Geometric Gamma Values and Zeta Values in Positive Characteristic

Abstract

Abstract. In analogy with values of the classical Euler Γ-function at rational numbers and the Riemann ζ-function at positive integers, we consider Thakur's geometric Γ-function evaluated at rational arguments and Carlitz ζ-values at positive integers. We prove that, when considered together, all of the algebraic relations among these special values arise from the standard functional equations of the Γ-function and from the Euler-Carlitz relations and Frobenius p-th power relations of the ζ-function.

Algebraic independence of arithmetic gamma values and Carlitz zeta values

“…This more complicated question will not be treated here however. We note that in another paper [CPY09b], the first, second, and fourth author have also established algebraic independence of geometric gamma and zeta values taken together. We leave the question of algebraic independence of arithmetic gamma and geometric gamma values taken together to a later work.…”

“…Adding the dimension of the torus with that of the vector group in question proves the desired algebraic independence result. Thus, the story of arithmetic gamma values unfolds through t-motives having arithmetic (cyclotomic) CM in this paper, just as t-motives having geometric (cyclotomic) CM provide the proper setting for special geometric gamma values [ABP04], [CPY09b].…”

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].…”

Algebraic relations among Goss’s zeta values on elliptic curves

“…This is a very powerful tool and has led to major transcendence results in the last decade. We refer the reader to [4, 17, 18, 19, 20, 21, 22, 23] for more details about transcendence applications.…”