# 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

^{[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 `n`_{D} 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`[`p`^{nD}] is isomorphic to `D`[`p`^{nD}],
then `C` is isomorphic to `D`. To the Dieudonné module of `D` we associate a non-negative integer ℓ_{D} which is
a computable upper bound of `n`_{D}. If `D` is a product ∏_{i ∈ I} `D`_{i} of isoclinic `p`-divisible groups, we
show that `n`_{D} = ℓ_{D}; if the set `I` has at least two elements we also show that `n`_{D} ≤ `n`_{Di}, `n`_{Di} + `n`_{Dj} − 1 | `i`, `j` ∈ `I`, `j` ≠ `i`}`n`_{D} ≤ 1 if and only if ℓ_{D} ≤ 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 `n`_{D}. 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