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I have a question about the proof of the following theorem:

A complex torus is conformally equivalent (so isomorphic as Riemann surface) to a complex elliptic curve

I used the book "N.Koblitz, Introduction to Elliptic Curves and Modular Forms". The author shows that the function

$$\theta:\mathbb C/\Lambda\rightarrow\gamma=\{(z_0:z_1:z_2)\in\mathbb P^2(\mathbb C): 4z_1^3-g_2z_0^2z_1-g_3z_0^3-z_0z_2^2=0\}$$ defined by

$\theta(z)=(1:\wp(z):\wp'(z))$ if $z\neq 0$ and $\theta(0)=(1:0:0)$

is bijective. Then he says that $\theta$ is biholomorphic but he gives no explanation of this statement. Reading this question, I tried to look Silverman's proof of the above theorem in his book "The Arithmetic of Elliptic Curves", but again the proof of the biholomorphism wasn't so detailed.

Can someone help me giving another reference or a more detailed proof?

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1 Answer 1

up vote 3 down vote accepted

A bijective holomorphic map $f:X\to Y$ between Riemann surfaces is an isomorphism, i.e. $f^{-1}:Y\to X$ is automatically holomorphic, and this answers your question.
The proof is very easy: locally near a point $x_0\in X$ the morphism $f$ has the form $z\mapsto z^n$ (in suitable coordinates) and the bijectivity hypothesis forces $n=1$, so that $f(z)=z$ is clearly an isomorphism.

It might interest some users to know that the exact same result is also true in arbitrary dimensions:
Any bijective holomorphic map $f:X\to Y$ between holomorphic manifolds is an isomorphism.
The proof is a bit more complicated and may be found on page 19 of Griffiths-Harris's Principles of Algebraic Geometry .

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