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

Abstract

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