Abstract
AbstractWe study the locally analytic vectors in the completed cohomology of modular curves and determine the eigenvectors of a rational Borel subalgebra of$\mathfrak {gl}_2(\mathbb {Q}_p)$. As applications, we prove a classicality result for overconvergent eigenforms of weight 1 and give a new proof of the Fontaine–Mazur conjecture in the irregular case under some mild hypotheses. For an overconvergent eigenform of weightk, we show its corresponding Galois representation has Hodge–Tate–Sen weights$0,k-1$and prove a converse result.
Highlights
This action satisfies all three properties as in the previous proposition. This action of Lie(Γ) on B∞ ⊗Q⊕p2: Γ → GL2 (Qp) V only depends on the restriction of the representation to any open subgroup of G. As it commutes with Γ, it induces a B-linear action of Lie(Γ) on (B∞ ⊗Qp V)Γ = B ⊗Qp V by Lemma 3.2.9
Let V be a finite-dimensional representation of G over Qp
B∞⊗BG,n (B∞)G0−an, pnΓ−an = B∞⊗Qp Can (G0, Qp). 0. It follows from Section 2.1.4 that this is equivariant with respect to the following actions of G0: the natural action on (B∞)G0−an, pnΓ−an, the right translation action on Can (G0, Qp) and trivial actions on both B∞
Summary
We obtain a (D , GL2 (Qp))module on Fl, which is very similar to the picture in the complex analytic setting (for example, the work of Kashiwara–Schmid [KS94]) As a corollary, this implies that on the locally analytic vectors in the completed cohomology, the infinitesimal character of GL2 (Qp) (i.e., action of Z (U (g))) and the infinitesimal character of GQp (Sen operator as an element in the centre of ‘C ⊗Qp Lie(GQp )’) are closely related. This implies that on the locally analytic vectors in the completed cohomology, the infinitesimal character of GL2 (Qp) (i.e., action of Z (U (g))) and the infinitesimal character of GQp (Sen operator as an element in the centre of ‘C ⊗Qp Lie(GQp )’) are closely related This directly implies the (Hodge–Tate) decomposition in Theorem 1.0.1.
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