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.



Algebraic relationships between elliptic functions
There exists an algebraic relation between two elliptic functions if and only if their
lattices are commensurable. This means that every period of one function is a rational
multiple of periods of another.


Proof. Let $f$ and $g$ be two elliptic functions, let $F(f,g)=0$ be an algebraic relation.
Choose a point $a$ such that for each solution $w$ of the equation $F(a,w)=0$ 
we have $F\_w(a,w)\neq 0$. Then there are finitely many holomorphic germs
$w=\phi\_j(z)$, that describe all solutions of $F(z,w)=0$ near $z=a$.


Let $T$ be a period of $f$. Let $z\_0$ be a point such that $f(a\_0)=a$.
Consider the sequence $z\_0, z\_0+T, z\_0+2T...$ Near these points we must have the relations
$g=\phi\_j(f)$. Because there are only finitely many $\phi\_j$ we will have
that the germs of $g$ at $z\_0+mT$ and at $z\_0+nT$ coinside. That is $(m-n)T$
will be a period of $g$.


