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

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.