Geodesic
Arithmetic Families of $(\varphi,\Gamma)$-Modules and Locally Analytic Representations of $GL_2(Q_p)$
Documenta mathematica, Tome 23 (2018), pp. 1313-1404.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Let $A$ be a $\bold{Q}_p$-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 $(\varphi,\Gamma)$-module of dimension $2$ over the relative Robba ring $\Cal R_A$ we construct a locally analytic $GL_2(Q_p)$-representation in $A$-modules.
Classification : 11S37, 11S25, 20G05, 22E50
Keywords: $p$-adic Langlands correspondence, $(\varphi,\Gamma)$-modules, affinoid algebras, locally analytic representations 
@article{DOCMA_2018__23__a23,
     author = {Gaisin, Ildar and Rodrigues Jacinto, Joaqu{\'\i}n},
     title = {Arithmetic {Families} of $(\varphi,\Gamma)${-Modules} and {Locally} {Analytic} {Representations} of $GL_2(Q_p)$},
     journal = {Documenta mathematica},
     pages = {1313--1404},
     publisher = {mathdoc},
     volume = {23},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2018__23__a23/}
}
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Gaisin, Ildar; Rodrigues Jacinto, Joaquín. Arithmetic Families of $(\varphi,\Gamma)$-Modules and Locally Analytic Representations of $GL_2(Q_p)$. Documenta mathematica, Tome 23 (2018), pp. 1313-1404. http://geodesic.mathdoc.fr/item/DOCMA_2018__23__a23/