\(p\)-adic representations of a local field
type: Item
instance of: scholarly article
title: \(p\)-adic representations of a local field
zbMATH Open document ID: 0961.11038
author: Pierre Colmez
published in: Documenta Mathematica
publication date: 1998-08-06T00:00:00Z
full work available at URL: https://eudml.org/doc/232707
full work available at URL: Colmez.MAN.html
review text: Let \(K\) be a finite extension of the field \(\mathbb Q_p\) of \(p\)-adic numbers and let \(\mathcal G_K= \text{Gal}(\overline{K}/K)\). Let \(V\) be a finite-dimensional vector space over \(\mathbb Q_p\) that is a representation space for \(\mathcal G_K\). Associated to \(V\) is a vector space \(D(V)\) over a field \textbf{B}\(_K\) constructed by Fontaine. The space \(D(V)\), called the \((\phi,\Gamma)\)-module of \(V\), is much easier to work with than the original \(V\), and it is possible to deduce properties of \(V\) from those of \(D(V)\). The present article surveys some applications of this fact. \textit{R. Coleman} [Invent. Math. 53, 91-116 (1979; Zbl 0429.12010)] constructed a map from the inverse limit of units in the cyclotomic tower over \(K\) to measures on \(\mathbb Z_p^*\), when \(K/\mathbb Q_p\) is unramified. \textit{B. Perrin-Riou} [Invent. Math. 115, 81-149 (1994; Zbl 0838.11071)] generalized this construction to crystalline representations of \(\mathcal G_K\), which allowed her to give a conjectural definition of the \(p\)-adic \(L\)-function attached to a motive having good reduction at \(p\). The author describes extensions of these maps to more general representations and describes resulting explicit reciprocity laws. More details are contained in the author's papers [Ann. Math. (2) 148, No. 2, 485-571 (1998; Zbl 0928.11045)] and [with \textit{F. Cherbonnier}, J. Am. Math. Soc. 12, No. 1, 241-268 (1999; Zbl 0933.11056)].
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11F30
Mathematics Subject Classification ID: 11S20
zbMATH DE Number: 1184272
zbMATH Keywords: \(p\)-adic representations
zbMATH Keywords: local fields
zbMATH Keywords: Iwasawa theory
zbMATH Keywords: explicit reciprocity laws
reviewed by: Lawrence C. Washington
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1184272