Deformation of Multiple Zeta Values and Their Logarithmic Interpretation in Positive Characteristic

TL;DR

This paper explores the deformation of multiple zeta values in positive characteristic, linking them to higher dimensional Drinfeld modules and expressing their deformations through multiple polylogarithms.

Contribution

It introduces a new connection between deformed multiple zeta values and higher dimensional Drinfeld modules, along with a representation via multiple polylogarithms.

Findings

Deformation of multiple zeta values related to higher dimensional Drinfeld modules.

Representation of deformed values as linear combinations of multiple polylogarithms.

Establishment of a logarithmic interpretation in positive characteristic.

Abstract

Pellarin introduced the deformation of multiple zeta values of Thakur as elements over Tate algebras. In this paper, we relate these values to a certain coordinate of a higher dimensional Drinfeld module over Tate algebras which we will introduce. Moreover, we define multiple polylogarithms in our setting and represent deformation of multiple zeta values as a linear combination of multiple polylogarithms.