# Computing Isomorphism Numbers of F-crystals Using the Level Torsions

Available at Journal of Number Theory, 2012

Abstract: The isomorphism number of an F-crystal (M, φ) over an algebraically closed field of positive characteristic is the smallest non-negative integer nM such that the nM-th level truncation of (M, φ) determines the isomorphism class of (M, φ). When (M, φ) is isoclinic, namely it has a unique Newton slope λ, we provide an efficiently computable upper bound for nM in terms of λ and the Hodge slopes of (M, φ). This is achieved by providing an upper bound for 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 interest (such as isoclinic F-crystals of K3 type).