What is the relevance of order 2 points for poles & zeros of Elliptic curves?

Question

I am reading up on poles & zeros & divisors of Elliptic Curves & I found some statements about points of order 2 which aren't well explained.

1. https://crypto.stanford.edu/pbc/notes/elliptic/divisor.html

Let $P = (a, b)$ be a point, not of order 2.

Why should it not be of order 2 here?

2. Mathematical Cryptography by Silverman

Suppose that the cubic polynomial used to define $E$ factors as $X^3 + AX + B = (X − \alpha_1)(X − \alpha_2)(X − \alpha_3)$

Then the points $P_1 = (\alpha_1, 0)$, $P_2 = (\alpha_2, 0)$, and $P_3 = (\alpha_3, 0)$ are points of order 2.

How do we get that? That these points are of order 2?

Answer 1

There's a subtlety in the passage you are reading: the author is attempting to define the order of vanishing of a function $g(X, Y) = (X - a)^k$ at the point $P = (a, b)$ on $E$. He defines the order of vanishing of $g$ to be $k$ at $P$. However, $P$ has order $2$ if and only if the tangent at $P$ is vertical, and for various algebraic-geometric reasons and the fact that $\frac{d}{dY}g(X, Y) = 0$, you need to define the order of a function at a point differently in this situation. Likely the author did not want to introduce this subtlety in this introduction, so they just ignore the problematic case.