Représentations localement analytiques de 𝐺𝐿₂(𝐐_{𝐩}) et (𝜑,Γ)-modules

Colmez, Pierre

doi:10.1090/ert/484

Cited by 14 publications

(69 citation statements)

References 20 publications

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“…By now the p-adic Langlands correspondence for GL 2 (Q p ) is very well understood through the work of Berger [1], Breuil [4,5,6], Colmez [12], [13], Emerton [15], Kisin [21], Paškūnas [24] (see [7] for an overview). The starting point of Colmez's work is Fontaine's [18] theorem that the category of modulo p h Galois representations of Q p is equivalent to the category of étale (ϕ, Γ)-modules over Z/p h ((X)).…”

Section: Background and Motivationmentioning

confidence: 99%

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Multivariable $$(\varphi ,\Gamma )$$ ( φ , Γ ) -modules and smooth o-torsion representations

Zábrádi¹

2016

Sel. Math. New Ser.

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Let G be a Q p -split reductive group with connected centre and Borel subgroup B = T N . We construct a right exact functor D ∨ ∆ from the category of smooth modulo p n representations of B to the category of projective limits of finitely generated étale (ϕ, Γ)-modules over a multivariable (indexed by the set of simple roots) commutative Laurentseries ring. These correspond to representations of a direct power of Gal(Q p /Q p ) via an equivalence of categories. Parabolic induction from a subgroup P = L P N P gives rise to a basechange from a Laurent-series ring in those variables with corresponding simple roots contained in the Levi component L P . D ∨ ∆ is exact and yields finitely generated objects on the category SP A of finite length representations with subquotients of principal series as Jordan-Hölder factors. Lifting the functor D ∨ ∆ to all (noncommuting) variables indexed by the positive roots allows us to construct a G-equivariant sheaf Y π,∆ on G/B and a G-equivariant continuous map from the Pontryagin dual π ∨ of a smooth representation π of G to the global sections Y π,∆ (G/B). We deduce that D ∨ ∆ is fully faithful on the full subcategory of SP A with Jordan-Hölder factors isomorphic to irreducible principal series.

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Section: Background and Motivationmentioning

confidence: 99%

“…My debt to the works of Christophe Breuil [8], Pierre Colmez [12] [13], Peter Schneider, and Marie-France Vignéras [25] [26] will be obvious to the reader. I would also like to thank Márton Erdélyi, Jan Kohlhaase, Vytautas Paškūnas, Peter Schneider, and Tamás Szamuely for discussions on the topic.…”

Section: Acknowledgementsmentioning

confidence: 99%

Multivariable $$(\varphi ,\Gamma )$$ ( φ , Γ ) -modules and smooth o-torsion representations

Zábrádi¹

2016

Sel. Math. New Ser.

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“…Inspired by the calculations of the p-adic local correspondence for trianguline 8 étale (ϕ, Γ)modules, Colmez ([11]) has recently given a direct construction, for a (not necessarily étale) (ϕ, Γ)module ∆ (of rank 2) over R L , of a locally analytic L-representation Π(∆) of GL 2 (Q p ). More precisely, we have the following theorem: Theorem 1.1 ( [11], Théorème 0.1). There exists a unique extension of ∆ to a GL 2 (Q p )-equivariant sheaf of Q p -analytic type 9 ∆ ⊠ ω P 1 over P 1 with central character ω.…”

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confidence: 99%

“…[22]) have been achieved by Emerton-Helm in [18]. The arguments in [11] strongly rely on the cohomology theory of locally analytic representations developed in [30], and specifically on Shapiro's lemma. Since the authors are not aware of any reference for these results in the relative setting, we develop, in an appendix (cf.…”

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Arithmetic families of (φ, Γ)-modules and locally analytic representations of GL_2(Q_p)

Gaisin¹,

Jacinto²

2017

Preprint

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Let A be a Q p -affinoid algebra, in the sense of Tate. We develop a theory of locally convex A-modules parallel to the treatment in the case of a field by Schneider and Teitelbaum. We prove that there is an integration map linking a category of locally analytic representations in A-modules and separately continuous relative distribution modules. There is a suitable theory of locally analytic cohomology for these objects and a version of Shapiro's Lemma. In the case of a field this has been substantially developed by Kohlhaase. As an application we propose a p-adic Langlands correspondence in families: For a regular trianguline (ϕ, Γ)-module of dimension 2 over the relative Robba ring RA we construct a locally analytic GL2(Qp)-representation in A-modules.

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Compairing Categories of Lubin–Tate $$(\varphi _L,\Gamma _L)$$-Modules

Schneider,

Venjakob

2023

Results Math

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In the Lubin–Tate setting we compare different categories of $$(\varphi _L,\Gamma )$$ ( φ L , Γ ) -modules over various perfect or imperfect coefficient rings. Moreover, we study their associated Herr-complexes. Finally, we show that a Lubin Tate extension gives rise to a weakly decompleting, but not decompleting tower in the sense of Kedlaya and Liu.

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Représentations localement analytiques de 𝐺𝐿₂(𝐐_{𝐩}) et (𝜑,Γ)-modules

Abstract: Abstract. We extend the p-adic local Langlands correspondence for GL 2 (Q p

Cited by 14 publications

References 20 publications

Multivariable $$(\varphi ,\Gamma )$$ ( φ , Γ ) -modules and smooth o-torsion representations

Multivariable $$(\varphi ,\Gamma )$$ ( φ , Γ ) -modules and smooth o-torsion representations

Arithmetic families of (φ, Γ)-modules and locally analytic representations of GL_2(Q_p)

Compairing Categories of Lubin–Tate $$(\varphi _L,\Gamma _L)$$-Modules

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