Let $\Lambda=\mathbb{Z}+\mathbb{Z} \lambda$ be a lattice in $\mathbb{C}$ where $\operatorname{Im}(\lambda)>0$, and let $X$ be the complex torus $\mathbb{C} / \Lambda .$

(i) Give the definition of an elliptic function with respect to $\Lambda$. Show that there is a bijection between the set of elliptic functions with respect to $\Lambda$ and the set of holomorphic maps from $X$ to the Riemann sphere. Next, show that if $f$ is an elliptic function with respect to $\Lambda$ and $f^{-1}\{\infty\}=\emptyset$, then $f$ is constant.

(ii) Assume that

$$f(z)=\frac{1}{z^{2}}+\sum_{\omega \in \Lambda \backslash\{0\}}\left(\frac{1}{(z-\omega)^{2}}-\frac{1}{\omega^{2}}\right)$$

defines a meromorphic function on $\mathbb{C}$, where the sum converges uniformly on compact subsets of $\mathbb{C} \backslash \Lambda$. Show that $f$ is an elliptic function with respect to $\Lambda$. Calculate the order of $f$.

Let $g$ be an elliptic function with respect to $\Lambda$ on $\mathbb{C}$, which is holomorphic on $\mathbb{C} \backslash \Lambda$ and whose only zeroes in the closed parallelogram with vertices $\{0,1, \lambda, \lambda+1\}$ are simple zeroes at the points $\left\{\frac{1}{2}, \frac{\lambda}{2}, \frac{1}{2}+\frac{\lambda}{2}\right\}$. Show that $g$ is a non-zero constant multiple of $f^{\prime}$.