Algebra structure of multiple zeta values in positive characteristic

This paper is a culmination of [CM21] 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 $\overline{k}$‑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 $\overline{k}$‑algebra homomorphism from the $\infty$‑adic MZV’s to the $v$‑adic MZV’s.

Keywords

multiple zeta values, $v$-adic multiple zeta values, Carlitz multiple star polylogarithms, logarithms of $t$-modules, $t$-motives

2010 Mathematics Subject Classification

11J93, 11R58

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The first and second named authors were partially supported by MOST Grant Number 107-2628-M-007-002-MY4.

The third named author was supported by JSPS KAKENHI Grant Number JP18K13398.

Received 16 December 2020

Published 21 October 2022