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Following the notations and notions developped in Katz and Mazur's book "The arithmetic moduli of elliptic curves", we denote $\mathbf{(Ell)}$ the category whose objects are elliptic curves $E/S$ over variable base-schemes, and whose morphisms are Cartesian squares. A moduli problem is then simply a contravariant functor $\mathbf{(Ell)}\rightarrow \mathbf{(Set)}$. We say that a moduli problem $\mathcal{P}$ is étale if for any elliptic curve $E/S$, the functor $\mathbf{(Sch/S)}\rightarrow \mathbf{(Set)}$ sending an $S$-scheme $T$ to $\mathcal{P}(E_T/T)$ is representable by an étale $S$-scheme $\mathcal{P}_{E/S}$.

Let $N\geq 1$ be an integer and let $\Gamma(N)$ denote the "naive" full level $N$ moduli problem, which takes an elliptic curve $E/S$ and sends it to the set of $S$-group-schemes isomorphisms $$\phi:\left(\mathbb{Z}/N\mathbb{Z}\right)_S^2\xrightarrow{\sim} E[N]$$ Question: Is $\Gamma(N)$ étale over $\mathbf{(Ell)}$?

Actually, I know that it is relatively representable (and even representable by an elliptic curve over a smooth affine curve over $\mathbb{Z}[1/N]$, usually denoted $Y(N)$). If $E/S$ is any elliptic curve, then the $S$-scheme $\mathcal{P}_{E/S}$ associated to $\Gamma(N)$ is finite over $S$, and étale when $N$ is invertible on $S$. Is it also the case when $N$ is not invertible on $S$?

In other words, I know that it is étale when seen as a moduli problem on the category $\mathbf{(Ell_{\mathbb{Z}[1/N]})}$ of elliptic curves where $N$ is invertible, but is it also étale over the whole of $\mathbf{(Ell)}$?

NB: For context, this is claimed in Katz and Mazur's book on page 110 (page 61 in the pdf), and it is used on page 121 (page 66 in the pdf) when they describe its representing object $\mathbb{E}/Y(N)$ as a modular family.

I thank you very much for you clarifications.

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  • $\begingroup$ @MarcPaul This comes from the fact that given an elliptic curve $E/S$ with a full level $N$ structure on it, $N$ must actually be invertible on $S$, right? I am also aware of this. However, I fail to see why, given an elliptic curve $E/S$ where $N$ is not invertible, then $\mathcal{P}_{E/S}\rightarrow S$ is étale. Even though there is no full level $N$ structure on $E/S$, $\mathcal{P}_{E/S}$ still exists. I think it is isomorphic as an $S$-scheme to $\operatorname{Isom}_{S\times Y(N)}(\operatorname{pr}_1^{\star}(E),\operatorname{pr}_2^{\star}(\mathbb{E}))$. But then...? $\endgroup$ – Suzet Aug 6 '18 at 10:12
  • $\begingroup$ Let me give a slightly imprecise explanation of what's going on here. Consider $N=p$ a prime number. Let $k$ be an algebraically closed field and let $E$ be an elliptic curve over $k$. If one adds level $p$ structure over $k$ to $E$, then one "trivializes" the group scheme $E[p]$ over $k$. Now, the "reason" that this gives an etale cover of the moduli problem is partly because, when $p\neq 0$ in $k$, the order of $E[p]$ is independent of $E$ (and equals $p^2$). But, if $p=0$ in $k$, then this is no longer true. Indeed, the order of $E[p]$ can be zero or $p$. $\endgroup$ – Ariyan Javanpeykar Aug 7 '18 at 8:03
  • $\begingroup$ Thank you very much for you clarifications @AriyanJavanpeykar. Could you explain a little more to what extent the order of $E[p]$ is related to the existence of an étale cover? I actually do not see, even intuitively, why your discussion about the order (on which I agree) is a "reason" why we have an étale cover. $\endgroup$ – Suzet Aug 7 '18 at 9:15
  • $\begingroup$ Let $n>0$, and consider the "forgetful" morphism of modular curves $Y(3n)\to Y(3)$ over $\mathbb{Q}$. Recall that the curve $Y(i)$ parametrizes pairs $(E,\phi)$, where $E$ is an elliptic curve and $\phi$ is a level $i$-structure on $E$, i.e., the choice of an isomorphism $\phi:E[i] \cong \mathbb{Z}/i\mathbb{Z}\times \mathbb{Z}/i\mathbb{Z}$. Now, if $Y(3n)\to Y(3)$ would want to be etale, then the cardinality of the fibre over a point in $Y(3)$ should be independent of that point.... $\endgroup$ – Ariyan Javanpeykar Aug 7 '18 at 19:02
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    $\begingroup$ Yes that's right. Probably, in characteristic $p$, adding level structure is finite flat over each Newton stratum, e.g., the ordinary locus. But I'm not really sure whether it is etale, because the group scheme $E[p^n]$ might be non-reduced. (The same amount of points in each fibre isn't always enough to guarantee etaleness. You also need the points to be actual points, instead of "thickened points".) $\endgroup$ – Ariyan Javanpeykar Aug 8 '18 at 6:53

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