Bernstein eigenvarieties

Abstract

We construct parabolic analogues of (global) eigenvarieties, of patched eigenvarieties and of (local) trianguline varieties, that we call respectively Bernstein eigenvarieties, patched Bernstein eigenvarieties, and Bernstein paraboline varieties. We study the geometry of these rigid analytic spaces, in particular (generalizing results of Breuil-Hellmann-Schraen) we show that their local geometry can be described by certain algebraic schemes related to the generalized Grothendieck-Springer resolution. We deduce several local-global compatibility results, including a classicality result (with no trianguline assumption at p), and new cases towards the locally analytic socle conjecture of Breuil in the non-trianguline case.(1) We have w F ≤ w x w 0 .(2) There exists a formal scheme X ,wx ̺,M• over E such that the associated reduced formal scheme (X ,wx ̺,M• ) red is formally smooth of dimension n 2 + dim p ℘ over the completion X wx,y pdR of X wx at y pdR , and formally smooth of dimensionBy Theorem 1.9 (2) and Theorem 1.7 (4), we deduce the following "R = T "-type result:rig and that ̺ is generic potentially crystalline with distinct Hodge-Tate weights. Then the embedding (1.3) induces an isomorphism after taking completions at x.We now discuss the problem of companion points on Bernstein eigenvarieties (resp. on patched Bernstein eigenvarieties, resp. on Bernstein paraboline varieties), which will be crucial to attack the socle conjecture. Let y be a point of (Spf R ρ,S ) rig (resp. of (Spf R ∞ ) rig , resp. of (Spf R ρ ℘ ) rig ), and ̺ be the Gal F℘ -representation associated to y. We assume ̺ is generic potentially crystalline with distinct Hodge-Tate weights. We let h be the (decreasing) Hodge-Tate weights of ̺ and λ = (λ i,τ ) i=1,...,n τ :F℘֒→E with λ i,τ = h i,τ + i − 1 (so λ is dominant with respect to B). Assume r(̺) ∼ = ⊕ r i=1 r x i with x = (x i ) ∈ (Spec Z Ω ) rig . Note that, as ̺ is generic, the P -filtration rAs usual, we denote by w • µ the dot action on a weight µ.Conjecture 1.11. Let w ∈ W P min,F℘ , then x ∈ E Ω,w•λ (U p , ρ) resp.x ∈ E ∞ Ω,w•λ (ρ), resp. x ∈ X Ω,w(h) (ρ ℘ ) if and only if ww 0 ≥ w F̺ .

Extensions of locally analytic generalized parabolic Steinberg representations

“…For certain potentially semistable Galois representation ρ L of Gal L which admits a special "parabolic" filtration (and hence not trianguline), we will define Fontaine-Mazur parabolic simple L -invariants of ρ L by using cohomology of (Φ, Γ)-modules over Robba ring. Moreover, we use Bernstein eigenvarieties (developed by [12]) to establish some local-global compatibility results on parabolic simple L-invariants.…”