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LOCALLY ANALYTIC VECTORS AND OVERCONVERGENT
$(\unicode[STIX]{x1D711},\unicode[STIX]{x1D70F})$-MODULES
Published online by Cambridge University Press:Â 03 May 2019
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.
MSC classification
- Type
- Research Article
- Information
- Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 1 , January 2021 , pp. 137 - 185
- Copyright
- © Cambridge University Press 2019
References
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