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

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?

-
Why do you think the fact that $Y(1)$ is a quotient of the upper-half plane implies that its universal cover is the upper-half plane? –  Bruno Joyal Oct 29 '11 at 2:38
I can't think of points where it is branched, which would make it a covering map... –  Nicole Oct 29 '11 at 2:40
Dear Nicole, There is branching at the points corresponding to $j=0$ and $j=1728$. Not coincidentally, these are the $j$-invariants corresponding to elliptic curves with extra automorphisms. Regards, –  Matt E Nov 25 '11 at 2:01

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.