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(elliptic curve $y^2 = x^3 + ax^2 + bx + c$)

Nagell-Lutz theorem:

If $p(x, y)$ is finite order on a given integer coefficient elliptic curve satisfy:

(1) x and y are integer

(2) y = 0 or y | D (D is discriminant)

(3) there is finite such points

here is what on the lecture note that I do not understand

"if point $p(x,y)$ satisfied $y = 0$, $P$ must have order 2, because $P + P = O$"

can someone explain above explanation?

Jyrki Lahtonen
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bsdshell
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1 Answers1

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You don't need the Lutz-Nagell Theorem to see this.

Let $P=(x,y)$ have order 2. Then $P=-P$ but we know $-P=(x,-y)$ and therefore we have $(x,y)=(x,-y)$ hence $y=0$.

Conversely, if $P$ is a point such that its $y$-coordinate is zero, then we again see that $P=-P$ so $P$ has order 2.

Matt B
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