Rational/meromorphic functions on curves

Question

Let, say, $E(\mathbb{C})$ be the set of affine points on an elliptic curve $y^2 = x^3 + ax + b$. Then $E(\mathbb{C})$, together with an additional point $\mathcal{O}$, can be viewed as a compact Riemann surface $X$. Let one call a meromorphic function $f : X \rightarrow \mathbb{C}$ rational if any point $P \in X$ has a (punctured) neighbourhood, on which $f$ can be expressed as a rational function of the coordinates $x, y$. Are all meromorphic functions on $X$ rational?

Answer 1

Any meromorphic function can be viewed in local charts as $$\frac{P}{Q}=\sum^{\infty}_{i=-k}a_{i}z^{i}$$where $P,Q$ are polynomial functions. However on an elliptic curve any meromorphic function can be expressed as rational functions of the Weistrauss $p$-function. Then you have a simpler description this way. I do not think $f=\frac{P}{Q}$ holds globally as this would force holomorphic $P,Q$ to be constants.

If I recall correectly, the reference on this is Neal Koblitz's book on elliptic curves and modular forms. This should be in Chapter $1$ or $2$.