Reconstructing <var>p</var>-divisible groups from their truncations of small level

Abstract

Let k be an algebraically closed field of characteristic p > 0. Let D be a p-divisible group over k. Let nD be the smallest non-negative integer for which the following statement holds: if C is a p-divisible group over k of the same codimension and dimension as D and such that C[pnD] is isomorphic to D[pnD], then C is isomorphic to D. To the Dieudonné module of D we associate a non-negative integer ℓD which is a computable upper bound of nD. If D is a product ∏i ∈ I Di of isoclinic p-divisible groups, we show that nD = ℓD; if the set I has at least two elements we also show that nD ≤ max{1, nDi, nDi + nDj − 1 | i, j ∈ I,  j ≠ i}. We show that we have nD ≤ 1 if and only if ℓD ≤ 1; this recovers the classification of minimal p-divisible groups obtained by Oort. If D is quasi-special, we prove the Traverso truncation conjecture for D. If D is F-cyclic, we explicitly compute nD. Many results are proved in the general context of latticed F-isocrystals with a (certain) group over k.