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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 |
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Vol. 5 (2011), No. 1, 111–129
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Abstract |
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By analogy with the Riemann zeta function at positive integers, for each finite field with fixed characteristic , we consider Carlitz zeta values at positive integers . Our theorem asserts that among the zeta values in the set , all the algebraic relations are those relations within each individual family . 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
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Mathematical Subject Classification 2000
Primary: 11J93
Secondary: 11M38, 11G09
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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
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Authors |
| Chieh-Yu Chang | |
| National Center for Theoretical
Sciences Mathematics Division National Tsing Hua University Hsinchu City 30042 Taiwan |
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| http://math.cts.nthu.edu.tw/~cychang/ | |
| Matthew A. Papanikolas | |
| Department of Mathematics Texas A&M University College Station, TX 77843-3368 United States |
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| http://www.math.tamu.edu/~map/ | |
| Jing Yu | |
| Department of Mathematics National Taiwan University Taipei City 106 Taiwan |
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| http://www.math.ntu.edu.tw/~yu/ | |
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