# Deriving Eulers Addition Theorem for Elliptic Integrals

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?

Thank you

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Having seen the relevant page: books.google.com/books?id=ovGVquPwo7oC&pg=PA113 I don't know what those covering parameters are, but this can be derived from the addition theorem for $\mathrm{sn}(u,k)$, since the addition theorem's compact restatement is $\mathrm{sn}^{-1}(a,k)+\mathrm{sn}^{-1}(b,k)=\mathrm{sn}^{-1}\left(\frac{b\sqrt{‌​\left(1-a^2\right)\left(1-k^2 a^2\right)}+a\sqrt{\left(1-b^2\right)\left(1-k^2 b^2\right)}}{\sqrt{1-k^2 a^2 b^2}},k\right)$ – J. M. Sep 5 '10 at 16:19
J. M., I am trying to prove the addition theorem for sn. – anon Sep 5 '10 at 16:23
As for proving the addition formula for $\mathrm{sn}$, books.google.com/books?id=zyxAb4ro-oMC&pg=PA30 shows one way of going about it. – J. M. Sep 5 '10 at 16:47

The elliptic curve is the image of $\mathbb{C}/\Lambda$ under a certain map. Now, each point $(\mathbf{x}_0,\mathbf{y}_0)$ is the image of some point $x$ of $\mathbb{C}/\Lambda$. This $x$ is the covering parameter. It's a complex number unique up to translation by elements of $\Lambda$.
Each of $\mathbf{x}$ and $\mathbf{y}$ can be regarded as a function on $\mathbb{C}/\Lambda$. We are interested in $f=\mathbf{y} -1-a\mathbf{x}-b\mathbf{x}^2$. This has four zeros in each fundamental region, and so four poles. If the zeros are $x_0,x_1,x_2,x_3$ and poles are $u_0,u_1,u_2,u_3$ then $x_0+x_1+x_2+x_3-u_0-u_1-u_2-u_3$ is an element of $\Lambda$. I suppose here, one must show that the poles of $f$ are independent of $a$ and $b$, which leads to $x_0+x_1+x_2+x_3$ being independent of $a$ and $b$ (here $x_0$ is the covering parameter for the point $(0,1)$).