Logarithmic Dieudonn\'e theory and overconvergent extensions

TL;DR

This paper proves a key property of overconvergent $F$-isocrystals related to Crew’s parabolicity conjecture using logarithmic Dieudonné theory, establishing an equivalence with $p$-divisible groups over curves.

Contribution

It provides an alternative proof of a slope property for certain $F$-isocrystals and links $p$-divisible groups with overconvergent $F$-isocrystals via logarithmic Dieudonné theory.

Findings

Established a slope property for $ abla$-modules in the context of Crew’s conjecture.

Proved an equivalence between potentially semi-stable $p$-divisible groups and overconvergent $F$-isocrystals on curves.

Extended the understanding of the relationship between $p$-divisible groups and overconvergent isocrystals.

Abstract

In the proof of Crew's parabolicity conjecture, we established a key property concerning the slopes of $\dagger$-hulls of $F$-isocrystals, extending a result of Tsuzuki. This article presents an alternative proof of this theorem for a specific class of $F$-isocrystals. The central ingredient is a local extension property for \'etale $p$-divisible subgroups. To relate $p$-divisible groups and overconvergent $F$-isocrystals, we employ logarithmic Dieudonn\'e theory, as introduced by Kato and further developed by Inoue. Over curves, this leads to an equivalence between the category of potentially semi-stable $p$-divisible groups and overconvergent $F$-isocrystals with slopes in the interval $[0,1]$.