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

Let $\mathrm{\Delta }$ 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 ${\prod }_{\alpha \in \mathrm{\Delta }}{\mathcal{G}}_{{\mathbb{F}}_q(\text{\hspace{0.17em}⁣}(X)\text{\hspace{0.17em}⁣})}$ to multivariable $\phi$-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 ${\prod }_{\alpha \in \mathrm{\Delta }}{\mathcal{G}}_K$, where $K\mathrm{\mid }{\mathbb{Q}}_p$ is a finite extension. We also obtain a plectic Fontaine equivalence and two equivalences for the subgroup ${\mathcal{G}}_{K,\mathrm{g}\mathrm{l}\mathrm{e}\mathrm{c}}$ of the plectic Galois group.