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I'm working on a problem,

Let $\mathcal{E}:y^2=x^3-2x+1$ be an elliptic curve over the field $\mathbb{F}_5$, and let $P=(0,1)$ be among the points on $\mathcal{E}$. Find the equation of the line on which $P$, $2P$ and $4P$ all lie. Deduce that $7P=O$, the neutral element of the group law (point at infinity).

I get $P=(0,1),2P=(1,3),4P=(3,2)$ all lie on $y=2x+1$, in agreement with the given solution. In showing $7P=O$, the solution uses the statement

The three intersection points with a line add up to $O$

without further explanation. I can't find this mentioned in the course notes and I can't figure out why this should be. Could someone explain?

mjc
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    This is the definition of the point addition for elliptic curves $E$, so that $E(K)$ becomes an additive group. See also this answer. – Dietrich Burde Mar 16 '24 at 13:14
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    How are you defining the group law on an elliptic curve? You can write out explicit formulas, but the usual method (e.g., section 3.2 in Silverman) is to use Riemann-Roch to show that defining the group law via collinearity gives you an abelian group structure. – anomaly Mar 16 '24 at 13:20
  • @DietrichBurde and anomaly : OK, thanks, I get it. Now that you put it like this, I think the definition given in my course notes immediately implies DietrichBurde's statement, though doesn't quite make it explicit. – mjc Mar 16 '24 at 13:29

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