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.

  • 6
    $\begingroup$ One way is to notice that at these points, the automorphisms act non-trivially on the tangent space to the stack. This implies that the differential of j is zero at these points, and therefore j is not etale. $\endgroup$ – t3suji Sep 22 '15 at 6:29
  • 1
    $\begingroup$ The map $\mathcal{M}_{1,1}\to\mathbb{A}^1$ is not representable. What's (your :) ) definition of "étale" and of "ramified"? mathoverflow.net/questions/224124/… $\endgroup$ – Qfwfq Nov 21 '15 at 17:28
  • $\begingroup$ @Qfwfq: The stack $\mathcal{M}_{1, 1}$ is DM, so etaleness of the map at a point $s \in \mathbb{A}^1$ is well-defined by requiring etaleness of the composed map $X \rightarrow \mathbb{A}^1$ at some (equivalently, any) preimage $x \in X$ of $s$ where $X \rightarrow \mathcal{M}_{1, 1}$ is an etale cover by a scheme. Then, a map is ramified over $s$ if it is not etale over $s$. $\endgroup$ – O-Ren Ishii Dec 24 '15 at 22:58

I would guess that the following argument should work.

By definition, the map $\mathbb{H} \to \mathcal{M}_{1,1} = [SL_2\mathbb{Z} \setminus \mathbb{H}]$ is unramified. However, the map $\mathbb{H} \to M_{1,1} = SL_2\mathbb{Z} \setminus \mathbb{H}$ is ramified precisely at those two points. Consequently, the ramification has to come from the map $\mathcal{M}_{1,1} \to M_{1,1}$.

  • 2
    $\begingroup$ This argument would work if the question were phrased in the language of stacks over complex analytic spaces, instead of schemes over the integers. If you replace the upper half-plane with the $SL_2(\mathbb{Z}/3\mathbb{Z})$-cover that comes from full level 3 structure, then I think the argument works more generally (but I certainly wouldn't say "by definition"). $\endgroup$ – S. Carnahan Sep 22 '15 at 6:29
  • $\begingroup$ Oh, silly me, I didn't notice the $\mathbb{Z}$ in the question... $\endgroup$ – Simon Rose Sep 22 '15 at 7:07
  • $\begingroup$ @S.Carnahan forgive my ignorance, but what if 3 is not invertible? $\endgroup$ – David Roberts Sep 22 '15 at 7:54
  • 2
    $\begingroup$ @DavidRoberts If 3 is not invertible, then the cover is not étale. In that case, you might as well calculate the $j$-invariant of a first-order deformation of the Artin-Schreier curve $y^2 = x^3 - x$. $\endgroup$ – S. Carnahan Sep 23 '15 at 9:00

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.