Past exam of the geometry course of the University of Oxford / 2011 / 2 / b

Let $\Lambda =\mathbb{Z}{\omega }_1+\mathbb{Z}{\omega }_2$ and $T=\mathbb{C}/\Lambda$. Its fundamental parallelogram with opposite sides identified is a torus, so $\chi (T)=0$. The Weierstrass elliptic function is even and doubly periodic. One way to verify the latter from its series is to differentiate: the locally normally convergent derivative series iswhich is absolutely convergent away from the lattice and periodic by reindexing. Thus $\wp (z+{\omega }_j)-\wp (z)$ is constant. Evaluating at $z=-{\omega }_j/2$ and using evenness shows that this constant is zero for each generating period.

Therefore $[z]\mapsto \wp (z)$ descends to a meromorphic function on $T$, with one double pole at $[0]$. Viewed as a map to the Riemann sphere, a pole is a holomorphic point in the target coordinate $1/w$. The map is consequently holomorphic everywhere and

The Riemann-Hurwitz formula gives total ramification four because $0=2\cdot 2-R$. The double pole at $[0]$ has local degree two and contributes one. Since ${\wp }^{\prime }$ is odd and periodic, it vanishes at each nonzero point satisfying $[h]=[-h]$. These areThey are three distinct finite points, each contributing at least one to ramification. Together with $[0]$ they exhaust the total of four. Each therefore has local degree exactly two, meaning its derivative has a simple zero. There are no other zeros of ${\wp }^{\prime }$, and their translates by $\Lambda$ give the full list in the complex plane.