I am dealing with the proof of the following Theorem, taken from Dale Husemöller's book Elliptic Curves:
I have trouble to understand the following underlined section of the proof:
Could you please elaborate this section for me?
- I only know that the tangent has something to do with the derivative:
$$L: \frac{\partial F}{\partial x}(P)(x-x_P) + \frac{\partial F}{\partial y}(P)(y-y_P)=0$$ where $P = (x_P,y_P)$ and $F(x,y) = y^2 - x^3 - ax$. However, I do not see what is has to do with double roots.
- For real differential functions $f:\mathbb{R} \to \mathbb{R}$ in one variable, I know that something has a double root $a$ if and only if $f(a) = f'(a) = 0$. But I do not know how this can be applied to functions with more than one variable.
Thank you in advance!
Solutions of the system
\begin{align} y^2 &= x^3 + ax\\ y &= \lambda x \end{align}
correspond to intersections of the line and the curve. Substitute $y$ into the first equation to get $$x(x^2-\lambda^2x + a) = 0.$$ We know one solution of the system, $(0,0)$, so the other two correspond to solutions of $x^2-\lambda^2 x + a = 0$. Now, we know that we can have two distinct real roots, two complex roots or double root. These correspond to two intersections, no intersections and one intersection of the line and the curve. Thus, double root corresponds to tangent line.
Let's show this more formally. Let $F(x,y) = 0$ be a smooth algebraic curve and let $y = ax + b$ be a tangent line at $(x_0,y_0)$. Define $g(x) = F(x,ax+b)$. Obviously, $g(x_0) = F(x_0,y_0) = 0$.
On the other hand $g'(x) = \partial_xF(x,ax+b) + a\partial_y F(x,ax+b)$. Also, $a = -\frac{\partial_xF(x_0,y_0)}{\partial_yF(x_0,y_0)}$ since we assumed $y = ax + b$ to be tangent. It follows that $$g'(x_0) = \partial_xF(x_0,y_0) -\frac{\partial_xF(x_0,y_0)}{\partial_yF(x_0,y_0)} \partial_yF(x_0,y_0) = 0.$$
In your example, $g(x) = x(x^2-\lambda^2x + a)$.