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

Question

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.

Answer 1

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 $\mathbb Z / p \mathbb Z$ for $p=3, 5, 7, 11, 13, 17$.

Answer 2

The connected–étale sequence holds over $\mathbb{Z}_p$ (see for example Tate's "Finite flat group schemes" article), thus $X$ splits over $\mathbb{Z}_p$.

Combining the above with the fact $X$ is étale over $\mathbb{Z}[\frac{1}{p}]$, we find $\mathbb{Q}(X) = \mathbb{Q}(\mu_p)$. As $X$ is killed by $p$ over $\mathbb{Z}_p$, it follows by flatness that $X$ is killed by $p$ over $\mathbb{Z}$, and in particular over $\mathbb{Z}[\frac{1}{p}]$. These two statements give us that $X$ is split over $\mathbb{Z}[\frac{1}{p}]$ (see Section 3.6 in Tate's article).

We now need some gluing argument to conclude $X$ is split over $\mathbb{Z}$. This can be done using Corollary 2.4 in Schoof's "Abelian varieties over cyclotomic fields with good reduction everywhere" paper.