0

I have an understanding problem that I want to geometrically solve regarding elliptic curves.

It is clear how two points are added: Draw the line that passess through the points (if it is the same point, just draw the tangent) and reflect the third point that intersects this line with the elliptic curve.

Let me go to the special cases:

  1. If one wants to add one point $P$ to itself and it happens that the tangent line does not intersect to any other point that means that $P + P = 2P = \mathcal{O}$. In other words, the point is inverse to itself (is it this statement true?). This usually happens when the $y$-coordinate of this point is $y_P = 0$.
  2. Now comes the main problem: There are points such that the line that passess trhough them does not intersect (geometrically) a third point (look at the following image). What is going on (both geometrically and analytically) with these points? What would happen if I substitute the line equation $y = cm +d$ to the equation of the curve $y^2 = x^3 + ax +b$? They are clearly not inverse points, but they aswell sum to the infinity point $\mathcal{O}$.
    Curve
Bean Guy
  • 321
  • 1
    The horizontal line in you figure definitely passes through three points on the curve. The point of tangency counts double. If the line $y=cx+d$ is tangent to some point, then what happens when you plug it into $y^2=x^3+ax+b$ is that the resulting cubic has a double root corresponding to the point of tangency. – Jyrki Lahtonen Jun 16 '21 at 10:23
  • @JyrkiLahtonen Aha! I was thinking that adding these two points would end up being the infinity point. But it is just the reflexion of one or the other (the point that counts double). So, is adding a point with its reflexion the only way to obtain the infinity point after an addition? I.e., $P + Q = \mathcal{O}$ if and only if $P = -Q$? – Bean Guy Jun 16 '21 at 13:55
  • 1
    Yes, that is correct (given that the defining equation is $y^2=x^3+ax+b$ so that inverses are reflections). – Douglas Molin Jun 17 '21 at 14:07
  • 1
    You need to study torsion-2 group – kelalaka Jun 17 '21 at 19:20

0 Answers0