Determination of algebraic relations among special zeta values in positive characteristic. (English) Zbl 1123.11025

Chang, Chieh-Yu; Yu, Jing

Adv. Math. 216, No. 1, 321-345 (2007).

Let \(k\) be a finite field of cardinality \(q\) and characteristic \(p\). Let \(k[\theta]\) the polynomial ring in one variable \(\theta\) over \(k\) and \(k((\theta^{-1}))\) the Laurent series field in \(\theta^{-1}\). Denote by \(k(\theta)\) the fraction field of the former in the latter. For a positive integer \(n\), define \[ \zeta(n) = \sum_f f^{-n} \in k((\theta^{-1})), \] the \(f\) ranging over all monic elements of \(k[\theta]\). Note that this sum converges for all positive integers \(n\), including \(n=1\). Clearly \(\zeta(pn)=\zeta(n)^p\). It was shown by L. Carlitz [Duke Math. J. 1, 137–168 (1935; Zbl 0012.04904)] that \(\zeta((q-1)n)\) equals \(\zeta(q-1)^n\) up to some factor in \(k(\theta)^\times\). In the present paper it is shown that these are the only algebraic relations over \(k(\theta)\) amongst the \(\zeta(n)\).
A few words on the proof. G. W. Anderson and D. S. Thakur [Ann. Math. (2) 132, No. 1, 159–191 (1990; Zbl 0713.11082)] have interpreted the \(\zeta(n)\) as periods of certain \(t\)-motives. This allows the present authors to apply M. A. Papanikolas’ results [Invent. Math. 171, No. 1, 123–174 (2008)] on the \(t\)-motivic analogue to Grothendieck’s period conjecture.

Reviewer: Lenny Taelman (Leiden)

MSC:

11J93	Transcendence theory of Drinfel’d and \(t\)-modules
11M38	Zeta and \(L\)-functions in characteristic \(p\)
11G09	Drinfel’d modules; higher-dimensional motives, etc.

Keywords:

t-motives; function fields; zeta values; transcendence; algebraic independence; periods

Citations:

Zbl 0012.04904; Zbl 0713.11082

References:

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