Reconstructing $p$-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 $n_D$ 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 Dieudonn\'e module of $D$ we associate a non-negative integer $\ell_D$ which is a computable upper bound of $n_D$. If $D$ is a product $\prod_{i\in I} D_i$ of isoclinic $p$-divisible groups, we show that $n_D=\ell_D$; if the set $I$ has at least two elements we also show that $n_D\le\max\{1,n_{D_i},n_{D_i}+n_{D_j}-1|i,j\in I, j\neq i\}$. We show that we have $n_D\Le 1$ if and only if $\ell_D\Le 1$; 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 compute explicitly $n_D$. Many results are proved in the general context of latticed $F$-isocrystals with a (certain) group over $k$.