Cristallinity of rigid flat connections revisited

TL;DR

This paper extends the theory of Frobenius isocrystals and Fontaine-Laffaille modules to non-proper rigid flat connections using a novel $p$-adic approach, bypassing traditional Simpson correspondence methods.

Contribution

It introduces a new topological strategy to establish the existence of these structures in a broader non-proper setting, advancing the understanding of rigid flat connections.

Findings

Generalized the existence theorem to non-proper cases

Developed a $p$-adic proof avoiding Simpson correspondence

Provided new insights into the structure of flat connections

Abstract

We generalise a theorem on the existence of Frobenius isocrystal and Fontaine-Laffaille module structures on rigid flat connections to the non-proper setting. The proof is based on a new strategy of a point-set topological flavour, which allows us to produce a purely $p$-adic statement and thereby to avoid the classical Simpson correspondence.