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.
We describe the completed local rings of the trianguline variety at certain points of integral weights in terms of completed local rings of algebraic varieties related to Grothendieck's simultaneous resolution of singularities. We derive several local consequences at these points for the trianguline variety: local irreducibility, description of all local companion points in the crystalline case, combinatorial description of the completed local rings of the fiber over the weight map, etc. Combined with the patched Hecke eigenvariety (under the usual Taylor-Wiles assumptions), these results in turn have several global consequences: classicality of crystalline strictly dominant points on global Hecke eigenvarieties, existence of all expected companion constituents in the completed cohomology, existence of singularities on global Hecke eigenvarieties. for discussions or answers to questions. E. H. is partially supported by SFB-TR 45 and SFB 878 of the D.F.G., B. S. and C. B. are supported by the C.N.R.S. 1 4.3. A locally analytic "Breuil-Mézard type" statement 58 5. Global applications 64 5.1. Classicality 64 5.2. Representation theoretic preliminaries 68 5.3. Companion constituents 71 5.4. Singularities on global Hecke eigenvarieties 78References 80are the Kazhdan-Lusztig polynomials and L(w ′ ) are certain finite type O Xp(ρ) wt(δ) ,x -modules such that:Formula (1.2) essentially comes from representation theory (in particular the structure of Verma modules) and doesn't use Theorem 1.5. By an argument analogous to the one for Theorem 1.8 (using Theorem 1.5), we have nonzeroMoreover we know that the cycle C(w 0 ) is irreducible and that [L(w 0 )] ∈ Z ≥0 C(w 0 ) (roughly because the support of the locally Q p -algebraic vectors lies in the locus of crystalline deformations). Consequently we can deduce Theorem 1.3 from the fact that P 1,w 0 w ′ (1) = 0, if we know that C(w ′ ) is contained in the support of L(w ′ ) for w x w ′ .We prove this last assertion by a descending induction on the length of the Weyl group element w x . Assume first that lg(w x ) = lg(w 0 ) − 1. In that case x is smooth on X p (ρ) and then M ∞ is locally free at x. Hence M ∞,wt(δ),x ≃ O r Xp(ρ) wt(δ) ,x for some r > 0 and we can combine (1.3) (multiplied by the integer r) with (1.2). Using C(w 0 ) = 0, C(w x ) = 0 and [L(w 0 )] ∈ Z ≥0 C(w 0 ), it is then not difficult to deduce [L(w x )] = 0, hence L(w x ) = 0 and then. By a Zariski-density argument analogous to the one in the proof of Theorem 1.7, we can then deduce [L(w ′ )] = 0 for any w ′ w x such that lg(w ′ ) ≥ lg(w 0 ) − 1 and any w x such that lg(w x ) ≤ lg(w 0 ) − 1. In particular we have the companion points x w ′ on Y (U p , ρ) for w ′ w x and lg(w ′ ) = lg(w 0 ) − 1 and formulas analogous to (1.2) and (1.3) localizing and completing atAssume now lg(w x ) = lg(w 0 ) − 2, we can repeat the argument of the case lg(w x ) = lg(w 0 ) − 1 but using the analogues of (1.2), (1.3) at x w ′ = (ρ, δ w ′ ) for w ′ w x and lg(w ′ ) = lg(w 0 ) − 1 = lg(w x ) + 1. The results on the local geometry o...
Abstract. Let p be a prime number and f an overconvergent p-adic automorphic form on a definite unitary group which is split at p. Assume that f is of "classical weight" and that its Galois representation is crystalline at p, then f is conjectured to be a classical automorphic form. We prove new cases of this conjecture in arbitrary dimensions by making crucial use of the patched eigenvariety constructed in [14].
We consider stacks of filtered ϕ-modules over rigid analytic spaces and adic spaces. We show that these modules parametrize p-adic Galois representations of the absolute Galois group of a p-adic field with varying coefficients over an open substack containing all classical points. Further we study a period morphism (defined by Pappas and Rapoport) from a stack parametrising integral data and determine the image of this morphism. E. HELLMANNThe following diagram summarizes the stacks that appear in this paper in the case of a miniscule cocharacter ν.
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