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The following is what Mazur wrote on page 91 of his paper, Modular curves and the Eisenstein ideal, published in Publ. IHES in 1977, DOI: 10.1007/BF02684339 (freely available at eudml):

Let $m$ be an integer. Let $J[m]_{/\mathbb{Z}}$ denote the scheme-theoretic kernel of multiplication by $m$ in the Neron model of $J_{/\mathbb{Z}}$. Since $J$ is semi-stable, $J[m]_{/\mathbb{Z}}$ is a quasi-finite flat group scheme, whose restriction to $S'=\text{Spec} \mathbb{Z}[1/N]$ is finite and flat.

Here, $J=J_0(N)$ with $N$ prime.

I have two questions:

  1. It seems that his claim is still true when $N$ is squarefree because $J$ is still semi-stable. But I don't know why. Can someone explain this?

  2. Suppose that $N$ is not squarefree, so $J$ does not have semistable reduction at a prime whose square divides $N$. Is the same statement true if $\text{gcd}(m, N)$ is squarefree? If not, is it true that if we assume that $\text{gcd}(m, N)=1$?

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  • $\begingroup$ @MartinSleziak thanks for providing a link where you can get for an expensive price a scan of a 1977 paper, which you can get freely (and legally) here numdam.org/issues/PMIHES_1977__47_ $\endgroup$ – YCor Nov 23 '17 at 13:52
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    $\begingroup$ @YCor In fact, if you look at my edit, you can see that I have included both doi (with link) and eudml link (Which is free and links to the same copy as your comment. I have even explicitly mentioned this in the edit summary.) Of course, you are by far more experienced MO user than me, feel free to make further improvements to the post, if needed. $\endgroup$ – Martin Sleziak Nov 23 '17 at 13:54
  • $\begingroup$ Oh, indeed. The problem seems to be that the DOI points to the paying site. $\endgroup$ – YCor Nov 23 '17 at 13:55
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    $\begingroup$ @YCor I understand that. Still, some users might have a subscription, so they can get (possibly) a better copy from the doi-link. I considered adding only eudml, but I opted to add also doi, since doi link should be stable. Of course, if you think including only eudml link or some other link is preferrable, feel free to edit the post. Here is a link to somewhat related discussion on meta: edits with links to material under restricted access. $\endgroup$ – Martin Sleziak Nov 23 '17 at 13:59

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