Talk:Elliptic curve
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I'm confused here. In the text, it states:
- By adding a point "at infinity", we obtain the projective version of this curve; every straight line intersects this curve in three points (if the line is tangent to the curve at a point, then that point is counted twice). It is then possible to introduce a group operation on the curve with the following property: if a straight line intersects the curve at the points P, Q and R, then P + Q + R = 0 in the group.
But, for example, the x-axis intersects the curve y2 = x3 - x + 1 at only 1 point (call it P); if we consider the other two points as being infinity, this seems to require that P + inf + inf = 0. However, the x-axis is not alone in this property; several lines parallel to the x-axis also intersect at only 1 point, if we select one of these and call the point of intersection Q, then Q + inf + inf = P + inf + inf = 0 implies P = Q. Bzzzt!!!! Is the group operation restricted to those lines which actually intersect at at least two points (where tangency counts as 2 points)? Chas zzz brown 02:00 Jan 22, 2003 (UTC)
My statement above was wrong: it only that way only for algebraically closed base fields. You're right: if the fields isn't algebraically closed, we only consider lines that are tangent to the curve or intersect it in two points. AxelBoldt 17:23 Jan 22, 2003 (UTC)
I'd like to see a bit more about the group aspect of elliptic curves over finite fields (with an eye towards ECC); especially more describing an algebraic (as opposed to geometric) approach to calculating P + Q (see [1] (http://www.certicom.com/research/online.html) for a nice explanation). Should that be done at this article, at the article for elliptic curve cryptography, or at a new article? Chas zzz brown 20:04 Feb 7, 2003 (UTC)
I think it can still fit here. Can't we give the general group law formula which works for all base fields? AxelBoldt 07:37 Feb 9, 2003 (UTC)
- Perhaps the best way to handle this would be to emphasise that the geometric picture is only valid for the real field, and then to explain that nevertheless we can make sense of things over any field by working out some algebraic formulation. The really important point here is that if two given points have rational coordinates (i.e. belong to some field), then so does the third point. This is why the group operation makes sense when defined over a given field. --Dmharvey 21:00, 28 May 2005 (UTC)
Added the restriction that K not have characteristic 3 - observe that in the diagrams in the article, there is a point P with the property P + P + P = 3P = 0 (in the y2 = x3 - x - 1 example, it's the intersection of the y-axis and the curve). Chas zzz brown 19:57 Feb 12, 2003 (UTC)
Page needs sections now. Charles Matthews 09:51, 21 Mar 2005 (UTC)
nother picture
Missing image
Elliptic_curve_simple.png
Image:Elliptic curve simple.png
—Sean κ. + 23:33, 27 May 2005 (UTC)
graphs of singularities
Would be nice to have graphs of singular weierstrass equations, illustrating both cusps and intersections.
On this point, it would also be nice to mention that the group law can still be defined for a singular curve, as long as we leave out the singular point (of which there can only be one -- I think). Dmharvey 17:00, 30 May 2005 (UTC)