How to construct a second point $Q$ and the third point on a cubic curve?

Question

It is written on Milne's elliptic curve book that

Let $C_F$ be a nonsingular cubic Projective curve over $\Bbb Q$. From any point $P \in C_F(\Bbb Q)$ we can construct a second point in $C_F(\Bbb Q)$ as the point of intersection of the tangent line at $P$ with $C_F(\Bbb Q)$ and from any pair of points $P,Q \in C_F(\Bbb Q)$ we can construct a third point of intersection of the chord through $P, Q$ with $C_F(\Bbb Q)$

Here I am not getting the construction of $Q$ and the third point because we have $C_F(\Bbb Q)$ nonsingular, so is $P$ i.e $m_P(F)=1$ and the tangent line at $P$ say $L$. Now $I(P, C_F \cap L)\geq m_P(F)m_P(L)= 1$ again Bezout's theorem says that $3=\sum_{R\in C_F(\Bbb Q) \cap C_L(\Bbb Q)}I(P, C_F \cap L)$. If we have $I(P, C_F \cap L)=3$ then what will happen? Am I missing anything? Please help by putting some light on the construction of $Q$ and the third point.

Answer 1

Imho, both cases should be thought of as finding the "third" point. By Bezout, a line meets the curve three times (counted with multiplicity). So whenever you pick two points $P$ and $Q,$ there is a unique third point on the line connecting them. If you picked the same point $P = Q$ twice, then the line is the tangent line.

In the case you're worried about, the line meets the curve at the "three" points $P, P,$ and $P.$ So when you take the tangent line (the line through $P$ and $P$), the third point is again $P.$

Answer 2

He's getting at the group structure of points on an elliptic curve in both cases. To add a point $P$ to itself, you look at the tangent line to the point and see where it intersects the curve (it will intersect it in exactly one point, one way to see this is Bezout). You then have to reflect to get $2P$.

Given any two points on a nonsingular elliptic curve if you look at the chord through those two points, it will always intersect the elliptic curve in exactly one other point (once again you could see this by Bezout). The reflection of this point over the $x$-axis is the sum of the two original points.

It's possible in both cases that the intersection point is the point at infinity which acts as the identity element. This possibility is included in Bezout's theorem.

For example if you take any point with horizontal tangent line, it's tangent line will intersect your elliptic curve at the point at infinity.