Locally analytic vectors of some crystabelian representations of GL_2(Qp)

For V a 2-dimensional p-adic representation of G_Qp, we denote by B(V) the admissible unitary representation of GL_2(Qp) attached to V under the p-adic local Langlands correspondence of GL_2(Qp) initiated by Breuil. In this article, building on the works of Berger-Breuil and Colmez, we determine the locally analytic vectors B(V)an of B(V) when V is irreducible, crystabelian and Frobenius semi-simple with Hodge-Tate weights (0,k-1) for some integer k>=2; this proves a conjecture of Breuil. Using this result, we verify Emerton's conjecture that dim Ref^{\eta\otimes\psi}(V)=dim Exp^{\eta|\cdot|\otimes x\psi}(B(V)an\otimes(x|\cdot|\circ\det)) for those V which are irreducible, crystabelian and not exceptional.Comment: Refereed version. Title is changed. Minor changes in the conten

(\phi,\Gamma)-modules over noncommutative overconvergent and Robba rings

We construct noncommutative multidimensional versions of overconvergent power series rings and Robba rings. We show that the category of \'etale $(\varphi,\Gamma)$-modules over certain completions of these rings are equivalent to the category of \'etale $(\varphi,\Gamma)$-modules over the corresponding classical overconvergent, resp. Robba rings (hence also to the category of $p$-adic Galois representations of $\mathbb{Q}_p$). Moreover, in the case of Robba rings, the assumption of \'etaleness is not necessary, so there exists a notion of trianguline objects in this sense.Comment: 41 pages, revise

Triangulable \CO_F-analytic (φq,Γ)(\varphi_q,\Gamma)-modules of rank 2

The theory of $(\varphi_q,\Gamma)$-modules is a generalization of Fontaine's theory of $(\varphi,\Gamma)$-modules, which classifies $G_F$-representations on \CO_F-modules and $F$-vector spaces for any finite extension $F$ of \BQ_p. In this paper following Colmez's method we classify triangulable \CO_F-analytic $(\varphi_q,\Gamma)$-modules of rank 2. In this process we establish two kinds of cohomology theories for \CO_F-analytic $(\varphi_q,\Gamma)$-modules. Using them we show that, if $D$ is an \CO_F-analytic $(\varphi_q,\Gamma)$-module such that $D^{\varphi_q=1,\Gamma=1}=0$ i.e. $V^{G_F}=0$ where $V$ is the Galois representation attached to $D$, then any overconvergent extension of the trivial representation of $G_F$ by $V$ is \CO_F-analytic. In particular, contrarily to the case of F=\BQ_p, there are representations of $G_F$ that are not overconvergent.Comment: 35 page

Exactness of the reduction on \'etale modules

We prove the exactness of the reduction map from \'etale $(\phi,\Gamma)$-modules over completed localized group rings of compact open subgroups of unipotent $p$-adic algebraic groups to usual \'etale $(\phi,\Gamma)$-modules over Fontaine's ring. This reduction map is a component of a functor from smooth $p$-power torsion representations of $p$-adic reductive groups (or more generally of Borel subgroups of these) to $(\phi,\Gamma)$-modules. Therefore this gives evidence for this functor---which is intended as some kind of $p$-adic Langlands correspondence for reductive groups---to be exact. We also show that the corresponding higher \Tor-functors vanish. Moreover, we give the example of the Steinberg representation as an illustration and show that it is acyclic for this functor to $(\phi,\Gamma)$-modules whenever our reductive group is \GL_{d+1}(\mathbb{Q}_p) for some $d\geq 1$.Comment: 18 pages; some typos corrected and proof of Lemma 1 rewritten, to appear in Journal of Algebr

LOCALLY ANALYTIC VECTORS AND OVERCONVERGENT (phi, tau)-MODULES

Let p be a prime, let K be a complete discrete valuation field of characteristic 0 with a perfect residue field of characteristic p, and let G(K) be the Galois group. Let pi be a fixed uniformizer of K, let K-infinity be the extension by adjoining to K a system of compatible p(n) th roots of pi for all n, and let L be the Galois closure of K-infinity. Using these field extensions, Caruso constructs the (phi, tau)-modules, which classify p-adic Galois representations of G(K). In this paper, we study locally analytic vectors in some period rings with respect to the p -adic Lie group Gal(L/K), in the spirit of the work by Berger and Colmez. Using these locally analytic vectors, and using the classical overconvergent (phi, Gamma)-modules, we can establish the overconvergence property of the (phi, tau)-modules.Peer reviewe

The p-adic L-functions of Evil Eisenstein Series

We compute the $p$-adic $L$-functions of evil Eisenstein series, showing that they factor as products of two Kubota--Leopoldt $p$-adic $L$-functions times a logarithmic term. This proves in particular a conjecture of Glenn Stevens.Comment: 49 page

Filtered modules corresponding to potentially semi-stable representations

We classify the filtered modules with coefficients corresponding to two-dimensional potentially semi-stable $p$-adic representations of the absolute Galois groups of $p$-adic fields under the assumptions that $p$ is odd and the coefficients are large enough.Comment: 19 page