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.
Categories: math.NT