Gamma Functions for Function Fields and Drinfeld Modules

Linear independence of Gamma values in positive characteristic

“…When q = 2, this result had been observed by Thakur in [20]. Our goal is to extend Sinha's result, treating several values at once and evaluating Γ at more general arguments.…”

“…The Anderson-Deligne-Thakur relations on Gamma values (see [18] or [20]), derived from the formalism of brackets, may be expressed as follows. Fix f ∈ A + and define N f := {a ∈ A : a ≡ 0 mod f }, U f := {u ∈ A : (u, f ) = 1}, M f := {a ∈ N f : a ≡ m mod f, m monic, deg m < deg f }.…”

“…The Gamma function satisfies several functional equations, directly analogous to the functional equations of the classical Gamma function, which imply the following algebraic relations on special values [20]: For all r ∈ k A, a ∈ A, g in A + , deg g = d, Γ(r + a) ∼ Γ(r) , (1.1) the classical Gamma function itself is still unknown except for very special cases due to Th. Schneider and G.V.…”

“…After isolated results by Thakur in [20], S. Sinha established the first transcendence results for general classes of values of the Gamma function. Theorem 1.1.1 (Sinha [16]; [17], §6.2).…”

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.…”