Commentarii Mathematici Helvetici

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Volume 85, Issue 1, 2010, pp. 165–202
DOI: 10.4171/CMH/192

Published online: 2009-12-23

Reconstructing p-divisible groups from their truncations of small level

Adrian Vasiu[1]

(1) Binghamton University, USA

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 ∈ IDi of isoclinic p-divisible groups, we show that nD = ℓD; if the set I has at least two elements we also show that nDmax{1, nDi, nDi + nDj − 1 | i, jI,  ji}. 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.

Keywords: p-divisible groups, F-crystals, algebras, affine group schemes

Vasiu Adrian: Reconstructing p-divisible groups from their truncations of small level. Comment. Math. Helv. 85 (2010), 165-202. doi: 10.4171/CMH/192