A $v$-adic variant of Anderson-Brownawell-Papanikolas linear independence criterion and its application

TL;DR

This paper develops a $v$-adic linear independence criterion for Carlitz multiple polylogarithms and applies it to prove a function field analogue of a conjecture on multiple zeta values for places of degree one.

Contribution

It introduces a $v$-adic variant of the Anderson-Brownawell-Papanikolas criterion and applies it to establish new results on $v$-adic multiple zeta values in function fields.

Findings

All $ar{k}$-linear relations among $v$-adic Carlitz polylogarithms at algebraic points come from $k$-linear relations.

Established a function field analogue of Furusho-Yamashita's conjecture for $v$-adic multiple zeta values.

Proved the criterion for places of degree one.

Abstract

Let $\overline{k}$ be a fixed algebraic closure of $k$. When the finite place $v$ is of degree one, we show that all $\overline{k}$-linear relations among $v$-adic Carlitz multiple polylogarithms at algebraic points arise from $k$-linear relations among these values of the same weight. As an application, we establish a function field analogue of Furusho-Yamashita's conjecture for $v$-adic multiple zeta values whenever the degree of the place $v$ is one.