For example, elliptic curve $E:y^2＝x^3-x$ can be transformed
to move the identity element to the origin, $(z,w)=(0,0)$, we do the change of coordinates $z=-x/y$, $w=-1/y$. The equation of this curve then becomes
$w=z^3+zw^2\quad$
We often make this transformation to figure out formal group associated to elliptic curve.
But what do we do when $y＝0$?
I'm confused.
Thank you in advance.
Best Answer
This is easier if you write in projective coordinates. Let $E$ be given as $$Y^2Z = X^3 -XZ^2$$ then the rational map $(x,y) \mapsto (-x/y, -1/y)$ on the affine chart where $Z = 1$ can be extended to a morphism on the projective closure ($E$ above) by $$(X : Y:Z) \mapsto (-X: -Z : Y)$$
In particular $(0:0:1)$ maps to $(0:1:0)$ - the point at infinity on your curve. This may also be seen in an analagous way to complex analysis by looking at the order of the poles of the functions $-x/y$ and $-1/y$ at $(0,0)$.