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arXiv · 2410.09475

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

Nataniel Marquis
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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.

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Nataniel Marquis. 2024-10-12. \'Equivalences de Fontaine multivariables Lubin-Tate et plectiques pour un corps local $p$-adique. https://arxiv.org/abs/2410.09475

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