Algebraic independence of values of Weierstrass elliptic and zeta functions

Let upper Omega Ω Ω be a lattice in upper C C C with invariants g 2 comma g 3 g₂,g₃ g 2 , g 3 and let script upper P left parenthesis z right parenthesis comma zeta left parenthesis z right parenthesis (z), ζ(z) ℘ ( z ) , ζ ( z ) be the associated Weierstrass elliptic and zeta functions, respectively. In this paper, we prove that if omega ω ω is any non-zero period of script upper P left parenthesis z right parenthesis (z) ℘ ( z ) and u 1 comma u 2 u₁,u₂ u 1 , u 2 complex numbers such that u 1 comma u 2 comma omega