Algebra structure of multiple zeta values in positive characteristic

TL;DR

This paper demonstrates that multiple zeta values over function fields in positive characteristic have consistent algebraic relations across different valuations, linking their $v$-adic and $ty$-adic forms through a shared algebraic structure.

Contribution

It establishes that $v$-adic MZVs satisfy the same algebraic relations as $ty$-adic MZVs, revealing a unified algebraic structure in positive characteristic.

Findings

$v$-adic MZVs satisfy the same algebraic relations as $ty$-adic MZVs

The algebra of $v$-adic MZVs is given by the $q$-shuffle product

Existence of a $ar{k}$-algebra homomorphism from $ty$-adic to $v$-adic MZVs

Abstract

This paper is a culmination of [CM20] on the study of multiple zeta values (MZV's) over function fields in positive characteristic. For any finite place $v$ of the rational function field $k$ over a finite field, we prove that the $v$-adic MZV's satisfy the same $\bar{k}$-algebraic relations that their corresponding $\infty$-adic MZV's satisfy. Equivalently, we show that the $v$-adic MZV's form an algebra with multiplication law given by the $q$-shuffle product which comes from the $\infty$-adic MZV's, and there is a well-defined $\bar{k}$-algebra homomorphism from the $\infty$-adic MZV's to the $v$-adic MZV's.