There has been significant progress in recent years in the study of solutions to linear difference equations involving shifts, q-difference or Mahler operators, over the field C(x) of rational functions. Last year, Adamczewski, Dreyfus, Hardouin and Wibmer showed that two formal power series solving difference equations with respect to two such independent commuting

operators are algebraically independent over C(x), unless one of them is itself a rational function.

I shall report on my work concerning the same question over fields of elliptic functions. The focus will be on the rather surprising aspects that are not present over the field of rational functions.