Algebraic independence and difference equations over elliptic function fields

Abstract

For a lattice \Lambda in the complex plane, let K_{\Lambda} be the field of \Lambda-elliptic functions. For two relatively prime integers p (respectively q) greater than 1, consider the endomorphisms \psi (resp. \phi) of K_{\Lambda} given by multiplication by p (resp. q) on the elliptic curve \mathbb{C}/\Lambda. We prove that if f (resp. g) are complex Laurent power series that satisfy linear difference equations over K_{\Lambda} with respect to \phi (resp. \psi) then there is a dichotomy. Either, for some sublattice \Lambda' of \Lambda, one of f or g belongs to the ring K_{\Lambda'}[z,z^{-1},\zeta(z,\Lambda')], where \zeta(z,\Lambda') is the Weierstrass zeta function, or f and g are algebraically independent over K_{\Lambda}. This is an elliptic analogue of a recent theorem of Adamczewski, Dreyfus, Hardouin and Wibmer (over the field of rational functions).