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.
Elliptic curves, inflection points and divisors
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elliptic-curves
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0That'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 $P$. – 2016-11-15
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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.
Edit: D'oh. The OP asked for examples in other characteristics. I'm apparently at a my dullest. Doesn't the same example work in any characteristic? (Except at char 3, because then the curve has a singular point). See a figure of the real points below. y^2+y=x^3">
The origin looks like an inflection point to me :-)
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1@Jasiu: The line is always the tangent. – 2011-09-12