Parabolic Crystalline Representations

TL;DR

This paper develops a parabolic extension of crystalline representation theory, introducing new modules and functors, and uses Higgs-de Rham flows to construct numerous crystalline representations, generalizing previous twisted cases.

Contribution

It introduces a parabolic framework for Fontaine-Faltings modules and Faltings' D-functor, expanding the scope of crystalline representation construction.

Findings

Established a parabolic version of crystalline representation theory.

Constructed infinitely many crystalline representations using Higgs-de Rham flows.

Unified twisted cases as special instances of the new theory.

Abstract

The theory of crystalline representations was established by Fontaine and Laffaille, Faltings, and others. In this paper, we develop a parabolic version of this theory. The key point is the construction of the parabolic version of Fontaine-Faltings modules and Faltings' $\mathbb D$-functor. The theory of Higgs-de Rham flows can be used to efficiently construct crystalline representations. We have established a parabolic version and utilized it to construct infinitely many crystalline representations. The twisted versions discussed in Sun, Yang, and Zuo's work can be seen as a special case, where the parabolic weights are equal at every infinity point.