Todd's relation conjecture and binary relations for multiple zeta values in positive characteristic

RESUMO

We prove Todd's relation conjecture: all $\mathbb{F}_q(\theta)$-linear relations of Thakur's multiple zeta values are generated from the fundamental binary relation by the operators $\mathcal{B}^{\mathrm{S}}$, $\mathcal{C}^{\mathrm{S}}$, $\mathcal{B}^{\mathrm{S}}\circ \mathcal{C}^{\mathrm{S}}$; moreover they are also generated by $\mathcal{B}^{\ast\mathrm{S}}$, $\mathcal{C}^{\mathrm{S}}$, $\mathcal{B}^{\ast\mathrm{S}}\circ\mathcal{C}^{\mathrm{S}}$. The $\mathcal{B}^{\ast\mathrm{S}}$-part of this conjecture has been proved by Chang, Chen and Mishiba. We prove the whole conjecture for Carlitz multiple polylogarithm values, which implies the $\mathcal{B}^{\mathrm{S}}$-part for multiple zeta values. We also determine all fixed relations and binary relations. Let $\mathfrak{BR}^{\mathrm{S}}_w$ be the $\mathbb{F}_q(\theta)$-linear space spanned by binary relations of weight $w$, and let $\operatorname{Fix}^{\mathrm{S}}_w$ be the $\mathbb{F}_q(\theta)$-linear space spanned by fixed relations. We derive generating functions $\sum_{w\ge 1}\bigl(\dim_{\mathbb{F}_q(\theta)}\operatorname{Fix}^{\mathrm{S}}_w\bigr)x^w=\frac{x^{q+1}(1-x)}{(1-2x)(1-2x+x^{q+1})}$ and $\sum_{w\ge 1}\bigl(\dim_{\mathbb{F}_q(\theta)}\mathfrak{BR}^{\mathrm{S}}_w\bigr)x^w=\frac{x^q(1-x)}{(1-2x)(1-2x+x^{q+1})}.$ Our results are based on the recent work of Im-Kim-Ngo Dac on the $\mathbb{F}_q$-linear relations of Thakur's multiple zeta values, and a system of transfer theorems between multiple zeta values and multiple polylogarithm values.