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

TL;DR

This paper develops a foundational theory of p-adic multiple zeta values, introducing p-adic multiple polylogarithms via Coleman's iterated integration and exploring their relation to the p-adic KZ equation and Drinfel'd associator.

Contribution

It establishes the connection between p-adic multiple polylogarithms, the p-adic KZ equation, and the p-adic Drinfel'd associator, providing new insights into their properties and relationships.

Findings

p-adic multiple polylogarithms are coefficients of solutions to the p-adic KZ equation

p-adic multiple zeta values are coefficients of the p-adic Drinfel'd associator

properties of p-adic multiple zeta values are sometimes analogous and sometimes unique compared to the complex case

Abstract

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.