Unipotency and semistability of overconvergent F-crystals

TL;DR

This paper introduces and relates the concepts of semistability and potential semistability for overconvergent F-crystals over local fields, establishing their equivalence with unipotency notions and reformulating a key conjecture in the field.

Contribution

It defines new notions of semistability for overconvergent F-crystals and proves their equivalence with existing unipotency concepts, connecting to a major conjecture.

Findings

Semistability and potential semistability are equivalent to unipotency and quasi-unipotency.

Reformulation of the conjecture that all overconvergent crystals are quasi-unipotent.

Extension of de Jong's theorem to the logarithmic case for F-crystals.

Abstract

In this paper (part of the author's PhD thesis), we introduce the notions of semistability and potential semistability of overconvergent F-crystals over an equal characteristic local field. We establish their equivalence with the notions of unipotency and quasi-unipotency given by Crew, and to recast the conjecture that every overconvergent crystal is quasi-unipotent in terms of potential semistability. Consequences, including an extension of de Jong's extension theorem for F-crystals to the logarithmic case, will appear in a subsequent paper.