# Dimensions of Automorphism Group Schemes of Finite Level Truncation of `F`-Cyclic `F`-Crystals

Available at *Transactions of the American Mathematical Society*, 2021

Abstract: Let `M _{π}` 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

`γ`of the automorphism group scheme of

_{Mπ}(m)`M`at finite level

_{π}`m`and the number of connected components of the endomorphism group scheme of

`M`at finite level

_{π}`m`. As an application, we show that if

`M`is a nonordinary DieudonnÃ© module defined by a cycle

_{π}`π`, then

`γ`for all

_{Mπ}(m+1) - γ_{Mπ}(m) < γ_{Mπ}(m) - γ_{Mπ}(m-1)`1 ≤ m ≤ n`where

_{Mπ}`n`is the isomorphism number of

_{Mπ}`M`.

_{π}Recommended citation: Zeyu Ding, Xiao Xiao, Dimensions of automorphism group schemes of finite level truncations of `F`-cyclic `F`-crystals, Transactions of American Mathematical Society, No. 374, Pages 269 â€“ 302, 2021.