(i) Show that the open unit $\operatorname{disc} D=\{z \in \mathbb{C}:|z|<1\}$ is biholomorphic to the upper half-plane $\mathbb{H}=\{z \in \mathbb{C}: \operatorname{Im}(z)>0\}$.

(ii) Define the degree of a non-constant holomorphic map between compact connected Riemann surfaces. State the Riemann-Hurwitz formula without proof. Now let $X$ be a complex torus and $f: X \rightarrow Y$ a holomorphic map of degree 2 , where $Y$ is the Riemann sphere. Show that $f$ has exactly four branch points.

(iii) List without proof those Riemann surfaces whose universal cover is the Riemann sphere or $\mathbb{C}$. Now let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a holomorphic map such that there are two distinct elements $a, b \in \mathbb{C}$ outside the image of $f$. Assuming the uniformization theorem and the monodromy theorem, show that $f$ is constant.