Minimal slope conjecture of $F$-isocrystals

TL;DR

This paper proves the minimal slope conjecture for overconvergent $F$-isocrystals on curves and certain varieties, confirming that isomorphic minimal slope filtrations imply isomorphism of the isocrystals.

Contribution

It provides a proof of the minimal slope conjecture for specific classes of overconvergent $F$-isocrystals, advancing understanding in $p$-adic cohomology.

Findings

Confirmed the conjecture for overconvergent $F$-isocrystals on curves

Proved the conjecture for overconvergent $ar{Q}_p$-$F$-isocrystals on smooth varieties over finite fields

Established isomorphism of isocrystals from isomorphic minimal slope filtrations

Abstract

The minimal slope conjecture, which was proposed by K.Kedlaya, asserts that two irreducible overconvergent $F$-isocrystals on a smooth variety are isomorphic to each other if both minimal slope constitutions of slope filtrations are isomorphic to each other. We affirmatively solve the minimal slope conjecture for overconvergent $F$-isocrystals on curves and for overconvergent $\bar{\mathbb Q}_p$-$F$-isocrystals on smooth varieties over finite fields.