#### Vol. 5, No. 1, 2011

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Frobenius difference equations and algebraic independence of zeta values in positive equal characteristic

### Chieh-Yu Chang, Matthew A. Papanikolas and Jing Yu

Vol. 5 (2011), No. 1, 111–129
##### Abstract

By analogy with the Riemann zeta function at positive integers, for each finite field ${\mathbb{F}}_{{p}^{r}}$ with fixed characteristic $p$, we consider Carlitz zeta values ${\zeta }_{r}\left(n\right)$ at positive integers $n$. Our theorem asserts that among the zeta values in the set ${\bigcup }_{r=1}^{\infty }\phantom{\rule{0.3em}{0ex}}\left\{{\zeta }_{r}\left(1\right),{\zeta }_{r}\left(2\right),{\zeta }_{r}\left(3\right),\dots \phantom{\rule{0.3em}{0ex}}\right\}$, all the algebraic relations are those relations within each individual family $\left\{{\zeta }_{r}\left(1\right),{\zeta }_{r}\left(2\right),{\zeta }_{r}\left(3\right),\dots \phantom{\rule{0.3em}{0ex}}\right\}$. These are the algebraic relations coming from the Euler–Carlitz and Frobenius relations. To prove this, a motivic method for extracting algebraic independence results from systems of Frobenius difference equations is developed.

##### Keywords
Algebraic independence, Frobenius difference equations, $t$-motives, zeta values
##### Mathematical Subject Classification 2000
Primary: 11J93
Secondary: 11M38, 11G09
##### Milestones
Received: 27 January 2010
Revised: 18 October 2010
Accepted: 21 November 2010
Published: 22 August 2011

Proposed: Brian Conrad
Seconded: Ravi Vakil, Michael F. Singer
##### Authors
 Chieh-Yu Chang National Center for Theoretical Sciences Mathematics Division National Tsing Hua University Hsinchu City 30042 Taiwan http://math.cts.nthu.edu.tw/~cychang/ Matthew A. Papanikolas Department of Mathematics Texas A&M University College Station, TX 77843-3368 United States http://www.math.tamu.edu/~map/ Jing Yu Department of Mathematics National Taiwan University Taipei City 106 Taiwan http://www.math.ntu.edu.tw/~yu/