Let $\Lambda$ be a lattice in $\mathbb{C}, \Lambda=\mathbb{Z} \omega_{1}+\mathbb{Z} \omega_{2}$, where $\omega_{1}, \omega_{2} \neq 0$ and $\omega_{1} / \omega_{2} \notin \mathbb{R}$. By constructing an appropriate family of charts, show that the torus $\mathbb{C} / \Lambda$ is a Riemann surface and that the natural projection $\pi: z \in \mathbb{C} \rightarrow z+\Lambda \in \mathbb{C} / \Lambda$ is a holomorphic map.

[You may assume without proof any known topological properties of $\mathbb{C} / \Lambda$.]

Let $\Lambda^{\prime}=\mathbb{Z} \omega_{1}^{\prime}+\mathbb{Z} \omega_{2}^{\prime}$ be another lattice in $\mathbb{C}$, with $\omega_{1}^{\prime}, \omega_{2}^{\prime} \neq 0$ and $\omega_{1}^{\prime} / \omega_{2}^{\prime} \notin \mathbb{R}$. By considering paths from 0 to an arbitrary $z \in \mathbb{C}$, show that if $f: \mathbb{C} / \Lambda \rightarrow \mathbb{C} / \Lambda^{\prime}$ is a conformal equivalence then

$$f(z+\Lambda)=(a z+b)+\Lambda^{\prime} \quad \text { for some } a, b, \in \mathbb{C}, \text { with } a \neq 0$$

[Any form of the Monodromy Theorem and basic results on the lifts of paths may be used without proof, provided that these are correctly stated. You may assume without proof that every injective holomorphic function $F: \mathbb{C} \rightarrow \mathbb{C}$ is of the form $F(z)=a z+b$, for some $a, b \in \mathbb{C}$.]

Give an explicit example of a non-constant holomorphic map $\mathbb{C} / \Lambda \rightarrow \mathbb{C} / \Lambda$ that is not a conformal equivalence.

Part II 2004