Une interprétation modulaire de la variété trianguline

Abstract

Résumé. En utilisant le système de Taylor-Wiles-Kisin construit dans un travail récent de Caraiani, Emerton, Gee, Geraghty, Paškūnas et Shin, nous construisons un analogue de la variété de Hecke. Nous montrons que cette variété coïncide avec une union de composantes irréductibles de l'espace des représentations galoisiennes triangulines. Nous précisons les relations de cette construction avec les conjectures de modularité dans le cas cristallin ainsi qu'avec une conjecture de Breuil sur le socle des vecteurs localement analytiques de la cohomologie complétée. Nous donnons également une preuve d'une conjecture de Bellaïche et Chenevier sur l'anneau local complété en certains points des variétés de Hecke.Abstract. Using a patching module constructed in recent work of Caraiani, Emerton, Gee, Geraghty, Paškūnas and Shin we construct some kind of analogue of an eigenvariety. We can show that this patched eigenvariety agrees with a union of irreducible components of a space of trianguline Galois representations. Building on this we discuss the relation with the modularity conjectures for the crystalline case, a conjecture of Breuil on the locally analytic socle of representations occurring in completed cohomology and with a conjecture of Bellaïche and Chenevier on the complete local ring at certain points of eigenvarieties.

On the density of supercuspidal points of fixed regular weight in local deformation rings and global Hecke algebras

“…Breuil, Hellmann, and Schraen [3] have made a careful study of the theory of the Jacquet module as it applies to M ∞ , and we will use their results. 1 Following the notation of [3], we write Π ∞ := Hom cont O (M ∞ , L); then Π ∞ is an R ∞ -admissible Banach space representation of G. We may pass to its R ∞ -locally analytic vectors, see [3, §3.1], and then form the locally analytic Jacquet module…”

“…We recall (again from [3]; see their Definition 2.4) that there is another important Zariski closed rigid analytic subspace…”

“…In order to modify these arguments so as to be sensitive to the support of M ∞ , we use the locally analytic Jacquet module of [17], which provides a representationtheoretic approach to the study of finite slope families. Breuil, Hellmann, and Schraen [3] have made a careful study of the theory of the Jacquet module as it applies to M ∞ , and we use their results.…”

“…While studying their paper, we realised that the second part our proof can be proved using the 'infinite fern' arguments by constructing what is now known as the patched eigenvariety. The construction of this patched eigenvariety appeared in the paper by Breuil-Hellmann-Schraen [3], although both ME and Hellmann explained their ideas to VP independently at around the same time.…”

A Note on Critical p-adic L-functions

“…4.2.36], one can actually reformulate the construction of S using the spectral theory of compact operators (e.g. see [8,Lem. 3.10]).…”