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LOCALLY ANALYTIC VECTORS AND OVERCONVERGENT $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$-MODULES

Part of: Algebraic number theory: local and $p$-adic fields Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  03 May 2019

Hui Gao  and
LĂ©o Poyeton
Hui Gao
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, FI-00014, Finland (hui.gao@helsinki.fi)
LĂ©o Poyeton
Affiliation:
UMPA, École Normale SupĂ©rieure de Lyon, 46 allĂ©e d’Italie, 69007Lyon, France (leo.poyeton@ens-lyon.fr)
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Abstract

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 $\unicode[STIX]{x1D70B}$ be a fixed uniformizer of $K$, let $K_{\infty }$ be the extension by adjoining to $K$ a system of compatible $p^{n}$th roots of $\unicode[STIX]{x1D70B}$ for all $n$, and let $L$ be the Galois closure of $K_{\infty }$. Using these field extensions, Caruso constructs the $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$-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 $\operatorname{Gal}(L/K)$, in the spirit of the work by Berger and Colmez. Using these locally analytic vectors, and using the classical overconvergent $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules, we can establish the overconvergence property of the $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$-modules.

Keywords

locally analytic vectorsoverconvergence(𝜑, 𝜏)-modules

MSC classification

Primary: 11F80: Galois representations 11S20: Galois theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 1 , January 2021 , pp. 137 - 185
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Copyright
© Cambridge University Press 2019

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