Integrality of $v$-adic Multiple Zeta Values

Abstract

In this article, we prove the integrality of $v$-adic multiple zeta values (MZVs). For any index $\mathfrak{s}\in {\mathbb{N}}^r$ and finite place $v\in A:={\mathbb{F}}_q[\theta ]$, Chang and Mishiba introduced the notion of the $v$-adic MZVs ${\zeta }_A(\mathfrak{s}{)}_v$, which is a function field analogue of Furusho's $p$-adic MZVs. By estimating the $v$-adic valuation of ${\zeta }_A(\mathfrak{s}{)}_v$, we show that ${\zeta }_A(\mathfrak{s}{)}_v$ is a $v$-adic integer for almost all $v$. This result can be viewed as a function field analogue of the integrality of $p$-adic MZVs, which was proved by Akagi–Hirose–Yasuda and Chatzistamatiou.