(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?
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.