Define what is meant by the degree of a non-constant holomorphic map between compact connected Riemann surfaces, and state the Riemann-Hurwitz formula.

Let $E_{\Lambda}=\mathbb{C} / \Lambda$ be an elliptic curve defined by some lattice $\Lambda$. Show that the map

$$\psi: z+\Lambda \in E_{\Lambda} \rightarrow-z+\Lambda \in E_{\Lambda}$$

is biholomorphic, and that there are four points in $E_{\Lambda}$ fixed by $\psi$.

Let $S=E_{\Lambda} / \sim$ be the quotient surface (the topological surface obtained by identifying $z+\Lambda$ and $\psi(z+\Lambda)$, for each $z)$ and let $\pi: E_{\Lambda} \rightarrow S$ be the corresponding projection map. Denote by $E_{\Lambda}^{0} \subset E_{\Lambda}$ the complement of the four points fixed by $\psi$, and let $S^{0}=\pi\left(E_{\Lambda}^{0}\right)$. Describe briefly a family of charts making $S^{0}$ into a Riemann surface, so that $\pi: E_{\Lambda}^{0} \rightarrow S^{0}$ is a holomorphic map.

Now assume that the complex structure of $S^{0}$ extends to $S$, so that $S$ is a Riemann surface, and that the map $\pi$ is in fact holomorphic on all of $E_{\Lambda}$. Calculate the genus of $S$.