Correspondence between a complex torus and an elliptic curve

Question

In The Arithmetic of elliptic Curves, on page 156, Silverman considers the map $$\phi\colon\mathbb C/\Lambda\longrightarrow E\subset \mathbb P^2(\mathbb C)$$ $$z\longmapsto [\wp(z),\wp'(z),1]$$ where $\Lambda$is a lattice, $E$ the corresponding elliptic curve $ZY^2=4X^3-g_2Z^2X-Z^3g_3$, and $g_2=g_2(\Lambda),g_3=g_3(\Lambda),\wp=\wp(\Lambda)$ as usual. The point at infinity of $E$ is $[0,1,0]$, and its preimage under the map $\phi$ should be $0$. However we cannot substitute $z=0$ into $\wp(z)$ and $\wp'(z)$ because these functions have poles at $0$. I do not understand how we get from $0\mapsto[\infty,\infty,1]$ to $0\mapsto[0,1,0]$. Must we treat this case separately? How to connect the multiplicity of poles to the order of the point at infinity?

Answer 1

A priori, the map $\phi$ is only defined on the punctured torus $\mathbb{C}/\Lambda-\{[0]\}$. This holds of course since both the Weierstrass $\wp$-function and its derivative have poles precisely on the lattice $\Lambda$. However, the map $\phi$ can be uniquely extended to a continuous map on the whole torus $\mathbb{C}/\Lambda$ (even to a holomorphic mapping by employing Riemann's Removable Singularities Theorem). This is not spelled out explicitely in Silverman's book. Let me therefore do it instead.

Assume that you have already convinced yourself that $\phi$ is continuous on $\mathbb{C}/\Lambda-\{[0]\}$. Let $z_n$ be a sequence in $\mathbb{C}$ converging to $0$. Since $\wp'$ is meromorphic with poles only at the lattice points, the function $\wp'$ is nowhere-vanishing in a small neighbourhood of $0$. You can even explicitely give a neighborhood by recalling that the set of zeroes of $\wp'$ is $\{\lambda_1/2,\lambda_2/2,(\lambda_1+\lambda_2)/2\}+\Lambda$, where $\Lambda=\mathbb{Z}\lambda_1+\mathbb{Z}\lambda_2$. Thus, as Daniel Fischer suggests in the comments, near $[0]\in\mathbb{C}/\Lambda$ the map $\phi$ can be written as $\phi([z])=[\wp(z)/\wp'(z),1,1/\wp(z)].$ Note that $\lim_{n \to \infty} \wp(z_n)/\wp'(z_n)=0$ since $0$ is a double pole of $\wp$, while $0$ is triple pole of $\wp'(z)=-2\sum_{\lambda\in \Lambda}\frac{1}{(z-\lambda)^3}$. Since the projection $\pi\colon \mathbb{C}^3-\{0\}\rightarrow \mathbb{CP}^1$ is continuous, it follows that $$\lim_{n \to \infty}\phi\big([z_n]\big)= \lim_{n \to \infty} \pi\big(\wp(z_n)/\wp'(z_n),1,1/\wp(z_n)\big)=\pi\big(\lim_{n \to \infty} \wp(z_n)/\wp'(z_n),1, \lim_{n \to \infty}1/ \wp(z_n)\big)=[0,1,0].$$ This shows that $\phi$ extends uniquely to a continuous function on $\mathbb{C}/\Lambda$ by defining $\phi([0]):=[0,1,0]$.

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You ask:

How to connect the multiplicity of poles to the order of the point at infinity?

I am not sure, why you would expect such a connection. There are two different notions of order at play here. On the one hand, you have the order of a pole. On the other hand, there is the order of an element of a group.

Concretely, the Weierstrass $\wp$-function has a double pole at each lattice point, while its derivative has a triple pole at each lattice point. The "point at infinity" $[0,1,0]\in E(\mathbb{C})$ has order $1$, however. Namely, the group structure on $E(\mathbb{C})$ is precisely the one induced from the torus $\mathbb{C}/\Lambda$ via the homeomorphism $\phi$. Thus, since $\phi([0])=[0,1,0]$, the point $[0,1,0]$ is the neutral element of the abelian group $E(\mathbb{C})$.