Structures de Hodge-Pink pour les $\varphi/\mathfrak S$-modules de Breuil et Kisin

TL;DR

This paper applies Fontaine's theory in equal characteristics to Breuil and Kisin's $S$-modules, providing a new elementary proof that weakly admissible modules are admissible, based on Hodge-Pink structures.

Contribution

It introduces Hodge-Pink structures to $S$-modules and offers a simplified proof of the Colmez-Fontaine theorem using these methods.

Findings

New proof of 'weakly admissible implies admissible' theorem

Application of Hodge-Pink structures to $S$-modules

Simplification of existing proofs in $p$-adic Hodge theory

Abstract

In this article, we apply the methods of our work on Fontaine's theory in equal characteristics to the $\varphi/\mathfrak S$-modules of Breuil and Kisin. Thanks to a previous article of Kisin, this yields a new and rather elementary proof of the theorem "weakly admissible implies admissible" of Colmez-Fontaine.