Computing isomorphism numbers of F-crystals by using level torsions

Abstract

The isomorphism number of an $F$-crystal $(M, ϕ)$ over an algebraically closed field of positive characteristic is the smallest non-negative integer $n_M$ such that the $n_M$-th level truncation of $(M, ϕ)$ determines the isomorphism class of $(M, ϕ)$. When $(M, ϕ)$ is isoclinic, namely it has a unique Newton slopes $λ$, we provide an efficiently computable upper bound of $n_M$ in terms of the Hodge slopes of $(M, ϕ)$ and $λ$. This is achieved by providing an upper bound of the level torsion of $(M, ϕ)$ introduced by Vasiu. We also check that this upper bound is optimal for many families of isoclinic $F$-crystals that are of special interests (such as isoclinic $F$-crystals of K3 type).