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Multivariable Lubin–Tate Fontaine equivalence

January 31 at 13:3014:30 CET

Nataniel Marquis (IMJ Paris)

In 1991 J.-M. Fontaine proved an equivalence between continuons representations of $\mathcal{G}_{\mathbb{Q}_p}$ of finite type over $\mathbb{Z}_p$ and the category of étale $(\varphi,\Gamma)$-modules over the ring of fonctions on a ghost circle. Recent developments in the mod $p$ Langlands program encouraged the search for similar equivalences for modules over multivariable rings. Work by Z\’abr\’adi and Carter-Kedlaya-Z\’abr\’adi fulfilled part of this expectation by establishing an equivalence between representations of finite products of $\mathcal{G}_{\mathbb{Q}_p}$ and multivariable cyclotomic $(\varphi,\Gamma)$-modules.

The first goal of this talk is to sketch a proof of a Lubin-Tate variant for a $p$-adic local field $K$. Namely, for a finite set $\Delta$, we obtain an equivalence between continuous representations of $\prod_{\Delta} \mathcal{G}_K$ and a category called the étale $(\Phi_{\Delta, q}\times \Gamma_{\Delta,K,\lt})$-modules over $\mathcal{O}_{\mathcal{E}_{K,\Delta}}$ with finite projective dévissage. On the way to characterise the essential image of the functor $\mathbb{D}_{\Delta,\lt}$, we will explain which properties of finite type representations over $\mathbb{Z}_p$ are preserved by a Fontaine type functor. This will allow to give a theorem similar to the structure of finite type $\mathbb{Z}_p$-modules for the underlying $\mathcal{O}_{\mathcal{E}_{K,\Delta}}$ appearing in the previous equivalence. Finally, we will motivate how Lubin-Tate multivariable $(\varphi,\Gamma)$-modules should be more useful than cyclotomic ones to obtain a Colmez functor for $\gl{n}{K}$.

Details

Date:
January 31
Time:
13:30 – 14:30 CET

Organizer

Otmar Venjakob

Venue

Heidelberg, Mathematikon, SR A and Livestream