Example of a Riemann surface

Question

Let $\Gamma = \omega_1 \mathbb{Z} + \omega_2 \mathbb{Z}$ with $\omega_1, \omega_2$ independent over $\mathbb{R}$. Let $E_{\Gamma} = \mathbb{C}{/ \Gamma}$. Show that $E_{\Gamma}$ is a Riemann surface.

We have that $E_{\Gamma}$ is a topological manifold since it is homeomorphic to the torus. Furthermore, if we represent it on the complex plane, it coincides with the parallelogram generated by $\omega_1, \omega_2$. Now, as an open cover for $E_{\Gamma}$, we take a tessellation (for example, with open rectangles), and as charts, we consider those that map these open sets to themselves up to translation. Then, taking two open sets that intersect, the gluing function maps the intersection to itself up to translation, making it a holomorphic function. I would like to know if this is acceptable and if there is a more formal way to express this properly...

Answer 1

More generally, see 5. Riemann surfaces as quotients

Theorem 4.13. Let $\Omega \subset \mathbb{C}_{\infty}$ and $G \subset \operatorname{Aut}\left(\mathbb{C}_{\infty}\right)$ with the property that

• $g(\Omega) \subset \Omega$ for all $g \in G$
• for all $g \in G, g \neq \mathrm{id}$, all fixed points of $g$ in $\mathbb{C}_{\infty}$ lie outside of $\Omega$
• let $K \subset \Omega$ be compact. Then the cardinality of $\{g \in G \mid g(K) \cap K \neq \emptyset\}$ is finite.

Under these assumptions, the natural projection $\pi: \Omega \rightarrow \Omega / G$ is a covering map which turns $\Omega / G$ canonically into a Riemann surface.

$\Bbb C/Γ$ in your question is in the list of examples after this theorem.