\'Equivalences de Fontaine multivariables Lubin-Tate et plectiques pour un corps local $p$-adique

TL;DR

This paper develops multivariable Fontaine equivalences connecting Galois representations over local fields with multivariable -modules, extending classical p-adic Hodge theory to a multivariable and plectic setting.

Contribution

It introduces a multivariable Fontaine equivalence for Galois representations and -modules, generalizing existing theories to multivariable and plectic contexts.

Findings

Established a multivariable Fontaine equivalence for _q-representations.

Derived a multivariable Lubin-Tate Fontaine equivalence for _q-representations over local fields.

Obtained plectic Fontaine equivalence and related subgroup equivalences.

Abstract

Let $\Delta$ be a finite set. We adapt the techniques of Carter-Kedlaya-Z\'abr\'adi to obtain a multivariable Fontaine equivalence which relates continuous finite dimensional $\mathbb{F}_q$-representations of $\prod_{\alpha \in \Delta} \mathcal{G}_{\mathbb{F}_q(\!(X)\!)}$ to multivariable $\varphi$-modules over a $\mathbb{F}_q$-algebra which is a domain. From this, we deduce a multivariable Lubin-Tate Fontaine equivalence for continuous finite type $\mathcal{O}_K$-representations of $\prod_{\alpha \in \Delta} \mathcal{G}_K$, where $K|\mathbb{Q}_p$ is a finite extension. We also obtain a plectic Fontaine equivalence and two equivalences for the subgroup $\mathcal{G}_{K,\mathrm{glec}}$ of the plectic Galois group.