Seminar des SFB/TRR 326 GAUS Multivariable Lubin-Tate Fontaine Equivalence

  • Termin in der Vergangenheit
  • Freitag, 31. Januar 2025, 13:30 - 14:30 Uhr
  • INF 205, SR A
    • Nataniel Marquis (Institut de Mathématiques de Jussieu – Paris Rive Gauche)
Livestream SR A
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    INF 205, SR A

  • Livestream

  • Veranstalter

  • Veranstaltungstyp

In 1991 J.-M. Fontaine proved an equivalence between continuons representations of G_Qp of finite type over Zp and the category of étale (φ,Γ)-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 Zábrádi and Carter-Kedlaya-Zábrádi fulfilled part of this expectation by establishing an equivalence between representations of finite products of G_Qp and multivariable cyclotomic (φ,Γ)-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 Δ, we obtain an equivalence between continuous representations of ΠΔ GK and a category called the étale (ϕΔ,p x ΓΔ,K)-modules over O_EK,Δ with finite projective dévissage. On the way to characterise the essential image of the functor DΔ, we will explain which properties of finite type representations over Zp  are preserved by a Fontaine type functor. This will allow to give a theorem similar to the structure of finite type Zp -modules for the underlying O_EK,Δ appearing in the previous equivalence. Finally, we will motivate how Lubin-Tate multivariable(φ,Γ)-modules should be more useful than cyclotomic ones to obtain a Colmez functor for nK.

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