ALGEBRAIC INDEPENDENCE AND DIFFERENCE EQUATIONS OVER ELLIPTIC FUNCTION FIELDS

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