Base change of twisted Fontaine-Faltings modules and Twisted Higgs-de Rham flows over very ramified valuation rings

TL;DR

This paper establishes a correspondence between stable twisted Higgs bundles and crystalline representations over very ramified valuation rings, extending previous results to a stronger version.

Contribution

It proves a stronger version of a previous theorem, linking stable twisted Higgs bundles to crystalline representations over ramified valuation rings.

Findings

Stable twisted Higgs bundles correspond to crystalline representations.

The representations are absolutely irreducible when restricted to the geometric fundamental group.

The result applies to smooth log schemes over ramified valuation rings.

Abstract

In this short notes, we prove a stronger version of Theorem 0.6 in our previous paper arXiv:1709.01485: Given a smooth log scheme $(\mathcal{X} \supset \mathcal{D})_{W(\mathbb{F}_q)}$, each stable twisted $f$-periodic logarithmic Higgs bundle $(E,\theta)$ over the closed fiber $(X \supset D)_{\mathbb{F}_q}$ will correspond to a $\mathrm{PGL}_r(\mathbb{F}_{p^f})$-crystalline representation of $\pi_1((\mathcal{X} \setminus \mathcal{D})_{W(\mathbb{F}_q)[\frac{1}{p}]})$ such that its restriction to the geometric fundamental group is absolutely irreducible.