${\mathbb{F}}_q$-linear relations among Thakur's multiple zeta values in positive characteristic

Abstract

Let ${\mathcal{Z}}_w^{({\mathbb{F}}_q)}$ be the ${\mathbb{F}}_q$-linear subspace of ${\mathbb{F}}_q(({\theta }^{-1}))$ spanned by Thakur's multiple zeta values ${\zeta }_A(\mathfrak{s})$ of weight $w$. We prove that ${\sum }_{w=1}^{\infty }\left({\dim }_{{\mathbb{F}}_q}{\mathcal{Z}}_w^{({\mathbb{F}}_q)}\right)x^w=\frac{x(1-x^q)(1-2x+x^q)}{(1-2x+x^{q+1}{)}^2}$. Moreover, we construct an explicit ${\mathbb{F}}_q$-basis of ${\mathcal{Z}}_w^{({\mathbb{F}}_q)}$, and prove that any ${\mathbb{F}}_q$-linear relation among Carlitz multiple polylogarithm values ${\mathrm{Li}}_A(\mathfrak{s})$ is an ${\mathbb{F}}_q$-linear combination of quadruple-carry relations. This result can be regarded as an ${\mathbb{F}}_q$-analogue of the corresponding ${\mathbb{F}}_q(\theta )$-theorem proved by Chang--Chen--Mishiba and independently by Im--Kim--Le--Ngo Dac--Pham. Our discovery of the quadruple-carry relations is inspired by the recent work of Im--Kim--Ngo Dac. These relations may be viewed as ${\mathbb{F}}_q$-analogues of the double-shuffle relations among classical multiple zeta values $\zeta (\mathfrak{s})$.

40 pages, 2 figures