The Carlitz module and a differential Ax-Lindemann-Weierstrass theorem for the Euler gamma function

We prove an Ax-Lindemann-Weierstrass differential transcendence result for Euler's gamma function, namely that the functions Γ(ν-ζ1(ν)),…, Γ(ν-ζn(ν)) are differentially independent over the field of rational functions in the variable ν, with coefficients in the field k of 1-periodic meromorphic functions over the complex numbers, as soon as ζ1,…, ζn determine a set of algebraic functions over k, stable by conjugation and pairwise distinct modulo the integers. 

To prove this result we use both the Galois theory of difference equations and the theory of a characteristic zero analog of the Carlitz module introduced by the second author in 2013. As an intermediate result we give an explicit description of the Picard-Vessiot rings and of the Galois groups associated to the operators in the image of the Carlitz module, using techniques inspired by the Carlitz-Hayes theory. This is a joint work with Federico Pellarin.