62 research outputs found
The infinite fern in higher dimensions
No full textIf is an automorphic modulo Galois representation, it is natural to wonder if automorphic points are Zariski dense in the deformation space of . We prove new results in this direction in the case of a unitary group split (and unramified) at . Namely, if is associated to an automorphic form for a unitary group (which contributes to coherent cohomology), we prove that the "infinite fern" (i.e. the image of an appropriate Eigenvariety) in the polarised deformation space of is Zariski dense in a non-empty union of irreducible components. This generalises in particular results of Gouv\^ea-Mazur for , Chenevier for and recently Hellmann-Margerin-Schraen. The novelty is that we use the local model of Breuil-Hellmann-Schraen to control tangent spaces in the local deformation rings, and a geometric argument on the Eigenvariety originally due to Bella\"iche-Chenevier and Ta\"ibi to reduce to points with enormous image. At those points, we can use a recent result of Newton-Thorne to control the vanishing of a Selmer group. In particular, we do not need to assume any "Taylor-Wiles" hypothesis on , which can in particular be irreducible. If we moreover add Taylor-Wiles hypothesis on and an extra hypothesis at , we have by a result of Allen the Zariski density everywhere
The infinite fern in higher dimensions
No full textIf is an automorphic modulo Galois representation, it is natural to wonder if automorphic points are Zariski dense in the deformation space of . We prove new results in this direction in the case of a unitary group split (and unramified) at . Namely, if is associated to an automorphic form for a unitary group (which contributes to coherent cohomology), we prove that the "infinite fern" (i.e. the image of an appropriate Eigenvariety) in the polarised deformation space of is Zariski dense in a non-empty union of irreducible components. This generalises in particular results of Gouv\^ea-Mazur for , Chenevier for and recently Hellmann-Margerin-Schraen. The novelty is that we use the local model of Breuil-Hellmann-Schraen to control tangent spaces in the local deformation rings, and a geometric argument on the Eigenvariety originally due to Bella\"iche-Chenevier and Ta\"ibi to reduce to points with enormous image. At those points, we can use a recent result of Newton-Thorne to control the vanishing of a Selmer group. In particular, we do not need to assume any "Taylor-Wiles" hypothesis on , which can in particular be irreducible. If we moreover add Taylor-Wiles hypothesis on and an extra hypothesis at , we have by a result of Allen the Zariski density everywhere
The infinite fern in higher dimensions
No full textIf is an automorphic modulo Galois representation, it is natural to wonder if automorphic points are Zariski dense in the deformation space of . We prove new results in this direction in the case of a unitary group split (and unramified) at . Namely, if is associated to an automorphic form for a unitary group (which contributes to coherent cohomology), we prove that the "infinite fern" (i.e. the image of an appropriate Eigenvariety) in the polarised deformation space of is Zariski dense in a non-empty union of irreducible components. This generalises in particular results of Gouv\^ea-Mazur for , Chenevier for and recently Hellmann-Margerin-Schraen. The novelty is that we use the local model of Breuil-Hellmann-Schraen to control tangent spaces in the local deformation rings, and a geometric argument on the Eigenvariety originally due to Bella\"iche-Chenevier and Ta\"ibi to reduce to points with enormous image. At those points, we can use a recent result of Newton-Thorne to control the vanishing of a Selmer group. In particular, we do not need to assume any "Taylor-Wiles" hypothesis on , which can in particular be irreducible. If we moreover add Taylor-Wiles hypothesis on and an extra hypothesis at , we have by a result of Allen the Zariski density everywhere
The infinite fern in higher dimensions
No full textIf is an automorphic modulo Galois representation, it is
natural to wonder if automorphic points are Zariski dense in the deformation
space of . We prove new results in this direction in the case of a
unitary group split (and unramified) at . Namely, if is
associated to an automorphic form for a unitary group (which contributes to
coherent cohomology), we prove that the "infinite fern" (i.e. the image of an
appropriate Eigenvariety) in the polarised deformation space of is
Zariski dense in a non-empty union of irreducible components. This generalises
in particular results of Gouv\^ea-Mazur for , Chenevier for
and recently Hellmann-Margerin-Schraen. The novelty is that we use the
local model of Breuil-Hellmann-Schraen to control tangent spaces in the local
deformation rings, and a geometric argument on the Eigenvariety originally due
to Bella\"iche-Chenevier and Ta\"ibi to reduce to points with enormous image.
At those points, we can use a recent result of Newton-Thorne to control the
vanishing of a Selmer group. In particular, we do not need to assume any
"Taylor-Wiles" hypothesis on , which can in particular be
irreducible. If we moreover add Taylor-Wiles hypothesis on and an
extra hypothesis at , we have by a result of Allen the Zariski density
everywhere
The infinite fern in higher dimensions
No full textIf is an automorphic modulo Galois representation, it is natural to wonder if automorphic points are Zariski dense in the deformation space of . We prove new results in this direction in the case of a unitary group split (and unramified) at . Namely, if is associated to an automorphic form for a unitary group (which contributes to coherent cohomology), we prove that the "infinite fern" (i.e. the image of an appropriate Eigenvariety) in the polarised deformation space of is Zariski dense in a non-empty union of irreducible components. This generalises in particular results of Gouv\^ea-Mazur for , Chenevier for and recently Hellmann-Margerin-Schraen. The novelty is that we use the local model of Breuil-Hellmann-Schraen to control tangent spaces in the local deformation rings, and a geometric argument on the Eigenvariety originally due to Bella\"iche-Chenevier and Ta\"ibi to reduce to points with enormous image. At those points, we can use a recent result of Newton-Thorne to control the vanishing of a Selmer group. In particular, we do not need to assume any "Taylor-Wiles" hypothesis on , which can in particular be irreducible. If we moreover add Taylor-Wiles hypothesis on and an extra hypothesis at , we have by a result of Allen the Zariski density everywhere
The infinite fern in higher dimensions
No full textIf is an automorphic modulo Galois representation, it is natural to wonder if automorphic points are Zariski dense in the deformation space of . We prove new results in this direction in the case of a unitary group split (and unramified) at . Namely, if is associated to an automorphic form for a unitary group (which contributes to coherent cohomology), we prove that the "infinite fern" (i.e. the image of an appropriate Eigenvariety) in the polarised deformation space of is Zariski dense in a non-empty union of irreducible components. This generalises in particular results of Gouv\^ea-Mazur for , Chenevier for and recently Hellmann-Margerin-Schraen. The novelty is that we use the local model of Breuil-Hellmann-Schraen to control tangent spaces in the local deformation rings, and a geometric argument on the Eigenvariety originally due to Bella\"iche-Chenevier and Ta\"ibi to reduce to points with enormous image. At those points, we can use a recent result of Newton-Thorne to control the vanishing of a Selmer group. In particular, we do not need to assume any "Taylor-Wiles" hypothesis on , which can in particular be irreducible. If we moreover add Taylor-Wiles hypothesis on and an extra hypothesis at , we have by a result of Allen the Zariski density everywhere
Trianguline representations with G-structure
No full textSoit |G| un groupe réductif déployé sur |mathbf{Q}_p|, |B subset G| un sous-groupe de Borel de |G| et |Tsubset B| un tore maximal déployé. Dans cette thèse nous définissons la notion de représentation trianguline |G|-valuée de |text{Gal}(overline{mathbf{Q}_p}/mathbf{Q}_p)| et initions l'étude de la variété trianguline correspondante en suivant les travaux de Breuil, Hellmann et Schraen. Pour cela nous introduisons la notion de |(G,varphi,Gamma)|-module sur l'anneau de Robba. Ce sont des |G|-torseurs sur l'anneau de Robba munis d'actions continues d'un Frobenius et du groupe |Gamma = mathbf{Z}_p^times| commutant entre elles. Comme pour les |(varphi,Gamma)|-modules classiques il existe un foncteur pleinement fidèle |D_{text{rig}}^dagger| de la catégorie des représentations |p|-adiques |G|-valuées de |text{Gal}(overline{mathbf{Q}_p}/mathbf{Q}_p)| vers celle des |(G,varphi,Gamma)|-modules sur l'anneau de Robba.On dit qu'une représentation |p|-adique |G|-valuée |rho : text{Gal}(overline{mathbf{Q}_p}/mathbf{Q}_p) to G(mathbf{Q}_p)| est trianguline si il existe un |(B,varphi,Gamma)|-module |Theta| tel que |D_{text{rig}}^dagger(rho)| provient de |Theta| par fonctorialité le long de l'inclusion |B subset G|. Un tel |Theta| est appelé une triangulation de |rho|. A une telle triangulation on peut associer un caractère |delta_Theta : T^vee(mathbf{Q}_p) to mathbf{Q}_p^times| que l'on appelle le paramètre de |Theta|. Le premier résultat de la thèse est le fait que les représentations |G|-valuées cristallines sont triangulines. Nous donnons une description explicite des paramètres correspondants.Soit |overline{rho} : text{Gal}(overline{mathbf{Q}_p}/mathbf{Q}_p) to G(mathbf{F}_p)| une représentation résiduelle, |mathfrak{X}_{overline{rho}}| l'espace rigide analytique paramétrant les déformations de |overline{rho}| et |widehat{T^vee}| l'espace rigide analytique paramétrant les caractères de |T^vee|. On définit |U^{text{tri}}_{text{vreg}} subset mathfrak{X}_{overline{rho}} times widehat{T^vee}| comme étant l'ensemble des couples |(rho,delta)| tel que |delta| est suffisamment régulier et |rho| admet une triangulation de paramètre |delta|. La variété trianguline |X^{text{tri}}_{text{vreg}}(overline{rho})| est définie comme étant l'adhérence de Zariski de |U^{text{tri}}_{text{vreg}}(overline{rho})| dans |mathfrak{X}_{overline{rho}} times widehat{T^vee}|. Le résultat principal de la thèse est le fait que |U^{text{tri}}_{text{vreg}}(overline{rho})| est un ouvert lisse de |X^{text{tri}}_{text{vreg}}(overline{rho})| généralisant ainsi un résultat de Breuil, Hellmann et Schraen au cas des représentations |G|-valuées.Let |G| be a split reductive group over |mathbf{Q}_p|, |B subset G| be a Borel subgroup and |T subset B| be a split maximal torus. In this thesis we define the notion of a |G|-valued trianguline representation of |text{Gal}(overline{mathbf{Q}_p}/mathbf{Q}_p)| and initiate the study of the corresponding trianguline variety following the works of Breuil, Hellmann and Schraen. For this we introduce the notion of a |(G,varphi,Gamma)|-module over the Robba ring. These are |G|-torsors over the Robba ring endowed with continuous commuting actions of a Frobenius automorphism and of |Gamma = mathbf{Z}_p^times|. As it is the case with classical |(varphi,Gamma)|-modules there exists a fully faithful functor |D_{text{rig}}^dagger| from the category of |G|-valued |p|-adic representations of |text{Gal}(overline{mathbf{Q}_p}/mathbf{Q}_p)| to the one the |(G,varphi,Gamma)|-modules over the Robba ring.We say that a |p|-adic |G|-valued representation |rho : text{Gal}(overline{mathbf{Q}_p}/mathbf{Q}_p) to G(mathbf{Q}_p)| is trianguline if there exists a |(B,varphi,Gamma)|-module |Theta| such that |D_{text{rig}}^dagger(rho)| comes from |Theta| by functoriality along the inclusion |B subset G|. Such a |Theta| is called a triangulation of |rho|. To such a triangulation we can associate a caracter |delta_Theta : T^vee(mathbf{Q}_p) to mathbf{Q}_p^times| which we call the parameter of |Theta|. The first result of this thesis is that |G|-valued crystalline representations are trianguline. We also give a concrete description of the corresponding parameters.Let |overline{rho} : text{Gal}(overline{mathbf{Q}_p}/mathbf{Q}_p) to G(mathbf{F}_p)| be a residual representation, |mathfrak{X}_{overline{rho}}| be the rigid analytic space parametrizing deformations of |overline{rho}| and |widehat{T^vee}| be the rigid analytic space parametrizing caracters of |T^vee|. We define |U^{text{tri}}_{text{vreg}} subset mathfrak{X}_{overline{rho}} times widehat{T^vee}| to be the set of couples |(rho,delta)| such that |delta| is sufficiently regular and such that |rho| admits a triangulation of parameter|delta|. The trianguline variety |X^{text{tri}}_{text{vreg}}(overline{rho})| is defined as the Zariski closure of |U^{text{tri}}_{text{vreg}}(overline{rho})| in |mathfrak{X}_{overline{rho}} times widehat{T^vee}|. The principal result of the thesis is that |U^{text{tri}}_{text{vreg}}(overline{rho})| is Zariski open and smooth inside |X^{text{tri}}_{text{vreg}}(overline{rho})|. This generalizes a result of Breuil, Hellmann and Schraen to the setting of |G|-valued representations
Variété trianguline et variété de Hecke aux points de poids de Hodge-Tate non-réguliers
No full textCette thèse porte sur les formes automorphes p-adiques et l'aspect localement analytique du programme de Langlands p-adique. Nous étudions la géométrie locale de la variété trianguline et le problème des points compagnons sur la variété de Hecke aux points de poids de Hodge-Tate non-réguliers. Il s'agit d'une généralisation des travaux de Breuil-Hellmann-Schraen qui avaient traité le cas régulier. Nous établissons d'abord les modèles locaux et l'irréductibilité locale de la variété triangulines aux points génériques, points qui peuvent avoir des poids de Hodge-Tate non-réguliers et même non-entiers. Nous démontrons alors l'existence, dans le cas cristallin Frobenius générique, des points compagnons de raffinement fixé mais de poids différents sur la variété de Hecke, en mettant en relation des propriétés partiellement classiques des formes automorphes p-adiques et certaines propriétés de cycles sur les modèles locaux. De plus, nous démontrons l'existence des points compagnons correspondant aux autres raffinements sur la variété de Hecke en utilisant un argument d'approximation. En conséquence, sous l'hypothèse de Taylor-Wiles, nous prouvons la conjecture de socle localement analytique de Breuil pour les représentations galoisiennes cristallines génériques de poids de Hodge-Tate non-réguliers.This thesis concerns p-adic automorphic forms and the locally analytic aspect of the p-adic Langlands program. We study the local geometry of the trianguline variety and the companion points problem on the eigenvariety at points with non-regular Hodge-Tate weights. This generalizes the works of Breuil-Hellmann-Schraen who have dealt with regular cases. We first establish the local models and local irreducibility for generic points on the trianguline variety, the points that might have non-regular or non-integral Hodge-Tate weights. Then in the crystalline Frobenius generic cases, we obtain the existence of companion points with fixed refinements but with different weights on the eigenvariety, by relating partially classical properties of p-adic automorphic forms and certain properties of cycles on the local models. Furthermore, we can find companion points corresponding to other refinements on the eigenvariety using an approximation argument. As a result, under the Taylor-Wiles hypothesis, we prove Breuil's locally analytic socle conjecture for generic crystalline Galois representations with non-regular weights
Représentations localement analytiques de
No full textInternational audienceNous construisons un complexe de représentations localement analytiques de , associé à certaines représentations semi-stables de dimension 3 du groupe de Galois absolu de . Nous montrons ensuite que l'on peut retrouver le -module filtré de la représentation galoisienne en considérant les morphismes, dans la catégorie dérivée des -modules, de ce complexe dans le complexe de de Rham de l'espace de Drinfel'd de dimension 2. La preuve requiert le calcul de certains espaces de cohomologie localement analytiques de sous-groupes unipotents à coefficients dans des séries principales localement analytiques
Représentations p-adiques de GL2(L) et Catégories Dérivées
No full textInternational audienceWe construct some locally -analytic representations of , a finite extension of , associated to some -adic representations of the absolute Galois group of . We prove that the space of morphisms from these representations to the de Rham complex of Drinfel'd's upper half space has a structure of rank 2 admissible filtered -module. Finally, we prove that this filtered module is associated, via Fontaine's theory, to the initial Galois representation
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