In the paper of Mazur, Modular curves and the Eisenstein ideal (1977), he showed the following (at page 57):

Lemma (3.4) (Criterion for constancy) Let $G$ be an etale admissible $p$-group over $\text{Spec} (\mathbb{Z}[1/N])$. If $N \not\equiv 1 \mod p$, then $G$ is constant. In general, $G$ is constant if and only if there is a prime number $\ell \neq N$ such that:

a) $\ell$ is not a $p$-th power modulo $N$;

b) The action of $\varphi_\ell$ in the Galois representation of $G$ is trivial.

Here, $p$ and $N$ are distinct prime numbers, and $\varphi_\ell$ denotes the Frobenius morphism.

What happens if $N$ is no longer a prime?

For instance, let $N$ be a positive integer not divisible by an odd prime number $p$. Let $G$ be an etale admissible $p$-group over $\text{Spec} (\mathbb{Z}[1/N])$. Suppose that $\varphi_\ell$ acts trivially on $G$ for all prime numbers $\ell$ such that $\ell \not\equiv 1 \mod p$ and $\ell \nmid Np$. Is then $G$ constant?

  • $\begingroup$ What does "admissible" mean in this context? $\endgroup$ – Joël Nov 7 '17 at 19:09
  • $\begingroup$ @Joël : Here, ''admissible'' means that it has a filtration by finite flat subgroup schemes such that the successive quotients are $\text{Spec}(\mathbb{Z}[1/N])$-isomorphic to one of the two group schemes: $\mathbb{Z}/{p\mathbb{Z}}$ or $\mu_p$. $\endgroup$ – user116950 Nov 8 '17 at 0:45
  • $\begingroup$ In addition to the doi-link in the post (which might be paywalled for some users), here is a free eudml link. $\endgroup$ – Martin Sleziak Nov 23 '17 at 14:24

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