Minimal $F$-crystals and isomorphism numbers of isosimple $F$-crystals

TL;DR

This paper extends the concept of minimal $p$-divisible groups to $F$-crystals, providing structural insights, explicit formulas, and bounds on isomorphism numbers for isosimple $F$-crystals in positive characteristic.

Contribution

It introduces the notion of minimal $F$-crystals, derives their structural properties, and establishes bounds on their isomorphism numbers based on ranks and slopes.

Findings

Explicit formula for Frobenius endomorphism of minimal $F$-crystals.

Definition of minimal height as an invariant for $F$-crystals.

Upper bounds on isomorphism numbers in terms of ranks and slopes.

Abstract

In this paper we generalize minimal $p$-divisible groups defined by Oort to $F$-crystal over an algebraically closed field of positive characteristic. We prove a structural theorem and give an explicit formula of the Frobenius endomorphism of the isosimple minimal $F$-crystals that are the building blocks of minimal $F$-crystals. We then define an invariant called the minimal height for $F$-crystals using minimal $F$-crystals and give an upper bound of the isomorphism numbers of isosimple $F$-crystals in terms of their ranks, Hodge slopes and Newton slopes.