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

Chang, Chieh-Yu

Camb. J. Math. 4, No. 3, 289-331 (2016).

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.

MSC:

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

Keywords:

double zeta values; \(t\)-motives; Carlitz tensor powers; periods; logarithms; Siegel’s lemma