How does the Weierstrass Elliptic Function convert solutions to Elliptic Curves to algebraic form?

Question

I've been reading up on elliptic functions and such, however I am still not completely familiar with them. On Wolfram Mathworld, it describes that the $\wp(\tau)$ can convert solutions to elliptic curves from the topological space of a torus, to algebraic form. Obviously, this sounds like a very important property of this function, but how exactly is that evident?

Answer 1

In the Wikipedia article Weierstrass elliptic functions is the important differential equation $\;\wp'(z)^2=4\wp(z)^3-g_2\wp(z)-g_3\;$ which means that $\;(\wp(z),\wp'(z))\;$ is a point on the elliptic curve $\;y^2=4x^3-g_2x-g_3\;$ in Weierstrass form, and thus determines a parametrization of the curve. Because both $\wp(z)$ and $\wp'(z)$ are elliptic functions, they are doubly periodic and thus are determined by their values on the period parallelogram with boundary identified to give a topological torus.

An Elliptic curve can be also written as $y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$ The important thing is that the genus of the curve is $1$ which means that the degree of the curve is $3$ or $4$.