Determination of the algebraic relations among special Γ-values in positive characteristic

Abstract

We devise a new criterion for linear independence over function fields. Using this tool in the setting of dual t-motives, we find that all algebraic relations among special values of the geometric Γ-function over F q [T ] are explained by the standard functional equations.

Values of certain L-series in positive characteristic

“…Following the terminology of the authors, it is a rigid analytic trivialization of Carlitz's t-motive. The functions ω, Ω also appear, under several different notations, in the papers [1], [5], [4], [31]. In [1], the function ω is related to the theory of scattering matrices (see Section 3.1 of loc.…”

“…Of these functional equations, only the first one transfers directly to the function ω and to the values of the series L(χ t , α) for α ≡ 1 (mod q − 1) positive. However, Anderson, Brownawell and Papanikolas [4] gave evidence of the fact that the function ω is intimately related to the arithmetic of the values of Thakur's geometric gamma function, which satisfies yet three other classes of functional equations in analogy with the three shown above. At the time of writing the present paper, the exact connection between the values of the general functions f q k (t, X) and the values of Thakur's function is not completely elucidated.…”

“…The evaluation at η ∈ C ∞ of the exponential function exp Λz associated to the lattice 4 The Tate algebra Ts is the ring of series in monomials in t1, . .…”

Multizeta values for function fields: A survey

“…[5,16,37] proving roughly (analog of the Grothendieck period conjecture in this setting) that the algebraic relations between the periods of t-motives (e.g., gamma, zeta or multizeta values) come from the structural relations between the motives involved, and using it to get several interesting special values results.…”

Book Review: Number theory in function fields

“…Due to the recent work of G. Böckle [Boc2], [Go4] we now know that these functions give rise to L-series via "Hecke operators" just as with classical modular forms. Moreover, the standard formalism used to construct L-series mentioned above carries over very readily to Drinfeld modules, again leading to a theory of L-series [Boc1] and gamma functions [Th1], [ABP1] involving only finite characteristic analysis; an analog of Euler's formula on ζ(2n) is easily established in this context. While some general results are known about these functions and their zeroes [Wa1], [Go3], most of the general principles of the theory are still unknown.…”