This is a silly question, but I can't resolve this:
$Y(1)$ is defined to be $\mathbb{H}/PSL_2(\mathbb{Z})$. So it seems that its universal cover should be $\mathbb{H}$.
On the other hand $Y(1)$ is isomorphic to $\mathbb{A}^1_{\mathbb{C}}$, and therefore is its own universal cover.
What am I missing?