# Reconstructing $p$-divisible groups from their truncations of small level

### Adrian Vasiu

Binghamton University, USA

## Abstract

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 $C[p_{n_{D}}]$ is isomorphic to $D[p_{n_{D}}]$, 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 $n_{D}≤max{1,n_{D_{i}},n_{D_{i}} +n_{D_{j}} −1∣i, j∈I, j=i}$. We show that we have $n_{D}≤1$ 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 $k$.

## Cite this article

Adrian Vasiu, Reconstructing $p$-divisible groups from their truncations of small level. Comment. Math. Helv. 85 (2010), no. 1, pp. 165–202

DOI 10.4171/CMH/192