\(p\)-adic multiple zeta values. I: \(p\)-adic multiple polylogarithms and the \(p\)-adic KZ equation

DOI10.1007/S00222-003-0320-9zbMATH Open1061.11034arXivmath/0304085OpenAlexW2018176186MaRDI QIDQ1879013

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Publication date: 22 September 2004

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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.

Full work available at URL: https://arxiv.org/abs/math/0304085

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