Potentially good reduction of Barsotti–Tate groups

Let R be a complete discrete valuation ring of mixed characteristic (0,p) with perfect residue field, K the fraction field of R. Suppose G is a Barsotti–Tate group (p-divisible group) defined over K which acquires good reduction over a finite extension K′ of K. We prove that there exists a constant c⩾2 which depends on the absolute ramification index e(K′/Qp) and the height of G such that G has good reduction over K if and only if G[pc] can be extended to a finite flat group scheme over R. For abelian varieties with potentially good reduction, this result generalizes Grothendieck's “p-adic Néron–Ogg–Shafarevich criterion” to finite level. We use methods that can be generalized to study semi-stable p-adic Galois representations with general Hodge–Tate weights, and in particular leads to a proof of a conjecture of Fontaine and gives a constant c as above that is independent of the height of G.