Équivalences de Fontaine multivariables Lubin-Tate et plectiques pour un corps local $p$-adique

Abstract

Let $Δ$ be a finite set. We adapt the techniques of Carter-Kedlaya-Zábrádi to obtain a multivariable Fontaine equivalence which relates continuous finite dimensional $\mathbb{F}_q$-representations of $\prod_{α\in Δ} \mathcal{G}_{\mathbb{F}_q(\!(X)\!)}$ to multivariable $φ$-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 $\prod_{α\in Δ} \mathcal{G}_K$, 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.