relationship between elliptic curves over $\mathbb{C}$ & Weierstrass $\wp$-function

Question

I'm studying about elliptic curves over $\mathbb{C}$, then at some point the Weierstrass $\wp$-function appear. It appears to show that:

1) $\wp$ and $\wp'$ generate the field of meromorphic functions on $\mathbb{C}/\Lambda$;

2)$\wp'(z)^{2}=4\wp(z)^{3}-g_{2}\wp(z) -g_{3}$;

3)$z \rightarrow (1,\wp(z),\wp'(z))$ is an isomorphism.

My question is that I can not see why exactly this function appears. What is the reason for proving these facts 1), 2) and 3) in relation to Weierstrass $\wp$-function.It is not clear to me the purpose of proving 1), 2) and 3) ...

I sorry if this question is trivial

Answer 1

At the simplest level it is a parameterisation. For example the circle $$x^2+y^2=1$$ is parameterized by $$x=\cos z$$ $$y=\sin z.$$

Similarity, the elliptic curve $$y^2=4x^3-g_2x-g_3$$ is parameterised by

$$x=\wp(z)$$ $$y=\wp^{\prime}(z)$$

More sophisticated point of view I that the $\wp$ function arises by integrating the only holomorphic differential over the curve. Historically the two ideas grew up together. In other words all statements about curves have their function theoretic analogs, for example addition of points is a rational addition property for elliptic functions.