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arXiv · 0804.0038

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

Chieh-Yu Chang·Matthew A. Papanikolas·Jing Yu
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Abstract

In analogy with the Riemann zeta function at positive integers, for each finite field F_p^r with fixed characteristic p we consider Carlitz zeta values zeta_r(n) at positive integers n. Our theorem asserts that among the zeta values in {zeta_r(1), zeta_r(2), zeta_r(3), ... | r = 1, 2, 3, ...}, all the algebraic relations are those algebraic relations within each individual family {zeta_r(1), zeta_r(2), zeta_r(3), ...}. These are the algebraic relations coming from the Euler-Carlitz relations and the Frobenius relations. To prove this, a motivic method for extracting algebraic independence results from systems of Frobenius difference equations is developed.

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Chieh-Yu Chang, Matthew A. Papanikolas, Jing Yu. 2008-03-31. Frobenius difference equations and algebraic independence of zeta values in positive equal characteristic. https://doi.org/10.2140/ant.2011.5.111

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