# A $t$-motivic interpretation of shuffle relations for multizeta values

## TL;DR

This paper develops a $t$-motivic framework to interpret shuffle relations among multizeta values over function fields, linking algebraic relations to torsion points on specially constructed $t$-modules.

## Contribution

It introduces a $t$-module construction that characterizes shuffle relations via torsion points, providing an effective criterion for their identification.

## Key findings

- Constructed a $t$-module $E'$ associated with multizeta shuffle relations.
- Established a criterion linking torsion points on $E'$ to shuffle relations.
- Provided an algorithm to determine the torsion property of points in $E'$.

## Abstract

Thakur (2010) showed that, for $r,$ $s\in \mathbb{N}$, a product of two Carlitz zeta values $\zeta_A(r)$ and $\zeta_A(s)$ can be expressed as an $\mathbb{F}_p$-linear combination of $\zeta_A(r+s)$ and double zeta values of weight $r+s$. Such an expression is called shuffle relation by Thakur. Fixing $r,$ $s\in \mathbb{N}$, we construct a $t$-module $E'$. To determine whether an $(r+s)$-tuple $\mathfrak{C}$ in $\mathbb{F}_q(\theta)^{r+s}$ gives a shuffle relation, we relate it to the $\mathbb{F}_q[t]$-torsion property of the point $\mathbf{v}_\mathfrak{C}\in E'(\mathbb{F}_q[\theta])$ constructed with respect to the given $(r+s)$-tuple $\mathfrak{C}$. We also provide an effective criterion for deciding the $\mathbb{F}_q[t]$-torsion property of the point $\mathbf{v}_\mathfrak{C}$.