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

Abstract

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