 Solution

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