Linear relations among double zeta values in positive characteristic. (English) Zbl 1421.11065

Summary: 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.


11M32 Multiple Dirichlet series and zeta functions and multizeta values
11G09 Drinfel’d modules; higher-dimensional motives, etc.
11J93 Transcendence theory of Drinfel’d and \(t\)-modules
11M38 Zeta and \(L\)-functions in characteristic \(p\)
11R58 Arithmetic theory of algebraic function fields
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