Elliptic functions surjective

Question

Is it true that every nonconstant elliptic function $f:\mathbf{C}/\Lambda\rightarrow\mathbf{P}^1$ is surjective? (I take elliptic functions to be defined on the torus $\mathbf{C}/\Lambda$) For how is it otherwise true that $\# f^{-1}(c)$ is independent of $c\in\mathbf{P}^1$?

Answer 1

Yes, it is true. An elliptic function is a holomorphic map $f\colon \mathbb{C}/\Lambda \to \mathbf{P}^1$. Since the torus is compact, the image of $f$ is compact, hence closed. If $f$ is not constant, it is an open mapping, so the image is then also open. The only nonempty set that is both open and closed in $\mathbf{P}^1$ is $\mathbf{P}^1$, so $f$ must be surjective.