The Weierstrass map between a torus and an elliptic curve is biholomorphic

Question

Let $\Lambda$ be a lattice in $\mathbb C$. We can build by this lattice the Weierstrass $\wp$-function in the following manner: $$ \wp(z) = \frac{1}{z^2}+\sum\limits_{l\notin\Lambda\setminus\{0\}}\left( \frac{1}{(z-l)^2} - \frac{1}{l^2}\right) $$ It can be shown that this function satisfies the following differential equation:

$$ \wp'(z)^2 = 4\wp(z)^3 - 60G_4\wp(z) - 140 G_6 $$ This function gives us a map frop $\mathbb C/\Lambda$ to the elliptic curve $y^2 = 4x^3 - 60G_4 x - 140G_6$ in such a way: $$ z \mapsto [\wp(z):\wp'(z):1] $$

My question is: why this map is biholomorphic?

Answer 1

For a fixed $\tau \in \mathbb{C} -\mathbb{R}$

• Define the curve $$E(\mathbb{C}) = \{ (x,y) \in \mathbb{C}^2, y^2 = 4 x^3 - g_2(\tau)x - g_3(\tau)\}$$ with a complex topology inherited from $\mathbb{C}^2$.

• Define the lattice $\Lambda = \mathbb{Z}+\tau \mathbb{Z}$ and look at $(\mathbb C- \Lambda) / \Lambda$ which is a complex torus minus one point. With the complex topology inherited from $\mathbb{C}$ it is naturally a non-compact Riemann surface.

• Define the map $$\varphi : (\mathbb C- \Lambda) / \Lambda \to E(\mathbb{C}), \qquad \varphi(z) = (\wp_\tau(z),\wp_\tau'(z))$$ $\varphi$ is well-defined and bijective because $\wp(z)$ takes all the complex values twice (if not then $\frac{1}{\wp(z)-a}$ would be a bounded entire function) and $(\wp_\tau(-z),\wp_\tau'(-z))=(\wp_\tau(z),-\wp_\tau'(z))$.

  $\varphi$ is also holomorphic as a map $(\mathbb C- \Lambda) / \Lambda \to \mathbb{C}^2$.

  Thus $\varphi$ is a bijective holomorphic map $(\mathbb C- \Lambda) / \Lambda \to E(\mathbb{C})$ which means it is biholomorphic and $E(\mathbb{C})$ is now a non-compact Riemann surface.

• Finally $\mathbb C / \Lambda $ is a compact Riemann surface (and an abelian group) by adding the point $z=0$ to $(\mathbb C- \Lambda) / \Lambda$.

  Thus $E(\mathbb{C})$ becomes a compact Riemann surface when adding one point : the image of $z=0$ by $\varphi$, that we call $O$, whose $\epsilon$- neighborhood is $\{O\} \cup \{(x,y) \in E(\mathbb{C}), |x| > \frac{1}\epsilon\}$ (the definition of 'holomorphic at $O$' follows from this thanks to the Riemann's theorem on removable singularities)

• Which is the same as defining $E(\mathbb{C})$ as the projective curve $$E(\mathbb{C})_{proj} = \{ (x:y:w) \in P_2(\mathbb{C}), y^2w = 4 x^3 -g_2(\tau)xw^2 - g_3(\tau)w^3\}$$ and $$\varphi(z) = (\wp_\tau(z):\wp_\tau'(z):1), \qquad\varphi(0) = (0:1:0)$$ is a biholomorphic map $\mathbb{C}/\Lambda \to E(\mathbb{C})_{proj}$