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

By analogy with the Riemann zeta function at positive integers, for each finite field Fpr with fixed characteristic p, we consider Carlitz zeta values ζr(n) at positive integers n. Our theorem asserts that among the zeta values in the set r=1{ζr(1),ζr(2),ζr(3),}, all the algebraic relations are those relations within each individual family {ζr(1),ζr(2),ζr(3),}. 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.

Algebraic independence, Frobenius difference equations, $t$-motives, zeta values
Mathematical Subject Classification 2000
Primary: 11J93
Secondary: 11M38, 11G09
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
Chieh-Yu Chang
National Center for Theoretical Sciences
Mathematics Division
National Tsing Hua University
Hsinchu City 30042
Matthew A. Papanikolas
Department of Mathematics
Texas A&M University
College Station, TX 77843-3368
United States
Jing Yu
Department of Mathematics
National Taiwan University
Taipei City 106