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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?

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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 $2$.

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  • $\begingroup$ Thank you. I am aware of the fact the Weierstrass function and its derivative generate the field of meromorphic functions on $E(\mathbb{C})$. What I do not see is why, for instance, the Weierstrass function (or any other meromorphic function) can locally be written as a rational function of the coordinates $x, y$ (and I don't find the latter in Koblitz's book). $\endgroup$
    – Albertas
    Feb 28, 2014 at 15:18
  • $\begingroup$ @Albertas: I think a meromorphic function over any Riemann Surface given by $f(z,w)=0$ can be turned into this form. But I do not recall the details (I think you need the correct local coordinates like $z$ or $z^{-1}$, then construct a meromorphic function locally as $f(z)$, etc). I think what you do is to map the elliptic curve into a torus using the Weistrauss function(see the wikiarticle en.wikipedia.org/wiki/Elliptic_curve). Then you have local coordinates in $z$ you can work with. $\endgroup$ Feb 28, 2014 at 17:20

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