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.

a fortiori$\langle p \omega_1, q \omega_2 \rangle$-periodic, and thus algebraically related with $g$. The same argument shows that more generally if the intersection of the period lattices of $f$ and $g$ is again a lattice then $f,g$ are algebraically dependent. The converse is true too (assuming as always that neither $f$ nor $g$ is constant) but not quite this easy. $\endgroup$