If $f$ and $g$ are two elliptic functions with the same periods, then there exists an algebraic relationship of the form $P(f,g)=0$, where $P$ a polynomial of two variables with constant coefficients.
This is a well known property. As a special case we have $P(f, f^{\prime})=0$, which is satisfied by the Weierstrass elliptic function $\mathfrak{D}$ which is a solution of the differential equation in $\Lambda$
$$(Y^{\prime})^{2}=4(Y)^{3}-g_{2}Y -g_{3}$$
where $\Lambda$ is the lattice generated by the two periods of $\mathfrak{D}$, and $g_{1},g_{2}$ are the invariants of the function $\mathfrak{D}$.
My question is: does there exist an algebraic relationship between two elliptic functions if they don't have the same periods, and if it exists, under which conditions (between periods)?
I have proved the existence of an algebraic relationship between two elliptic functions $f$ and $g$ if the periods of $f$ are $\omega_{1}$ and $\omega_{2}$, and the periods of $g$ are $p\omega_{1}$ and $q\omega_{2}$, where $p,q\in\mathbb{Q}$. This is a generalization of the case where the periods are equal.
The problem is still open for further generalizations. I welcome any suggestions.