Define the degree of an analytic map of compact Riemann surfaces, and state the Riemann-Hurwitz formula.

Let $\Lambda$ be a lattice in $\mathbb{C}$ and $E=\mathbb{C} / \Lambda$ the associated complex torus. Show that the $\operatorname{map}$

$$\psi: z+\Lambda \mapsto-z+\Lambda$$

is biholomorphic with four fixed points in $E$.

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

Now assume that, by adding finitely many points, it is possible to compactify $S^{\prime}$ to a Riemann surface $S$ so that $p$ extends to a regular map $E \rightarrow S$. Find the genus of $S$.