Here by algebraic stack I mean an algebraic stack over the etale site $\textbf{Sch}/\mathbb{C}$.


So I've read from various nonrigorous sources that the upper half plane $\mathcal{H}$ is a fine moduli space for framed elliptic curves. If the data of a framing can be expressed in algebraic geometry (maybe as some choice of basis for the first etale cohomology), then it ought to be able to be seen as an algebraic stack. However, in this case, it should be covered by schemes $U\_i$ etale over $\mathcal{H}$, and I don't really see what these schemes would be.



Is the upper half plane an algebraic stack?
The upper half plane is not an algebraic stack over ${\mathbb C}$. 


Suppose it were, then it would be in fact an algebraic space, since its points do not have any isotropy groups. But it is also smooth and one dimensional. It is easy to show that algebraic spaces over ${\mathbb C}$ which are smooth, separated, one dimensional are in fact schemes. 


