Dimensions of automorphism group schemes of finite level truncations of $F$-cyclic $F$-crystals

Abstract

Let ${\mathcal{M}}_{\pi }$ be an $F$-cyclic $F$-crystal ${\mathcal{M}}_{\pi }$ over an algebraically closed field defined by a permutation $\pi$ and a set of prescribed Hodge slopes. We prove combinatorial formulas for the dimension ${\gamma }_{{\mathcal{M}}_{\pi }}(m)$ of the automorphism group scheme of ${\mathcal{M}}_{\pi }$ at finite level $m$ and the number of connected components of the endomorphism group scheme of ${\mathcal{M}}_{\pi }$ at finite level $m$. As an application, we show that if ${\mathcal{M}}_{\pi }$ is a nonordinary Dieudonné module defined by a cycle $\pi$, then ${\gamma }_{{\mathcal{M}}_{\pi }}(m+1)-{\gamma }_{{\mathcal{M}}_{\pi }}(m)<{\gamma }_{{\mathcal{M}}_{\pi }}(m)-{\gamma }_{{\mathcal{M}}_{\pi }}(m-1)$ for all $1\le m\le n_{{\mathcal{M}}_{\pi }}$, where $n_{{\mathcal{M}}_{\pi }}$ is the isomorphism number of ${\mathcal{M}}_{\pi }$.