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One can classify (commutative) finite flat group schemes (with order of $p$-powers) over $\mathbb Z_p$ using semi-linear algebraic datas such as Breuil-Kisin modules. And we can fix the special fiber and form a moduli space of deformations.

How about the global story? Let $K$ be a number field, can we classify all finite flat group schemes over $O_K$ ? Can we classify finite flat group schemes over a smooth projective curve over a finite field?

The etale ones can be described in terms of Galois representations. There is an exact sequence between global Brauer group and local ones in class field theory, can we form a local-global story here similarly?

Related question: Nontrivial p-divisible groups over $\mathbb Z$ for general prime $p$ .

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    $\begingroup$ For groups of order p (prime) over the rings on integers in number fields, this is worked out in the last section of Tate, John; Oort, Frans. Group schemes of prime order. Ann. Sci. École Norm. Sup. (4) 3 1970 1--21. MR0265368 $\endgroup$
    – anon
    Jun 9, 2019 at 14:32
  • $\begingroup$ There has been some work done by Vasiu on this, see projecteuclid.org/euclid.ant/1510842466. $\endgroup$
    – Bombyx mori
    Jun 15, 2019 at 6:16

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