Geodesic
Arithmetic Families of $(\varphi,\Gamma)$-Modules and Locally Analytic Representations of $GL_2(Q_p)$
Documenta mathematica, Tome 23 (2018), pp. 1313-1404
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Let A be a Qp​-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 RA​ we construct a locally analytic GL2​(Qp​)-representation in A-modules.
DOI : 10.4171/dm/649
Classification : 11S25, 11S37, 20G05, 22E50
Mots-clés : (φ,Γ)-modules, p-adic Langlands correspondence, affinoid algebras, locally analytic representations 
@article{10_4171_dm_649,
     author = {Ildar Gaisin and Joaqu{\'\i}n Rodrigues Jacinto},
     title = {Arithmetic {Families} of $(\varphi,\Gamma)${-Modules} and {Locally} {Analytic} {Representations} of $GL_2(Q_p)$},
     journal = {Documenta mathematica},
     pages = {1313--1404},
     year = {2018},
     volume = {23},
     doi = {10.4171/dm/649},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/649/}
}
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Ildar Gaisin; Joaquín Rodrigues Jacinto. Arithmetic Families of $(\varphi,\Gamma)$-Modules and Locally Analytic Representations of $GL_2(Q_p)$. Documenta mathematica, Tome 23 (2018), pp. 1313-1404. doi: 10.4171/dm/649

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