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?

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    $\begingroup$ Near $0$, consider the parametrisation $$z \mapsto [\wp(z)/\wp'(z), 1, 1/\wp'(z)]\,.$$ $\endgroup$ – Daniel Fischer Aug 31 '18 at 20:13
  • $\begingroup$ Use $z\longmapsto [\wp(z):\wp'(z):1]=[z^3\wp(z):z^3\wp'(z):z^3]$. $\endgroup$ – dan_fulea Sep 10 '18 at 13:49

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