Arithmetic Families of $(\varphi,\Gamma)$-Modules and Locally Analytic Representations of $GL_2(Q_p)$

Ildar Gaisin; Joaquín Rodrigues Jacinto

doi:10.4171/dm/649

JournalsdmVol. 23pp. 1313–1404

Arithmetic Families of $(φ, Γ)$ -Modules and Locally Analytic Representations of $G L_{2} (Q_{p})$

Ildar Gaisin
University of Tokyo, 7 Chome-3-1 Hongo, Bunkyo, Tokyo 113-8654, Japan
Joaquín Rodrigues Jacinto
University College London, 25 Gordon street, WC1H 0AY, London, UK

$Arithmetic Families of $(\varphi,\Gamma)$-Modules and Locally Analytic Representations of $GL_2(Q_p)$ cover$

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Abstract

Let $A$ be a $Q_{p}$ -affinoid algebra, in the sense of Tate. We develop a theory of locally convex $A$ -modules parallel to the treatment in the case of a field by Schneider and Teitelbaum. We prove that there is an integration map linking a category of locally analytic representations in $A$ -modules and separately continuous relative distribution modules. There is a suitable theory of locally analytic cohomology for these objects and a version of Shapiro's Lemma, generalizing results of Kohlhaase. As an application we propose a $p$ -adic Langlands correspondence in families: For a regular trianguline $(φ, Γ)$ -module of dimension $2$ over the relative Robba ring $R_{A}$ we construct a locally analytic $G L_{2} (Q_{p})$ -representation in $A$ -modules.

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Ildar Gaisin, Joaquín Rodrigues Jacinto, Arithmetic Families of $(φ, Γ)$ -Modules and Locally Analytic Representations of $G L_{2} (Q_{p})$ . Doc. Math. 23 (2018), pp. 1313–1404

DOI 10.4171/DM/649