Here's the situation I'm in.
I have a map from the (closure of) Upper Half Plane ($\mathbb{U}$) into the punctured (closed) disk ($\mathbb{D}$) called $q$ that satisfies
$q(0) = 1$, and $q(z+1) = q(z)$.
EDIT: And $q$ is injective on the interval $(0,1)$ on the real line.
I also know that $q$ is a holomorphic covering map.
From this information, I can show that $q$ has to be $q(z) = exp(2\pi i z)$.The argument is as follows:
Let $p(z) = exp(2\pi i z)$. Let $h$ be the lift of $q$ through $p$, and I choose it so that $h(0) = 0$. By taking the lift of $p$ through $q$, one can see that $h \in PSL(2,\mathbb{R})$.
Because $h$ is a lift, it follows that $h(z+1) = h(z) + n$ for some integer $n$. This argument shows that $h(\infty) = \infty$. It follows that $h(z) = a*z$, and $a=1$.
I feel like I'm being very stupid here. Does the fact that $q(z) = exp(2\pi i z)$ follow from some more general principle?
For instance, if I have a (simply connected) space $M$, and two covering maps $p$ and $q$ into $S$, can one say anything about $p$ and $q$?
Or maybe what I'm missing is complex analysis. Is there some kind of a uniqueness statement about holomorphic maps that I could've used to conclude that $q$ has to be the exponential map?