How could $Y(1)$ be the quotient of the upper half plane?

Question

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?

Answer 1

To follow up on Bruno Joyal's comment, $PSL(2,\mathbb Z)$ does not act freely on $\mathbb H$. If you think of this group as being generated by $z\mapsto -1/z$ and $z\mapsto z+1$, then the complex number $i$ is fixed by the first map, so it has a nontrivial stabilizer.