Dimensions of automorphism group schemes of finite level truncation of F-cyclic F-crystals
Transactions of the American Mathematical Society, 2021
Abstract: Let Mπ be an F-cyclic F-crystal over an algebraically closed field defined by a permutation π and a set of prescribed Hodge slopes. We prove combinatorial formulas for the dimension γMπ(m) of the automorphism group scheme of Mπ at finite level m and the number of connected components of the endomorphism group scheme of Mπ at finite level m. As an application, we show that if Mπ is a nonordinary Dieudonné module defined by a cycle π, then γMπ(m+1) - γMπ(m) < γMπ(m) - γMπ(m-1) for all 1 ≤ m ≤ nMπ where nMπ is the isomorphism number of Mπ.