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I'm reading on divisors and elliptic curve pairing. For a field $F$ and a rational function $f(x,y) \in F(x,y)$ it's often written $f(P)$ for points $P$ on the curve. But what is $f(P)$ when $P = \infty$, i.e. the unit element for elliptic curve addition?

kuco 23
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  • If you're working with an elliptic curve in Weierstrass form, then $\infty$ is really the point $[0:1:0]$ in projective space. Letting $X,Y,Z$ be the projective coordinates of $\mathbb{P}^2$, you're probably working with an equation in the affine open set where $Z \neq 0$, with affine coordinates $x = X/Z$ and $y = Y/Z$. The point $[0:1:0]$ does not lie in this open set, so you should move to a different affine open containing $[0:1:0]$, say where $Y \neq 0$ with affine coordinates $u = X/Y$ and $v = Z/Y$, and compute $f(P)$ there. – Viktor Vaughn Sep 05 '22 at 19:53

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You have to first apply homogenization to $f$ by $Z^n f(\frac{X}{Z}, \frac{Y}{Z})$ to get $f_{\textrm{homog}}$ which we will also denote by $f$. Then $f(\infty) = f(0, 1, 0)$.

For example lets say $f = x^2 - y + 2$, then homogenizing we get the projective curve $f = X^2 - YZ + 2Z$, and evaluation $f(\infty) = 0$.

Ignatio Mobius
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