Question

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.

Answer 1

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 $g$.