-Invariants and Local–global Compatibility for the Group

Abstract

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.

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

“…Since z is Σ ℘ \ J-very classical, one has the bijection (34) (which is an analogue of [19, (4.20)], see also [24,Lem.3.27]). The proposition then follows from the multiplicity one result for automorphic representations of G(A), by the same argument as in the proof of [19,Thm.4.8] (see also [24, §3.4.3] especially the arguments after [24,Lem.3.27]).…”

“…the (ϕ, Γ)-module (of rank 2) over R k(z) associated to ρ z,℘ (we refer to [4] for B † rig,F℘ , B † rig and (ϕ, Γ)modules). The following proposition, which follows from the same argument as in [24,Cor.3.26], would be useful to apply the adjunction formula in families. (2) Hom Gal F ρ, H 1 et (K p , W (k Σp , w)) = 0 (in particular, ρ is associated to certain Hilbert eigenforms);…”

“…One important ingredient in [24] is Zhang's generalization [44, Thm.1.1] of Colmez-Greenberg-Stevens formula [22] (on L-invariants) in F ℘ -case. But the results in [44] are only for non-critical case.…”

Simple ℒ-invariants for GL_{𝓃}

“…The following theorem is the main result of the paper. The rest of the section is to prove Theorem 4.10, and we use the strategy of [18]. First note that the injectivity of (83) follows from…”

Derivatives of Frobenius and Derivatives of Hodge—Tate Weights

“…One is to the Exceptional Zero phenomenon, and the other is to the local-global compatibility in p-adic Langlands program. In the case of n = 2 the former is done in [19] and the latter is done in [15]. If δ is a multiplicative character of Q × p , the character of Q × p whose restriction to Z × p coincides with δ| Z × p and whose value at p is 1, is again denoted by δ| Z × p by abuse of notation.…”