TITLE: What even *are* elliptic functions?
QUESTION [17 upvotes]: I am just beginning to learn about elliptic functions. Wikipedia defines an elliptic function as a function which is meromorphic on $\Bbb C$, and for which there exist two non-zero complex numbers $\omega_1$ and $\omega_2$, with $\frac{\omega_1}{\omega_2}\not\in \Bbb R$, which satisfy $$f(z)=f(z+\omega_1)=f(z+\omega_2).$$
That's all fine and dandy, but what does this have to do with an ellipse?
I sort of know (but not really) about the Jacobi elliptic functions. I am told by the internet that the Jacobi elliptic functions can be defined as inverses of elliptic integrals, which relate to the arc lengths of ellipses. But other than that, I have no idea how elliptic functions relate to ellipses. 
I have looked at several sources, like this, this, and this. From what I can understand, any elliptic function can be expressed in terms of the Jacobi elliptic functions and Weierstrass elliptic functions, but I have yet to understand why that is true. Perhaps it has something to do with what ODE's elliptic functions satisfy? I do not know. 
I would really appreciate some help and/or a good source on the introduction to the study of elliptic functions in the context of elliptic integrals, because I do work best with integrals. Thanks!

REPLY [6 votes]: For a lattice $L = u\Bbb{Z}+v\Bbb{Z}$ in $\Bbb{C}$ let $ \wp_L(z)$ be the only $L$-periodic function with only one double pole at $z=0$ where $\wp_L(z)=z^{-2}+O(z^2)$. Since analytic $L$-periodic functions are constant then $$\wp_L'(z)^2=4\wp_L(z)^3+g_2(L)\wp_L(z)+g_3(L)$$ with $g_2,g_3$ found from the $z^4,z^6$ coefficient of $\wp_L(z)-z^{-2}$.
Thus $z\mapsto(\wp_L(z),\wp_L'(z))$ is an isomorphism from the complex torus $\Bbb{C}/L$ to the elliptic curve $E:y^2=4x^3+g_2(L)x+g_3(L)$.
$$K(k)=\int_{0}^{\pi/2}\frac{dt}{\sqrt{1-k^2\sin^2t}}= \int_0^1 \frac{dw}{\sqrt{(1-k^2 w^2) (1-w^2)}}$$ $$=\int_0^1 \frac{ds}{2\sqrt{(1-k^2 s) (1-s)s}}=\frac1{ik}\int_{-(k^{-2}+1)/3}^{1-(k^{-2}+1)/3} \frac{dx}{\sqrt{4x^3 + a(k)x+b(k)x}}$$ where $a(k)=4\frac{-k^4 + k^2 - 1}{3k^4},b(k)=4\frac{-2k^6 + 3k^4 + 3k^2 - 2}{27k^6}$.
Take the lattice $L$ such that $a(k)=g_2(L),b(k)=g_3(L)$. From the  $z\mapsto(\wp_L(z),\wp_L'(z))$  isomorphism we get $$\frac{dx}{ \sqrt{4x^3 + a(k)x+b(k)x}} = \frac{d\wp_L(z)}{\wp_L'(z)} = dz$$
In the group law of $E$: $-(x,y)=(x,-y)$ thus the points $(k^{-2}+1)/3,0), (1-(k^{-2}+1)/3,0)$ are in the 2-torsion of $E$. On the complex torus side it is obvious the 2-torsion is  :  $L,u/2+L,v/2+L,(u+v)/2+L$, altogether it means $$K(k) = \frac1{ik}\int_{u/2}^{(u+v)/2+\ell} dz = \frac{v/2+\ell}{ik}\qquad\text{ for some } \ell \in L$$


In other words the complete elliptic integrals are the way to recover the lattice from the elliptic curve $y^2=(1-k^2 s) (1-s)s$.