Dimensions of group schemes of automorphisms of truncated Barsotti--Tate groups

Abstract

Let $D$ be a $p$-divisible group over an algebraically closed field $k$ of characteristic $p>0$. Let $n_D$ be the smallest non-negative integer such that $D$ is determined by $D[p^{n_D}]$ within the class of $p$-divisible groups over $k$ of the same codimension $c$ and dimension $d$ as $D$. We study $n_D$, lifts of $D[p^m]$ to truncated Barsotti--Tate groups of level $m+1$ over $k$, and the numbers ${\gamma }_D(i):=\dim (Aut(D[p^i]))$. We show that $n_D\le cd$, $({\gamma }_D(i+1)-{\gamma }_D(i){)}_{i\in \mathbb{N}}$ is a decreasing sequence in $\mathbb{N}$, for $cd>0$ we have ${\gamma }_D(1)<{\gamma }_D(2)<...<{\gamma }_D(n_D)$, and for $m\in \{1,...,n_D-1\}$ there exists an infinite set of truncated Barsotti--Tate groups of level $m+1$ which are pairwise non-isomorphic and lift $D[p^m]$. Different generalizations to $p$-divisible groups with a smooth integral group scheme in the crystalline context are also proved.

52 pages. Final version as close to the galley proofs as possible. To appear in IMRN

References in corpus (6)

• Crystalline boundedness principle
• Reconstructing $p$-divisible groups from their truncations of small level
• Discrete invariants of varieties in positive characteristic
• Minimal truncations of supersingular p-divisible groups
• Specialization of $F$-Zips
• Level m stratifications of versal deformations of p-divisible groups

Cited by in corpus (2)

• Subtle Invariants of $F$-crystals
• Canonical Barsotti-Tate Groups of Finite Level