Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 4 / 23H (Riemann Surfaces)

For a nonconstant holomorphic map $f:X\to Y$ between compact connected Riemann surfaces, its degree of a holomorphic map is the number of preimages of a regular value. More generally it is the constant sum ${\sum }_{p\in f^{-1}(q)}{e}_p$, where $e_p$ is the local multiplicity. The Riemann-Hurwitz formula is

Negation is well-defined on the complex torus because $-\Lambda =\Lambda$. It is holomorphic with holomorphic inverse itself, so $\psi$ is biholomorphic. Its fixed points satisfy $2z\in \Lambda$. If $\Lambda =\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2$, they are the four distinct classes

Away from these points, choose a small coordinate disk $U$ disjoint from $\psi (U)$. The quotient map is a homeomorphism from $U$ onto $\pi (U)$, so transfer its complex coordinate to that open set of $S^{\prime }$. The transition between two such lifted charts is induced either by identity or by $\psi$, hence is holomorphic. These charts give the required Riemann-surface structure, and $\pi$ is locally a holomorphic bijection there.

The quotient has generic degree two. At a fixed point, a local coordinate centred there changes under $\psi$ as $u\mapsto -u$. The assumed holomorphic quotient map is invariant under this operation, so its local expansion is even and has multiplicity at least two. Since the total degree is two, it is exactly two. There are no other ramification points. The domain torus has genus one, so Riemann-Hurwitz formula gives