# Finite flat group schemes over $\mathbb{Z}$ that are extensions of $\mu_p$ by $\mathbb{Z}/p\mathbb{Z}$

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.

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 $$\mathbb Z / p \mathbb Z$$.
• 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. – Nulhomologous Sep 9 at 0:01
• 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... – Nulhomologous Sep 9 at 15:50