Description
In 1991, J.-M. Fontaine proved an equivalence between continuous representations of $G_{\mathbb{Q}_p}$ of finite type over $\mathbb{Z}_p$ and the category of étale $(\varphi,\Gamma)$-modules over the ring of functions on a ghost circle. This equivalence was crucially used by P. Colmez to construct representations of $G_{\mathbb{Q}_p}$ from representations of $GL_2(\mathbb{Q}_p)$. The first part of this talk will give some rough ideas of these constructions, then motivate how the multivariable (resp. Lubin-Tate) variant should be better suited for higher-dimensional $\mathbb{Q}_p$-groups (resp. for $K$-groups).
Such a multivariable Fontaine equivalence was proved by the work of Z\'abr\'adi and of Carter-Kedlaya-Z\'abr\'adi, linking multivariable cyclotomic $(\varphi,\Gamma)$-modules to representations of products of $G_{\mathbb{Q}_p}$. I will sketch how to adapt their work to obtain a multivariable Lubin-Tate Fontaine equivalence, namely, for a finite set $\Delta$, an equivalence between continuous representations of $\prod_{\Delta} \mathcal{G}_K$ and categories called the topological étale $(\Phi_{\Delta, q}\times \Gamma_{\Delta,K,\lt})$-modules over $\mathcal{O}_{\mathcal{E}_{K,\Delta}}$ with projective dévissage for various topologies. The projective dévissage condition, useful in the proof, happens to be automatic in our situation, but it obtains some structures on the underlying modules. If time allows, I will discuss how the previous multivariable setting appears from $GL_2(K)$, suggesting a plectic phenomenon.