Complétés universels de représentations de GL2(ℚp)

Colmez, Pierre; Dospinescu, Gabriel

doi:10.2140/ant.2014.8.1447

Cited by 29 publications

(47 citation statements)

References 27 publications

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“…. For a locally analytic representation W of H ′ , let W (n) denote the subrepresentation generated by r nanalytic vectors as in [13…”

Section: Adjunction Formulasmentioning

confidence: 99%

Some Results on the Locally Analytic Socle for $\textrm{GL}_{n}(\mathbb {Q}_{p})$

Ding

2018

International Mathematics Research Notices

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We study some closed rigid subspaces of the eigenvarieties, constructed by using the Jacquet-Emerton functor for parabolic non-Borel subgroups. As an application (and motivation), we prove some new results on Breuil's locally analytic socle conjecture for GLn(Qp).which is a Banach space over E equipped with a continuous action of G(Q p ) ∼ = GL 3 (Q p ) (this isomorphism depends on the choice of u), and a continuous action of (commutative) Hecke algebra H p outside p. The action of H p commutes with that of GL 3 (Q p ). Let ρ be a continuous representation of Gal(F /F ) over E associated to automorphic forms of G, and we associate to ρ a maximal ideal of H p (shrinking H p if needed). Suppose ( S(U p , E) lalg ) mρ = 0, where "lalg" denotes the locally algebraic vectors for GL 3 (Q p ), (·) mρ denotes the maximal E-vector space on which H p acts viawhich is an admissible unitary Banach representation of GL 3 (Q p ), and is supposed to be (a direct sum) of the right representation corresponding to ρ p := ρ| Gal(Fu/Fu) ∼ =Gal(Qp/Qp) in p-adic Langlands programme Jacquet-Emerton functors. Let P ⊃ B with the Levi subgroup L P = GL 2 × GL 1 (the case s = (12) would use the other maximal parabolic proper subgroup). Denote by L P (λ) the irreducible algebraic representation of L P with highest weight λ, for an admissible locally analytic representation V of GL 3 (Q p ), we put (cf. [20], and §1.3)denotes the algebraic dual of L P (λ). In fact, J B,(P,λ) (V ) is a closed subrepresentation (of T (Q p )) of the usual Jacquet-Emerton module J B (V ), and thus is an essentially admissible locally analytic representation of T (Q p ). We would use the subfunctor J B,(P,λ) (·) of J B (·) to construct a closed subspace of the eigenvariety. The adjunction property for J B,(P,λ) (·), which we discuss below, would allow us to get some nice properties of such closed subspace (cf. Thm.1.4, 1.5).Adjunction formulas. Suppose V is moreover very strongly admissible (cf. [18, Def.0.12]), we have an adjunction formula (cf. Thm.2.15) (obtained by combining Breuil's adjunction formula [6, Thm.4.3] and the adjunction formula for the classical Jacquet functor):where ψ is a finite length smooth representation of T (Q p ) over E, χ λ denotes the algebraic character of T (Q p ) with weight λ and we refer to §2.1 for the representations F G P (·, ·) etc.; meanwhile, recall that for J B (V ), by [6, Thm.4.3], one has (2) Hom GL 3 (Qp) F G B

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“…. For a locally analytic representation W of H ′ , let W (n) denote the subrepresentation generated by r nanalytic vectors as in [13…”

Section: Adjunction Formulasmentioning

confidence: 99%

Some Results on the Locally Analytic Socle for $\textrm{GL}_{n}(\mathbb {Q}_{p})$

Ding

2018

International Mathematics Research Notices

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“…L'énoncé peu appétissant suivant sera utilisé très souvent dans la suite. (24) de multiplication par z. On a les formules suivantes pour l'action de l'algèbre de Lie : a) Sur O(k)(Σ n )…”

Section: Numérologie Et Lissité De Hunclassified

Revĉtements du demi-plan de Drinfeld et correspondance de Langlands $p$-adique

Dospinescu

Bras

2017

Ann. of Math. (2)

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Résumé. -Nous décrivons le complexe de de Rham des revêtements du demi-plan de Drinfeld pour GL 2 (Qp). Cette description, conjecturée par Breuil et Strauch, fournit une réalisation géométrique de la correspondance de Langlands locale p-adique pour certaines représentations de de Rham de dimension 2 de Gal(Qp/Qp).Abstract. -We describe the de Rham complex of the étale coverings of Drinfeld's p-adic upper half-plane for GL 2 (Qp). Conjectured by Breuil and Strauch, this description gives a geometric realization of the p-adic local Langlands correspondence for certain two-dimensional de Rham representations of Gal(Qp/Qp).

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“…Démonstration. -C'est la réunion des théorèmes III.4, III.48, de la remarque III.31 et du corollaire III.46 de [14].…”

Section: Rappels Et Complementsunclassified

“…Cela demande d'étendre/adapter bon nombre de résultats de [15] et [11]. Nous reposons aussi de manière essentielle sur [14], qui généralise (souvent avec des preuves différentes) la plupart des résultats de [11] à la catégorie Rep L (δ) (et non seulement aux objets irréductibles). Dans cet article on étend (avec une preuve différente) une loi de réciprocité explicite de Colmez et on raffine sa théorie du modèle de Kirillov de Π alg , ainsi que le résultat principal de [15].…”

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Extensions de représentations de de Rham et vecteurs localement algébriques

Dospinescu¹

2015

Compositio Math.

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Soit Π une complétion unitaire irréductible d'une représentation localement algébrique de GL2(Q p ). On décrit les déformations infinitésimales Π1 de Π qui sont elles-mêmes complétions d'une représentation localement algébrique. Cela répond à une question de Paskunas et a des applications directes à la conjecture de Breuil-Mézard. Abstract. -Let Π be an irreducible unitary completion of a locally algebraic GL2(Qp)representation. We describe those first-order deformations of Π which are themselves completions of a locally algebraic representation. This answers a question of Paskunas and has direct applications to the Breuil-Mézard conjecture.c

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Complétés universels de représentations de GL₂(ℚ_p)

Cited by 29 publications

References 27 publications

Some Results on the Locally Analytic Socle for $\textrm{GL}_{n}(\mathbb {Q}_{p})$

Some Results on the Locally Analytic Socle for $\textrm{GL}_{n}(\mathbb {Q}_{p})$

Revĉtements du demi-plan de Drinfeld et correspondance de Langlands $p$-adique

Extensions de représentations de de Rham et vecteurs localement algébriques

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