Descent of morphisms of overconvergent F-crystals

TL;DR

This paper proves a restricted version of Tsuzuki's conjecture on the full faithfulness of the embedding of overconvergent F-crystals into convergent crystals, focusing on potentially semistable crystals and those with limited slopes.

Contribution

It establishes the conjecture for potentially semistable crystals and crystals with at most two slopes, extending previous partial results.

Findings

Full faithfulness proven for potentially semistable crystals

Full faithfulness proven for crystals with at most two slopes

Extends de Jong's partial results on the conjecture

Abstract

Tsuzuki has conjectured that for crystals with Frobenius and connection over a local field k((t)), the embedding of the category of overconvergent crystals into the category of convergent crystals is fully faithful. We prove Tsuzuki's conjecture restricted to the subcategory of potentially semistable (or quasi-unipotent) crystals, following de Jong's proof of a slightly weaker result. We also prove Tsuzuki's conjecture restricted to crystals with at most two distinct slopes.