Parabolic eigenvarieties via overconvergent cohomology

Salazar, Daniel Barrera; Williams, Chris

doi:10.1007/s00209-021-02707-9

Cited by 4 publications

(10 citation statements)

References 72 publications

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“…Remark 3.2.10. (1) Similar results (but in the setting of overconvergent cohomology) were obtained in [2].…”

Section: Density Of Classical Pointssupporting

confidence: 62%

“…Note that there was already some work in that direction, cf. [51], [66], [2] (see after Theorem 1.4 below for a brief comparison with [51]). These spaces parametrize certain p-adic automorphic (resp.…”

Section: Introductionmentioning

confidence: 99%

“…Similar spaces have been constructed in [51] (see also [2] for a construction via overconvergent cohomology, and also [66]), but the new feature in Theorem 1.4 is that we take into account the action of the full Bernstein centre (rather than just the action of Z L P (F ℘ )), obtained by applying Bushnell-Kutzko's theory of types. This allows to parametrize the full L P (F ℘ )-action, which is particularly important for our applications.…”

Section: Introductionmentioning

confidence: 99%

“…By an analogue of Theorem 1.5 (2), one can show that for each τ : F ℘ ֒→ E, the set of τ -Sen weights of ̺ is given by {k x,j,τ := h j,τ + wt(χ i(j) ) τ } j=1,...,n where i(j) ∈ {1, . .…”

Section: Introductionmentioning

confidence: 99%

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Bernstein eigenvarieties

Breuil,

Ding

2021

Preprint

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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 .

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“…Remark 3.2.10. (1) Similar results (but in the setting of overconvergent cohomology) were obtained in [2].…”

Section: Density Of Classical Pointssupporting

confidence: 62%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Bernstein eigenvarieties

Breuil,

Ding

2021

Preprint

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“…The main input of the current paper is a pairing of this construction with a systematic study of the eigenvariety parametrising Bianchi modular forms. The use of overconvergent cohomology in constructing eigenvarieties-generalising the pioneering work of Hida in the ordinary setting-was known to Stevens, later explored by Ash-Stevens [3], Urban [55] and more recently by Hansen [27] and the authors [17].…”

Section: Our Resultsmentioning

confidence: 99%

Families of Bianchi modular symbols: critical base-change p-adic L-functions and p-adic Artin formalism

Salazar

Williams

2021

Sel. Math. New Ser.

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Let K be an imaginary quadratic field. In this article, we study the eigenvariety for $$\mathrm {GL}_2/K$$ GL 2 / K , proving an étaleness result for the weight map at non-critical classical points and a smoothness result at base-change classical points. We give three main applications of this; let f be a p-stabilised newform of weight $$k \ge 2$$ k ≥ 2 without CM by K. Suppose f has finite slope at p and its base-change $$f_{/K}$$ f / K to K is p-regular. Then: (1) We construct a two-variable p-adic L-function attached to $$f_{/K}$$ f / K under assumptions on f that conjecturally always hold, in particular with no non-critical assumption on f/K. (2) We construct three-variable p-adic L-functions over the eigenvariety interpolating the p-adic L-functions of classical base-change Bianchi cusp forms. (3) We prove that these base-change p-adic L-functions satisfy a p-adic Artin formalism result, that is, they factorise in the same way as the classical L-function under Artin formalism.

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