Computing isomorphism numbers of F-crystals by using level torsions

TL;DR

This paper introduces an efficient method to compute the isomorphism number of F-crystals over algebraically closed fields of positive characteristic, especially for isoclinic cases, using level torsions and Hodge slopes.

Contribution

It provides a new upper bound for the isomorphism number of isoclinic F-crystals based on level torsions and Hodge slopes, with proven optimality in key cases.

Findings

Derived an upper bound for the isomorphism number in terms of Hodge slopes and Newton slopes.

Proved the upper bound is optimal for many isoclinic F-crystals, including those of K3 type.

Established a practical method for computing isomorphism numbers of F-crystals.

Abstract

The isomorphism number of an $F$-crystal $(M, \phi)$ 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 $(M, \phi)$ determines the isomorphism class of $(M, \phi)$. When $(M, \phi)$ is isoclinic, namely it has a unique Newton slopes $\lambda$, we provide an efficiently computable upper bound of $n_M$ in terms of the Hodge slopes of $(M, \phi)$ and $\lambda$. This is achieved by providing an upper bound of the level torsion of $(M, \phi)$ 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).