Logarithmic Crystalline Representations

TL;DR

This work explicitly constructs logarithmic Fontaine-Faltings modules and logarithmic crystalline representations, extending Faltings' $p$-adic Riemann–Hilbert framework to logarithmic settings. It develops both local and global $\mathbb{D}^{\log}$-functors, demonstrates their compatibility with the non-logarithmic theory on the open complement, and proves that $\mathbb{D}^{\log}$ is fully faithful. The approach relies on gluing local data across Frobenius lifts, towers that resolve logarithmic poles, and period-ring techniques to realize Galois representations from filtered de Rham data with logarithmic poles. The results yield a logarithmic analogue of crystalline representations, providing a robust link between logarithmic Fontaine-Faltings modules and representations of the étale fundamental group of the logarithmic generic fiber, with algebraization in the proper case.

Abstract

In 1989, Faltings proved the comparison theorem between étale cohomology and crystalline cohomology by studying Fontaine-Faltings modules and crystalline representations. In his paper, he mentioned these modules and representations can be extended to the logarithmic context, but without detail. This note aims to explicitly present the construction of logarithmic Fontaine-Faltings modules and logarithmic crystalline representations.

Theorems & Definitions (19)

• Theorem 1: \ref{['algebraicalize']}
• Definition 1
• Remark 1
• Proposition 1
• proof
• Theorem 2.1: Logarithmic version of Faltings' gluing theorem
• proof
• Remark 2
• Lemma 1
• proof