# Properties of Weierstrass's elliptic function.

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.

(Solution candidate promoted to answer, see below).

• $\frac1{\wp(z)-a}$ must have some pole as a doubly periodic entire function is bounded thus constant. Next, the number of zeros of $\wp(z)-a$ is given by the argument principle so it varies continuously with $a$, and hence it is equal to $2$ (on a fundamental parallelogram) for all $a\in \Bbb{C}$. Feb 28 at 18:26
• @reuns Is my solution correct? Feb 28 at 19:01
• Yes ${}{}{}{}{}$ Feb 28 at 19:02

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 $$\wp^{-1}(\{a\}) = \{z_a,-z_a\}$$.

• I forgot to mention this solution is based on @reuns's comment to my question, whom I have to thank for the help. Mar 1 at 12:44