Putting a Riemann surface structure on a set of equivalence classes in a torus I'm looking at the torus given by $X = \mathbb{C}/\Lambda$ where $\Lambda$ is the lattice spanned by $1$ and $\omega$ where $\omega$ is a primitive cube root of unity.
I've shown that $\sigma(z) = \omega z$ is a well-defined map on the torus and now I've been asked to explain how to put a Riemann surface structure on the set of equivalence classes $Y = \{z, \sigma(z), \sigma^2(z)\}$ such that the natural map $X \to Y$ is holomorphic.
I'm having quite a lot of difficulty with this. I'm not sure how to visualise $Y$ as a space, I can see that if $z$ is a fixed point then $Y$ is just a copy of $X$ and then the map $X \to Y$ is just the identity but I'm struggling to write down explicit local coordinates for $Y$ in general.
Thanks for any help
 A: $\newcommand{\Cpx}{\mathbf{C}}$Here's one possible diagram of a fundamental domain $R$ for $X = \Cpx/\Lambda$, the images of $R$ under the cyclic group generated by $\sigma$, and a polygon-with-edge-gluing for the quotient $Y = X/\langle\sigma\rangle$:

To fill in the remaining details, it may help to notice that the action of $\sigma$ on $R$ amounts to cutting $R$ into equilateral triangles along the segment from $0$ to $1 + \omega$ and rotating each triangle one-third of a turn counterclockwise about its center. (Edges joined in $X$ remain joined after this operation!)
There are three fixed points of $\sigma$ in $X$, namely $0$ and the two centers of the triangles. Consequently, the quotient map $X \to Y$ has three branch points, each of order three. The function $(z - z_{0})^{3}$ may be used as a local coordinate near the image of $z_{0}$ in $Y$, i.e., near each vertex of the fundamental domain of $Y$. (N.B., this fundamental domain has three vertices, since $0$ and $1 + \omega$ are identified.)
