator. We show various properties of $p$-adic multiple zetavalues, which are sometimes analogous to the complex case and are sometimespeculiar to the $p$-adic case, via the $p$-adic KZ equation.