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

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 $E^{\prime }$. 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 ${\mathbf{v}}_{\mathfrak{C}}\in E^{\prime }({\mathbb{F}}_q[\theta ])$ 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 ${\mathbf{v}}_{\mathfrak{C}}$.