Integrality of $v$-adic multiple zeta values

TL;DR

This paper proves that $v$-adic multiple zeta values in function fields are $v$-adic integers for almost all places, extending the concept of integrality from classical $p$-adic MZVs to the function field setting.

Contribution

It establishes the integrality of $v$-adic MZVs in function fields, providing a new analogue to known $p$-adic MZV integrality results.

Findings

$v$-adic MZVs are $v$-adic integers for almost all $v$

The result extends classical $p$-adic MZV integrality to function fields

Provides valuation estimates for $v$-adic MZVs

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.