\protect \mathcal{L}-invariants, partially de Rham families, and local-global compatibility

Trans. Amer. Math. Soc.

2019

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Let L be a finite extension of Q p , and ρ L be an n-dimensional semi-stable non crystalline p-adic representation of Gal L with full monodromy rank. Via a study of Breuil's (simple) L-invariants, we attach to ρ L a locally Q p -analytic representation Π(ρ L ) of GL n (L), which carries the exact information of the Fontaine-Mazur simple L-invariants of ρ L . When ρ L comes from an automorphic representation of G(A F + ) (for a unitary group G over a totally real filed F + which is compact at infinite places and GL n at p-adic places), we prove under mild hypothesis that Π(ρ L ) is a subrerpresentation of the associated Hecke-isotypic subspaces of the Banach spaces of p-adic automorphic forms on G(A F + ). In other words, we prove the equality of Breuil's simple L-invariants and Fontaine-Mazur simple L-invariants.

“…cit. also show the existence of a similar spectral sequence in the case without fixing the central character, which can be used to prove (19)):…”

Section: Extensions Of Locally Analytic Representations Imentioning

confidence: 85%

“…Let R L := B † rig,L , and R E := R L ⊗ Qp E. The cup-product induces a perfect pairing (e.g. see [19,Lem. 1.13]…”

Section: Fontaine-mazur Simple L-invariantsmentioning

confidence: 99%

Simple ℒ-invariants for GL_{𝓃}

Trans. Amer. Math. Soc.

2019

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“…This hypothesis is automatically satisfied when F ℘ = Q p by the weak admissibility of (D 0 , D). Note also that in critical case, one can still define Fontaine-Mazur L-invariants L σ for the embeddings σ with a σ = 0; we refer to [21] for results in this case (where a key ingredient is the Colmez-GreenbergStevens formula in critical case).…”

Section: This Is a Smooth Admissible Representation Of G(a ∞ ) Equippmentioning

confidence: 99%

-Invariants and Local–global Compatibility for the Group

Forum of Mathematics, Sigma

2016

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Let F be a totally real number field, ℘ a place of F above p. Let ρ be a 2-dimensional p-adic representation of Gal(F/F) which appears in theétale cohomology of quaternion Shimura curves (thus ρ is associated to Hilbert eigenforms). When the restriction ρ ℘ := ρ| D℘ at the decomposition group of ℘ is semistable noncrystalline, one can associate to ρ ℘ the so-called Fontaine-Mazur L-invariants, which are however invisible in the classical local Langlands correspondence. In this paper, we prove one can find these L-invariants in the completed cohomology group of quaternion Shimura curves, which generalizes some of Breuil's results [Breuil, Astérisque, 331 (2010), 65-115] in the GL 2 /Q-case.

“…Keep the above notation.(1) The rigid space X p (ρ, λ Σ L \J ) is smooth at the point x c J .(2) The following statements are equivalent:(a) σ ∈ Σ \ J; (b) the natural projection of complete noetherian local E-algebrasThe smoothness of X p (ρ, λ Σ L \J ) follows from the same arguments of [14] (see also [4]). A key point is obtaining (a bound for) the dimension of the tangent space of X p (ρ, λ Σ L \J ) at x c J via Galois cohomology calculation (in our case, it would in particular involve some partially de Rham Galois cohomology considered in [22]). For (2) (from which Theorem 1.2 follows), the direction (c) ⇒ (a) follows easily from global triangulation theory; (b) ⇒ (c) follows from some locally analytic representation theory (e.g.…”

mentioning

confidence: 99%

“…The smoothness of X p (ρ, λ Σ L \J ) follows from the same arguments of [14] (see also [4]). A key point is obtaining (a bound for) the dimension of the tangent space of X p (ρ, λ Σ L \J ) at x c J via Galois cohomology calculation (in our case, it would in particular involve some partially de Rham Galois cohomology considered in [22]).…”

mentioning

confidence: 99%

Companion points and locally analytic socle for GL2(L)

2019

Isr. J. Math.

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Let L be a finite extension of Qp, we prove under mild hypothesis Breuil's locally analytic socle conjecture for GL2(L), showing the existence of all the companion points on the definite (patched) eigenvariety. This work relies on infinitesimal "R=T" results for the patched eigenvariety and the comparison of (partially) de Rham families and (partially) Hodge-Tate families. This method allows in particular to find companion points of non-classical points.This work was supported by EPSRC grant EP/L025485/1. 1 Appendix B. Some locally analytic representation theory 47 References 50 1. IntroductionIn this paper, we prove (under mild technical hypotheses) Breuil's locally analytic socle conjecture for GL 2 (L) where L is a finite extension of Q p .Breuil's locally analytic socle conjecture for GL 2 (L). Let L 0 be the maximal unramified extension over Q p in L, d := [L : Q p ], d 0 := [L 0 : Q p ], E be a finite extension of Q p big enough to contain all the Q p -embeddings of L in Q p , and Σ L := Hom Qp (L, Q p ) = Hom Qp (L, E).Let ρ L be a two dimensional crystalline representation of Gal L over E with distinct Hodge-Tate weights (−k 1,σ , −k 2,σ ) σ∈Σ L (k 1,σ > k 2,σ ) (where we use the convention that the Hodge-Tate weight of the cyclotomic character is −1), let α, α be the eigenvalues of crystalline Frobeniuswhere unr(z) denotes the unramified character of L × sending uniformizers to z; we define δ, δ c J the same way as δ, δ c J by exchanging α and α. Recall ρ L is trianguline, and there exists Σ ⊆ Σ L (resp. Σ ⊆ Σ L ) such that δ c Σ is a trianguline parameter of ρ L , also called a refinement of ρ L . The refinement δ c Σ resp. δ c Σ is called non-critical if Σ = ∅ (resp. Σ = ∅). Note the information of Σ and Σ is lost when passing to the Weil-Deligne representation associated to ρ L , thus is invisible in classical local Langlands correspondence. In fact, in terms of filtered ϕ-modules, we haveFor a continuous very regular character χ (cf.(3)) of T (L) over E, putSuppose ρ L is the restriction of certain global modular Galois representation ρ (i.e. ρ is associated to classical automorphic representations; in this paper, we would consider the case of automorphic representations of definite unitary groups). Using global method, we can attach to ρ an admissible unitary Banach representation Π(ρ) of GL 2 (L) (e.g. in the case we consider) such thatis the modulus character of the Borel subgroup B (of upper triangular matrices). The representation Π(ρ) is expected to 2 be right representation of GL 2 (L) corresponding to ρ L in the p-adic Langlands program (see [8] for a survey). Let Π(ρ) an be the locally Q p -analytic subrepresentation of Π(ρ), which is dense in Π(ρ). We have the following Breuil's conjecture (cf. [9, Conj.8.1] and [11]) concerning the socle of Π(ρ) an .The "only if" part would be (in many cases) a consequence of global triangulation theory, while the "if" part is more difficult. In modular curve case (thus L = Q p ), this was proved in [12] using p-adic comparison theorems and the theory of o...