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

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

2008-03-31

arxiv.org/abs/0804.0038v2

Algebra & Number Theory 5 (2011), 111-129

Mathematics

Number Theory

14 pages

Scientific paper

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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