Suppose $X$ is a finite flat group scheme over $\mathbb Z$, killed by a prime number $p$ and such that there exists an extension as finite flat group schemes defined over $\mathbb Z$: $$0\to \mathbb{Z}/p\mathbb{Z}\to X \to \mu_p \to 1.$$

Question: Can we conclude that $X\cong \mathbb{Z}/p\mathbb{Z}\times \mu_p$ over $\mathbb{Z}$?

I know that the answer to this question is negative in general if you consider it over $\mathbb Q$, since you can take $X=E[7]$, the group scheme of $p$-torsion points of an elliptic curve with a $7$-torsion point defined over $\mathbb Q$, since we have such elliptic curves but no such curve with $E[7]\cong \mathbb{Z}/7\mathbb{Z}\times \mu_7$. Of course you can find examples easily for $p=2,3,5$, and probably for infinitely many prime numbers.

On the other hand, over the finite field $\mathbb{F}_p$ the answer is positive, since $\mu_p$ is connected and $\mathbb{Z}/p\mathbb{Z}$ étale, and one could use the connected-étale exact sequence of $X$ to get an splitting of the exact sequence above.

If the answer to the question is affirmative, I will be also interested for what other ring of integers the result is true. I suspect it should be related to the fact that $\mathbb{Q}$ has no unramified extensions.


It is proved in step 3 and 4 of section 3.4.3 in J-M. Fontaine. Il n’y a pas de variété abélienne sur Z. Invent. Math., 81(3):515– 538, 198 (using the ramification bound in that paper) that:

For $E=\mathbb Q$ and $\mathbb Q(\sqrt{-1})$, $\mathbb Q(\sqrt{-3})$, in the category of finite flat group schemes over $O_E$ killed by $p$, there is no non-trivial extension of $\mu_p$ by $\mathbb Z / p \mathbb Z$.

  • $\begingroup$ Many thanks for the reference. After reading it, it seems to me that Fontaine proved it for $E=\mathbb{Q}, \mathbb{Q}(\sqrt{-1}), \mathbb{Q}(\sqrt{-3})$ and $\mathbb{Q}(\sqrt{5})$ (it needs a more delicate argument for $p=3$ and this last field), not for a general number field... In the result it is under the hypothesis of his theorem 4. $\endgroup$ Sep 9 '20 at 0:01
  • $\begingroup$ @Nulhomologous You're right, I forgot to put the "small" restriction. But his method can be improved to give more results, see Abrashkin‘s works. $\endgroup$
    – sawdada
    Sep 9 '20 at 0:09
  • $\begingroup$ I am sorry to say that after reading in detail the proof by Fontaine, I must say that he only proves the result for $K=\mathbb{Q}$ and $p=3, 5, 7, 11, 13, 17$ (and for the other fields, for a more restrictive list of primes). This is because he needs his lemma 3.4.2., which in turn uses the tables by Diaz y Diaz only for that primes. I am not sure Abrashkin says something about this problem on some of his papers... $\endgroup$ Sep 9 '20 at 15:50
  • $\begingroup$ @Nulhomologous See Serre's Duke87 aghitza.org/pdf/translation-serre-duke.pdf 4.5 Group schemes of type (p, p) over Z, where he uses Fontaine's proof to prove the result for any p>=3. $\endgroup$
    – sawdada
    Sep 9 '20 at 17:23
  • $\begingroup$ I am surely confused, but Serre's proof of the result (lemma 3 in section 4.5) only says that Fontaine proved it and no other argument... $\endgroup$ Sep 10 '20 at 10:17

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