Pointwise criteria of p-adic local systems

TL;DR

This paper establishes criteria for when p-adic local systems are crystalline or semi-stable based on their behavior at classical points, introducing a crystalline Riemann-Hilbert functor and applications like a semi-stable comparison theorem.

Contribution

It introduces a crystalline Riemann-Hilbert functor and provides pointwise criteria for p-adic local systems to be crystalline or semi-stable.

Findings

Characterization of crystalline and semi-stable local systems via classical points

Introduction of a crystalline Riemann-Hilbert functor

A semi-stable comparison theorem in the relative setting

Abstract

Given a Z_p-linear local system over a smooth rigid space, we show that it is crystalline (resp. semi-stable) with respect to any smooth (resp. semi-stable) integral model if and only if its restrictions at many classical points are crystalline (resp. semi-stable) representations. To this end, we introduce a crystalline Riemann--Hilbert functor, and give several applications, including a semi-stable comparison theorem in the relative setting.