Meromorphic Functions that satisfy a first order algebraic differential equation

Question

I have been writing a project on elliptic functions and in this project I prove the following theorem: Suppose a meromorphic function $f$ satisfies an algebraic addition theorem, that is there exists a polynomial $F$ such that $$F(f(z_1),f(z_2),f(z_1+z_2))=0$$ for all $z_1,z_2\in\mathbb{C}$. Then $f$ is either elliptic, singly periodic or rational. I also heard that if $f$ satisfies $$P(f(z),f'(z))=0 $$ for some polynomial $P$ then $f$ also belongs to the three classes given. My question is, is the second theorem true, and if so does anyone know a resource containing a proof I can reference? Thanks.

Answer 1

N. I. Akhiezer proves a more general statement in his book:

Any two elliptic functions with the same periods are connected by an algebraic relation.

His proof relies on representing an arbitrary elliptic function as $R_1(\wp)+R_2(\wp)\wp^\prime$, where $R_1$ and $R_2$ are rational functions, and noting that one can always algebraically eliminate $\wp$ and $\wp^\prime$ through their relation ${\wp^\prime}^2=4\wp^3-g_2\wp-g_3$. From here, the case of relating an elliptic function and its derivative algebraically becomes a special case.

Answer 2

This is a form of uniformization theorem. An algebraic curve (a Riemann surface) is uniformized by the plane or the sphere if it has euler characteristic $\geq 0,$ which means that the Riemann surface is a sphere (corresponding to a rational function), a cylinder (periodic function) or a torus (elliptic function).