Canonical Barsotti-Tate Groups of Finite Level

Abstract

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $c,d\in \mathbb{N}$ be such that $h=c+d>0$. Let $H$ be a $p$-divisible group of codimension $c$ and dimension $d$ over $k$. For $m\in {\mathbb{N}}^{\ast }$ let $H[p^m]=\ker ([p^m]:H\to H)$. It is a finite commutative group scheme over $k$ of $p$ power order, called a Barsotti-Tate group of level $m$. We study a particular type of $p$-divisible groups $H_{\pi }$, where $\pi$ is a permutation on the set $\{1,2,\dots ,h\}$. Let $(M,{\phi }_{\pi })$ be the Dieudonné module of $H_{\pi }$. Each $H_{\pi }$ is uniquely determined by $H_{\pi }[p]$ and by the fact that there exists a maximal torus $T$ of $G{L}_M$ whose Lie algebra is normalized by ${\phi }_{\pi }$ in a natural way. Moreover, if $H$ is a $p$-divisible group of codimension $c$ and dimension $d$ over $k$, then $H[p]\cong H_{\pi }[p]$ for some permutation $\pi$. We call these $H_{\pi }$ canonical lifts of Barsotti-Tate groups of level $1$. We obtain new formulas of combinatorial nature for the dimension of $\mathbf{Aut}(H_{\pi }[p^m])$ and for the number of connected components of $\mathbf{End}(H_{\pi }[p^m])$.

arXiv admin note: text overlap with arXiv:math/0608032 by other authors

References in corpus (2)

• Crystalline boundedness principle
• Minimal truncations of supersingular p-divisible groups