Let $\lambda$ and $\mu$ be fixed, non-zero complex numbers, with $\lambda / \mu \notin \mathbb{R}$, and let $\Lambda=\mathbb{Z} \mu+\mathbb{Z} \lambda$ be the lattice they generate in $\mathbb{C}$. The series

$$\wp(z)=\frac{1}{z^{2}}+\sum_{m, n}\left[\frac{1}{(z-m \lambda-n \mu)^{2}}-\frac{1}{(m \lambda+n \mu)^{2}}\right]$$

with the sum taken over all pairs $(m, n) \in \mathbb{Z} \times \mathbb{Z}$ other than $(0,0)$, is known to converge to an elliptic function, meaning a meromorphic function $\wp: \mathbb{C} \rightarrow \mathbb{C} \cup\{\infty\}$ satisfying $\wp(z)=\wp(z+\mu)=\wp(z+\lambda)$ for all $z \in \mathbb{C}$. ( $\wp$ is called the Weierstrass function.)

(a) Find the three zeros of $\wp^{\prime}$ modulo $\Lambda$, explaining why there are no others.

(b) Show that, for any number $a \in \mathbb{C}$, other than the three values $\wp(\lambda / 2), \wp(\mu / 2)$ and $\wp((\lambda+\mu) / 2)$, the equation $\wp(z)=a$ has exactly two solutions, modulo $\Lambda$; whereas, for each of the specified values, it has a single solution.

[In (a) and (b), you may use, without proof, any known results about valencies and degrees of holomorphic maps between compact Riemann surfaces, provided you state them correctly.]

(c) Prove that every even elliptic function $\phi(z)$ is a rational function of $\wp(z)$; that is, there exists a rational function $R$ for which $\phi(z)=R(\wp(z))$.