Let $\Lambda$ be a lattice in $\mathbb{C}$, and $f: \mathbb{C} / \Lambda \rightarrow \mathbb{C} / \Lambda$ a holomorphic map of complex tori. Show that $f$ lifts to a linear map $F: \mathbb{C} \rightarrow \mathbb{C}$.

Give the definition of $\wp(z):=\wp_{\Lambda}(z)$, the Weierstrass $\wp$-function for $\Lambda$. Show that there exist constants $g_{2}, g_{3}$ such that

$$\wp^{\prime}(z)^{2}=4 \wp(z)^{3}-g_{2} \wp(z)-g_{3}$$

Suppose $f \in \operatorname{Aut}(\mathbb{C} / \Lambda)$, that is, $f: \mathbb{C} / \Lambda \rightarrow \mathbb{C} / \Lambda$ is a biholomorphic group homomorphism. Prove that there exists a lift $F(z)=\zeta z$ of $f$, where $\zeta$ is a root of unity for which there exist $m, n \in \mathbb{Z}$ such that $\zeta^{2}+m \zeta+n=0$.