In Tate's famous paper about $p$-divisible groups, for anya prime number $p$ he askedasks whether there exist nontrivialexists a $p$-divisible groupsgroup $G$ over $\mathbb Z$, i.e. such that $p$-divisible groups which are$G$ is not a product of powersdirect sum of $\mu_{p^\infty}$ and $\mathbb Q_p/ \mathbb Z_p $.
In works of V. A. Abrashkin, he claims that there exist nontrivial $p$-divisible groups for $p=2$ and some irregular primesi. What about the general case? Do we knowe a lotdirect sum of primes $p$a constant group and a diagonalizable group. He calls such that every $p$-divisible group over $\mathbb Z$ is trivial?groups nontrival.
Motivation: On the other hand,This question has some good applications. For instance we know there is no abelian scheme over $\mathbb Z$, whose proofkey idea involves analyzing $p$-divisible groups for small $p$ using discriminant bound. In particular Tate's question is true for $p=3,5,7,11,13,17$ (see theorem $4$ in Fontaine's paper Il n'y a pas de variété abélienne sur $\mathbb Z$ for a proof).
Edit: One answer below suspectsFor the works ofnegative side, the case $p=2$ is bad but also good up to isogeny. In an old paper $p$-divisible group over $\mathbb Z$ by V. A. Abrashkin might contain errors, maybe a further reference is good. Anywayhe claims that there also exist nontrivial $p$-divisible groups for some small irregular primes, the key questionbut this seems to be wrong (see one answer below).
Odlyzko's discriminant bound is what we knownot good for general primelarge $p$. But time has passed for more than $30$ years, iswhat about the general case now? Is there any progress towards Tate's question? With the technique of mordern $p$-adic Hodge theory, maybe such question could be solved. Indeed, some generalizations of Fontaine's result for abelian schemes rely on the Fontaine-Laffaille theory.