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

We prove a differential transcendence result of type “Ax-Lindemann-Weierstrass” for Euler’s gamma function. Given meromorphic functions ${\zeta }_1,\dots ,{\zeta }_n$ of a complex variable $\nu$ that are pairwise distinct modulo $\mathbb{Z}$ and algebraic over the field $k$ of meromorphic $1$-periodic functions, the functions $\mathrm{\Gamma }(\nu -{\zeta }_1(\nu )),\dots ,\mathrm{\Gamma }(\nu -{\zeta }_n(\nu ))$ are differentially independent over the field $k(\nu )$. We determine the structure of certain difference field extensions related to the torsion of an avatar of the Carlitz module over meromorphic functions. These extensions are abelian and purely transcendental, the latter property being crucial in our main result, and obtained applying a criterion of differential algebraicity of Hardouin and Singer.