Compact Riemann Surface

Question

Let's consider the $\mathbb{Z}-$Action on $\mathbb{C}^{\times}$ given by, for $k\in \mathbb{Z}$, $k\cdot z:= q^kz$, with a fixed $q\in \mathbb{C}^{\times}$ such that $0<|q|<1$.

I have to prove that $\mathbb{C}^{\times}/q^{\mathbb{Z}}$ is a compact Riemann surface that is isomorphic to a complex torus $\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$ for some $\tau\in \mathbb{C}$.

I really don't know how to start. Anyone knows how to prove this?

Answer 1

HINT:

Try to obtain a fundamental domain for this action, e.g. analyzing what is the orbit of the circle $S^1=\{z\,|\,||z||=1\}$.

Answer 2

It is supposedly obvious that $\Bbb{C/(Z+\tau Z)}$ is a compact Riemann surface (for $\tau\not \in \Bbb{R}$)

The isomorphism to $\Bbb{C^\times/q^Z}$ is $$z+\Bbb{Z+\tau Z}\to \exp(2i\pi z) q^\Bbb{Z}, \qquad q=\exp(2i\pi \tau)$$