Algebraic independence and difference equations over elliptic function fields

For a lattice Λin 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 \mathbb{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).

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