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

Abstract

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

Alternating multizeta values in positive characteristic

“…Next, we introduce another representation of CMZVs via certain pre-t-motives. Unlike the (A)MZV case, it needs the following variant, which was first introduced by Chang, Papanikolas and Yu [CPY11].…”

“…For the t-motivic interpretation of CMZVs, unlike the (A) MZV case, we need to use a generalization of the pre-t-motive introduced in [CPY11], which we recall in Definition 3.1. In [CPY11], they investigated Papanikolas theory for higher-level pre-t-motives together with a rigid analytic trivialization. The t-motivic interpretation of CMZVs is stated by the following theorem:…”

Algebraic independence of arithmetic gamma values and Carlitz zeta values

“…The main result of the present paper (Theorem 4.2.2) considers arithmetic gamma values and zeta values simultaneously and furthermore determines all algebraic relations among them. The earlier results of the fourth author [Y91] on transcendence of zeta values, already surpassing the parallel classical results, have recently been further improved [CY07], [CPY09a] to complete algebraic independence results for zeta values. Here we show that these techniques generalize to give algebraic independence of both arithmetic gamma and zeta values together.…”

“…We briefly mention some additional avenues of research one can now pursue in light of Theorem 4.2.2. In [CPY09a], specific techniques inspired from [Ch09] are introduced to deal with varying q, i.e. to obtain algebraic independence results for zeta values at positive integers with varying constant fields.…”

Hyperderivatives of periods and quasi-periods for Anderson $t$-modules

“…Other investigations by Anderson, Brownawell, Chang, Denis, Thakur, Yu, and many others have produced transcendence results on function field Γ-values [2], [14], [22], [23], [74], [76], [77]; Drinfeld logarithms and quasi-logarithms [8]- [10], [20], [21], [28], [32], [34], [62], [69], [82], [83], [85], [87]; zeta values and multiple zeta values [16]- [19], [24]- [26], [42], [44], [53], [86], [87]; and of particular interest to the present paper, hyperderivatives of Drinfeld logarithms and quasi-logarithms [10]- [13], [30]- [32], [54].…”