Interpolating Hodge–Tate and de Rham periods

Shah, Shrenik

doi:10.1007/s40687-018-0135-3

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“…The existence of X tri,J−dR (r L , |k J |) follows essentially from [3, Thm. 5.2.4] (see also [35]). Actually, one can modify the proof of [3,Thm.…”

Section: Trianguline Variety Revisitedmentioning

confidence: 99%

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Companion points and locally analytic socle for GL2(L)

Ding

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...

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“…The existence of X tri,J−dR (r L , |k J |) follows essentially from [3, Thm. 5.2.4] (see also [35]). Actually, one can modify the proof of [3,Thm.…”

Section: Trianguline Variety Revisitedmentioning

confidence: 99%

“…Keep the situation of Theorem 3.21, and let J ′ ⊆ J. The isomorphism (35) (with J replaced by J ′ ) induces an isomorphism…”

Section: ))mentioning

confidence: 99%