Parabolic log convergent isocrystals

TL;DR

This paper introduces parabolic log convergent isocrystals as a $p$-adic analogue of parabolic bundles, establishing equivalences between categories of $p$-adic representations and isocrystals, and relating log convergent isocrystals on stacks with parabolic structures.

Contribution

It defines parabolic log convergent isocrystals and proves categorical equivalences linking $p$-adic representations, fundamental groups, and isocrystals, extending classical results to the $p$-adic setting.

Findings

Establishes equivalence between $p$-adic representations and unit-root parabolic isocrystals.

Relates log convergent isocrystals on stacks to parabolic isocrystals.

Provides a $p$-adic interpretation of unit-rootness via semistability.

Abstract

In this paper, we introduce the notion of parabolic log convergent isocrystals on smooth varieties endowed with a simple normal crossing divisor, which is a kind of $p$-adic analogue of the notion of parabolic bundles on smooth varieties defined by Seshadri, Maruyama-Yokogawa, Iyer-Simpson, Borne. We prove that the equivalence between the category of $p$-adic representations of the fundamental group and the category of unit-root convergent $F$-isocrystals (proven by Crew) induces the equivalence between the category of $p$-adic representations of the tame fundamental group and the category of unit-root semisimply adjusted parabolic log convergent $F$-isocrystals. We also prove equivalences which relate categories of log convergent isocrystals on certain fine log algebraic stacks with some conditions and categories of adjusted parabolic log convergent isocrystals with some conditions. We also give an interpretation of unit-rootness in terms of the generic semistability with slope 0. Our result can be regarded as a $p$-adic analogue of the results of Seshadri, Mehta-Seshadri, Iyer-Simpson and Borne.