Holomorphic differential of elliptic curve over $\mathbb C$

Question

There are well known holomorphic differential of elliptic curve over $\mathbb C$.

That is, $ω＝dx/y＝dx/\sqrt{x(x-1)(x-λ)}$ on $E(\mathbb C)$

But this seems to have pole at $x＝0,1,λ,∞$.

Why these points are not pole?

Answer 1

Because $x$ (or $x-1$ or $x-\lambda$) isn’t a local coordinate at these points. The correct coordinate is $y$. Near e.g. $x=0$, $x=y^2\times f(x)$ with $f(x)$ defined and nonzero, so that $dx/y=2f(x)dy+yf’(x)dx$ is holomorphic. The same works for $x=1,x=\lambda$.

At infinity, it’s a little bit different. Write $z=1/x, v=x/y$, which are regular at infinity (and $v$ is the coordinate) and $x=v^{-2}f(z)$, $y=v^{-3}g(z)$ with $f,g$ defined and nonzero near zero. Then for $x$ near infinity $dx/y=f’(z)dz/(v(g(z))-2f(z)dv/g(z)$. So it’s enough to show $dz/v$ holomorphic at $z=0$. But $z=v^2h(z)$ with $h$ defined and nonzero near $0$ so it concludes.