moduli space of elliptic curves, $\mathbb{H}/SL(2,\mathbb{Z})$ or $\mathbb{H}/PSL(2,\mathbb{Z})$

Question

I see some (most) people saying the moduli space of elliptic curves is $\mathcal{M}_{1,1}=\mathbb{H}/SL(2,\mathbb{Z})$ which is an ineffective orbifold. But it is also fine to forget the trivial action of $-\mathrm{Id}_{2\times 2}\in SL(2,\mathbb{Z})$ and simply use the effective orbifold $\mathbb{H}/PSL(2,\mathbb{Z})$. Is there any reason for using $SL(2,\mathbb{Z})$ instead of $PSL(2,\mathbb{Z})$? The latter seems to make the story much simpler and does not lose essential information.

Answer 1

Maybe this is a silly question. One reason for the preference to $SL(2,\mathbb{Z})$ might just be it ''naturally'' arises in the moduli problem of elliptic curves, whereas $PSL(2,\mathbb{Z})$ is more ''artificial''. In the view point of Teichmüller theory, $\mathbb{H}$ is the Teichmüller space of the punctured torus, and $SL(2,\mathbb{Z})$ is its mapping class group. That global $\mathbb{Z}/2$ action also reflects the fact that every elliptic curve has a copy of $\mathbb{Z}/2$ in its automorphism group, induced by $z\mapsto -z$.