Linear relations among double zeta values in positive characteristic

For each integer $n \geq 2$, we study linear relations among weight $n$ double zeta values and the $n$th power of the Carlitz period over the rational function field $\mathbb{F}_q (\theta)$. We show that all the $\mathbb{F}_q (\theta)$-linear relations are induced from the $\mathbb{F}_q [t])$-linear relations among certain explicitly constructed special points in the $n$th tensor power of the Carlitz module. We then establish a principle of Siegel’s lemma for computing and determining the $\mathbb{F}_q [t])$-linear relations mentioned above, and thus obtain an effective criterion for computing the dimension of weight $n$ double zeta values space.

Keywords

double zeta values, $t$-motives, Carlitz tensor powers, periods, logarithms, Siegel’s lemma

2010 Mathematics Subject Classification

Primary 11J93, 11R58. Secondary 11G09, 11M32, 11M38.

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Published 7 September 2016