Properties of Weierstrass's elliptic function.

Question

In section 6.2 of An introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces by M. Schlichenmaier, the author wants to embed the torus $T = \mathbb{C}/L$ (for a conventional integer lattice $L = \{m + n z: m,n\in\mathbb{N},z\in\mathbb{C}\}$) in the complex projective space $\mathbb{P}^2$. To do so he introduces the map

$$\Psi:T\to\mathbb{P}^2:z\mapsto\begin{cases} (\wp(z):\wp'(z): 1), & z \neq 0 \\ (0:1:0), & z = 0\end{cases}$$

where $\wp$ stands for Weierstrass elliptic function. Most of the reasoning seems to me quite straight forward, but I fail catastrophically too understand two notions employed to prove the injectivity of $\Psi$. I cite the problematic passages.

$\wp$ is a [meromorphic] function on the torus, thus it takes every value of $\mathbb{P}^1$ equally often (calculated with multiplicity).

Why? The image of a non-constant meromorphic function over $\mathbb{C}$ is dense in $\mathbb{C}$, but how can we get from here to the conclusion that $\wp:T\to\mathbb{P}^1$ is surjective? And not only that, but moreover he says it takes every such value equally often! Does this really mean that $\exists n\in \mathbb{N}\; \forall z\in\mathbb{P}^1\;\exists \{z_1,...,z_n\}\subseteq\mathbb{N}:\;\;\wp(z_i) = z\; \forall i\in [\![1,n]\!]$? This is something I am completely unable to understand or prove. Then he continues:

It has a pole of order $2$ at $0\in T$ and nowhere else. Hence every value occurs two times.

I presume the reasoning behind this assertion (which follows right after the one cited before it) is very closely related to the preceding one, and I am equally unable to prove it myself.

I would really appreciate any help. Thank you in advance.

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(Solution candidate promoted to answer, see below).

Answer 1

Let $\bar{z} = \pi(z)$ for the canonical projection $\pi: \mathbb{C}\to T$. For any $a\in\mathbb{C}$ define

$$f_a:\mathbb{C}\to\mathbb{C}:z\mapsto \frac{1}{\wp(\bar{z}) - a}$$

By definition of $\wp$ we have that $f$ is doubly periodic and not constant. If $f$ doesn't have a pole, then $f$ is bounded and entire, hence constant, which is a contradiction.

Consider now a contour $C$ following the boundary of a parallelogram defined by the lattice $L$, but displaced as to include both the point $a$ and the vertex in the lattice closest to $a$. Since $\wp'$ has the same double periodicity as $\wp$, by the argument principle we calculate

$$ Z - P = \frac{1}{2i \pi}\oint_C \frac{\wp'(\bar{z})}{\wp({\bar{z}})-a} dz = 0$$

due to the periodicity of the integrand over the lattice. Now $z\mapsto\wp(\bar{z}) - a$ has one single pole of order $2$ inside the contour, and consequently the number of zeros (counted with multiplicity) of $f(z)^{-1}$ is $2$. Because this is valid for all $a\in\mathbb{C}$, the conclusion follows: $\wp:T\to\mathbb{C}$ is surjective and the preimage of any complex number has exactly two elements (counted with multiplicity).

In fact, because $\wp$ is even, the preimage of any $a\in\mathbb{C}$ is of the form $\wp^{-1}(\{a\}) = \{z_a,-z_a\}$.