Fix a base ring $R$. Is the rigidification of the modular "curve" $X_0(N)$ in the sense of Abramovich-Olsson-Vistoli representable iff $N\geq 5$ and $N$ is $0, 2, 3, 6, 8, 11 (\mathrm{mod}\: 12)$? There is no assumption on invertibility of $2$ and $3$ in $R$. The statement seems to be implied in the Eisenstein ideal paper but it is not explicitly said.

  • $\begingroup$ I am (maybe was) under the impression that for $\Gamma_0$-level on its own, this moduli problem was never representable due to the automorphism you get by $-1$. I've always either seen people talk about a stack in this case, or have some additional level structure to rigidify the problem. $\endgroup$ – Leray Jenkins Aug 5 at 18:24

Firstly, I believe your real question is not what you wrote. Indeed, the “modular curve” $X_0(N)$ is a scheme, so it is automatically "representable". What I believe you mean to ask about is the representability of a certain moduli problem/moduli stack. For this question, you need to first define the moduli problem/stack, since there are multiple moduli problems/stacks whose coarse moduli schemes are $X_0(N)$. The usual moduli problem for elliptic curves with $\Gamma_0(N)$-structures is one, but as you mentioned so is the rigidification of this moduli problem.

From now on I'll assume that what you mean to ask is if the moduli stack classifying generalized elliptic curves with cyclic subgroups of order $N$ (in the sense of Drinfeld in bad characteristics) is representable after rigidification by removing the $\mu_2$ in every automorphism group.

Usually the main obstruction to representability is the existence of automorphisms of the objects being classified. At least in the case of moduli problems finite etale over the moduli stack of elliptic curves, this is the only obstruction (c.f. for example section 4.7 in Katz-Mazur's book), so you're reduced to checking if the objects of your rigidified stack are in fact rigid (here I assume by "rigidification" you are removing the $\mu_2$ from each automorphism group).

Thus, at least in every (tame) characteristic $p\nmid N$ (these are the characteristics where the moduli problem is finite etale over the moduli stack of elliptic curves), your question is reduced to the problem of figuring out the automorphism groups of the objects of your rigidified stack. For generalized elliptic curves with automorphism group equal to $\mu_2$ (this includes all the true generalized elliptic curves at the cusps), this is trivially satisfied, so you are reduced to asking about the case of elliptic curves with larger automorphism groups. In characteristic 0 (or tame characteristics not equal to 2 or 3) this boils down to the question: Can an automorphism of order 3 or 4 stabilize a cyclic subgroup of order $N$? I don't have the time to figure out the answer right now but I'm pretty sure the answer is no - You could perhaps look up the proof for why $X_1(N)$ is rigid for $N\ge 4$). If $p = 2$ or $p=3$ (still assuming $p\nmid N$), you get bigger automorphism groups, but at least Deuring's lifting theorem allows you to lift individual automorphisms to characteristic 0, and so you can reduce to the characteristic 0 situation.

I don't know the answer for bad characteristics - in the smooth locus this should be pretty straightforwards to answer (though I'd need to take another look at Katz-Mazur to be certain). At the cusps, it becomes far from obvious what the correct notion of a “$\Gamma_0(N)$-structure” should be, at least for non-squarefree $N$. For this you'd likely need to take a look at this lovely paper of Cesnavicius.


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