Let $\mathcal{M}_{1, 1}$ be the stack of elliptic curves. Its coarse moduli space is $\mathbb{A}^1_{\mathbb{Z}}$ with the map $\mathcal{M}_{1, 1} \rightarrow \mathbb{A}^1_{\mathbb{Z}}$ given by the $j$-invariant. I have heard that this map is ramified (i.e. not etale) at points of $\mathbb{A}^1_{\mathbb{Z}}$ at which $j = 0$ or $j = 1728$. How does one prove this?

I know that these are precisely the points where the automorphism group jumps, but I don't see how to use this. I am familiar with the argument that proves that $\mathcal{M}_{1, 1} \rightarrow \mathbb{A}^1_{\mathbb{Z}}$ is etale away from $j = 0$ and $j = 1728$, the essential point being that the automorphism functor is the etale group $\mathbb{Z}/2\mathbb{Z}$ on this locus.

your:) ) definition of "étale" and of "ramified"? mathoverflow.net/questions/224124/… $\endgroup$ – Qfwfq Nov 21 '15 at 17:28