Trivial multiple zeta values in Tate algebras

TL;DR

This paper investigates the structure of trivial multiple zeta values in Tate algebras over positive characteristic fields, revealing their module structure and implications for relations among Thakur's multiple zeta values.

Contribution

It introduces a module structure for trivial multiple zeta values in Tate algebras and characterizes its generators, aiding in understanding relations among multiple zeta values.

Findings

Module structure over a non-commutative polynomial ring is established.

Explicit generators for the module are determined.

The structure helps detect linear relations among Thakur's multiple zeta values.

Abstract

We study trivial multiple zeta values in Tate algebras. These are particular examples of the multiple zeta values in Tate algebras in positive characteristic introduced by the second author. If the number of variables involved is 'not large' in a way that is made precise in the paper, we can endow the set of trivial multiple zeta values with a structure of module over a non-commutative polynomial ring with coefficients in the rational fraction field over a finite field. We determine the structure of this module in terms of generators and we show how in many cases, this is sufficient for the detection of some linear relations between Thakur's multiple zeta values.