Let $\Lambda=\langle\lambda, \mu\rangle \subseteq \mathbb{C}$ be a lattice. Give the definition of the associated Weierstrass $\wp$-function as an infinite sum, and prove that it converges. [You may use without proof the fact that

$$\sum_{w \in \Lambda \backslash\{0\}} \frac{1}{|w|^{t}}$$

converges if and only if $t>2$.]

Consider the half-lattice points

$$z_{1}=\lambda / 2, \quad z_{2}=\mu / 2, \quad z_{3}=(\lambda+\mu) / 2,$$

and let $e_{i}=\wp\left(z_{i}\right)$. Using basic properties of $\wp$, explain why the values $e_{1}, e_{2}, e_{3}$ are distinct

Give an example of a lattice $\Lambda$ and a conformal equivalence $\theta: \mathbb{C} / \Lambda \rightarrow \mathbb{C} / \Lambda$ such that $\theta$ acts transitively on the images of the half-lattice points $z_{1}, z_{2}, z_{3}$.