Let $f$ and $g$ be tow elliptic function with the same periods, then there exists an algebraic relationship of the form $P(f,g)=0$, where $P$ a polynomial of tow variables and constants coefficients.

it's a well known property, as special case we have $P(f, f^{\prime})=0$, which is verified by the Weirstrass elliptic function $\mathfrak{D}$:

the Weierstrass elliptic function $\mathfrak{D}$ is a solution of the differential equation in $\Lambda$

$$(Y^{\prime})^{2}=4(Y)^{3}-g_{2}Y -g_{3}$$

where $\Lambda$ 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: there exists an algebraic relationship between tow elliptic functions, if they don't have the same periods, and if there exists, under which conditions (between periods).

i have proved the existence of an algebraic relationship between tow elliptic functions $f$ and $g$ in this case: (which generalize the above property):

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 $p,q\in\mathbb{Q}$.

the problem still open for furthermore generalisation, you are welcome if you have 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. $\endgroup$ – Noam D. Elkies May 9 '12 at 5:14