In the book Elliptic Curves - McKean & Moll we are given the outline for a proof of Eulers addition theorem:
The (projective) quartic $\mathbf y^2 = (1-\mathbf x^2)(1-k^2 \mathbf x^2)$ has four intersections with the quadratic $\mathbf y = 1 + a \mathbf x + b \mathbf x^2$. One is located at $(\mathbf x, \mathbf y) = (0,1)$. Let the other three have covering parameters $x_1,x_2,x_3$. Check that their sum is independent of $a$ and $b$ and identify it, mod $\mathbb L$, as either $0$ or $2K$. Now compute $\mathbf x_1 + \mathbf x_2 + \mathbf x_3 = 2ab/c$ and $\mathbf x_1 \mathbf x_2 \mathbf x_3 = 2a/c$ with $c = k^2 - b^2$ and conclude that $\mathbf x_1 \mathbf y_2 + \mathbf x_2 \mathbf y_1 = \pm \mathbf x_3 (1 - k^2 \mathbf x_1^2 \mathbf x_2^2)$. What is the ambiguous sign?
It is too difficult for me to solve so I have some questions that I hope someone will be able to help me with:
- What are these covering parameters and how do we find information about their sum?
- Why does the information about the covering parameters matter?
- I have computed $\mathbf x_1 + \mathbf x_2 + \mathbf x_3$ and $\mathbf x_1 \mathbf x_2 \mathbf x_3$ but I don't see how to reach the conclusion from there. How is it done?