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In almost all texts concerning the group law on an elliptic curve it is first proven that any nonsingular cubic can be given by a Weierstrass equation and then the group law using the point $O$ at infinity is described (which in this case is an inflexion point).

Since I seem to be able to geometrically define addition of points $P$ and $Q$ for any nonsingular cubic and any point $O$ on the curve chosen as the identity element, I wonder why this is not mentioned more often.

I see that $P+Q+R=O$ for any collinear points $P$, $Q$ and $R$ only if $O$ is an inflexion point. Of course it all gets much easier with this property at hand, but is it 'needed'?

Maybe I did miss some counterexample, but isn't it possible to have a nice geometric addition law for all sorts of elliptic curves, not only for the ones in Weierstrass form with $O$ the (inflection) point at infinity?

Thank you for your insights.

Edit: Probably the title was a bit misleading. The more basic thing to ask seems to be Why do we require $P+Q+R=O$ for collinear points?

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2 Answers 2

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Actually, sometimes we don't. If you write an elliptic curve in Edwards normal form, the group law is not given by collinearity. Instead, it's given by is a polynomial function deforming the group law on a circle. Edwards normal form is used in cryptography, I think because certain operations are easier to perform with it.

(Curves in Edwards normal form can also be lifted in an appropriate sense to projective space in such a way that the group law is given in terms of coplanar rather than collinear points, where one of the four points is always the identity and can be arbitrary. I regret that I do not have any reference to give for this construction other than an old high school paper of mine; surely it is classical.)

In algebraic geometry textbooks, the form of the group law involving collinearity is a special case of a more general construction, namely that of the divisor class group or Picard group of an algebraic curve. This construction is an analogue of the construction of the class group of the ring of integers of a number field, and the role of collinear points is that they furnish principal divisors (analogous to principal ideals) which we quotient by to get the divisor class group. So one very good reason to present the group law this way is to aim towards this generalization. (The Picard group has the conceptual advantage of being defined without reference to a particular point on the curve.)

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Thank you for your great answer. The aim towards the Picard group really is a good motivation. Also thank you for your paper on the Edwards form; I learned something today. –  Gregor Bruns Aug 29 '12 at 10:46

If you have any genus one curve $E$, any point $O$ on $E$, then there is a unique algebraic group structure on $E$ making $O$ the identity element. This applies in particular when $E$ is a smooth plane cubic.

However, it is not hard to show that given $E$ and $O$, we can always embed $E$ as a plane cubic in such a way that $O$ is furthermore an inflection point of $E$.

Since the discussion of the group law is a little simpler when $O$ is an inflection point (and indeed becomes simplest when $O$ is moved to be a point at infinity, with its tangent line being the line at infinity), most books put themselves in that case when discussing the group law.

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