# Elliptic curves, inflection points and divisors

I'm studying basics of elliptic curves. I'm reading An Elementary Introduction to Elliptic Curves by Leonard Charlap and David Robbins. It is stated there that the divisor of a line (i.e. a polynomial of the form $ax + by + c$) can have only few forms, among them is $3\langle P \rangle - 3\langle \mathcal{O}\rangle$. I tried to find an example of a curve and a line on it that has such divisor, but to no avail. Can anyone provide an example? If it helps, they suggest that $P$ is an inflection point.

• That's actually an "if and only if". More precisely, assuming that $\mathcal O$ is an inflection point (which it is if you're using a standard Weierstrass equation), then the divisor of $ax+by+c$ has the form $3\langle P\rangle-3\langle\mathcal O\rangle$ if and only if $P$ is an inflection point and the line $ax+by+c$ is the tangent line at $P$. – Joe Silverman Nov 15 '16 at 18:54

Let the base field be $F_2$ (hopefully you're fine with a finite base field). Let the curve be $y^2+y=x^3$ and the line $y=0$. The function $y$ has a pole of order 3 at the point of infinity ${\mathcal O}$ and a triple zero at origin ${\mathcal P}=(0,0)$, so the divisor of $y$ is $3{\mathcal P}-3{\mathcal O}$ as prescribed.
• @Jasiu: This kind of examples abound in all characteristics. The condition is equivalent to finding a point ${\mathcal P}$ of order 3. I would just have to think a bit harder to find one, as my experience is mostly with char 2 (for reasons similar to yours:-) – Jyrki Lahtonen Sep 12 '11 at 8:25