(a) Let $\Lambda=\mathbb{Z}+\mathbb{Z} \tau$ be a lattice in $\mathbb{C}$, where the imaginary part of $\tau$ is positive. Define the terms elliptic function with respect to $\Lambda$ and order of an elliptic function.

Suppose that $f$ is an elliptic function with respect to $\Lambda$ of order $m>0$. Show that the derivative $f^{\prime}$ is also an elliptic function with respect to $\Lambda$ and that its order $n$ satisfies $m+1 \leqslant n \leqslant 2 m$. Give an example of an elliptic function $f$ with $m=5$ and $n=6$, and an example of an elliptic function $f$ with $m=5$ and $n=9$.

[Basic results about holomorphic maps may be used without proof, provided these are accurately stated.]

(b) State the monodromy theorem. Using the monodromy theorem, or otherwise, prove that if two tori $\mathbb{C} / \Lambda_{1}$ and $\mathbb{C} / \Lambda_{2}$ are conformally equivalent then the lattices satisfy $\Lambda_{2}=a \Lambda_{1}$, for some $a \in \mathbb{C} \backslash\{0\}$.

[You may assume that $\mathbb{C}$ is simply connected and every biholomorphic map of $\mathbb{C}$ onto itself is of the form $z \mapsto c z+d$, for some $c, d \in \mathbb{C}, c \neq 0$.]