Logarithmic classes of number fields
type: Item
instance of: scholarly article
title: Logarithmic classes of number fields
zbMATH Open document ID: 0827.11064
author: Jean-François Jaulent
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1995-06-11T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93606
full work available at URL: jtnb6-2.html
review text: Fix a prime \(\ell\). Let \(K\) be a number field with associated local field \(K_{\mathfrak p}\) at the prime \({\mathfrak p}\). Motivated by an explicit formula for certain Hilbert symbols one is led to consider a \(\mathbb{Z}_\ell\)-valued valuation map \(\widetilde {v}_{\mathfrak p}\) on \(K_{\mathfrak p}\) defined via the \(\ell\)-adic (Iwasawa) logarithm \(\text{Log}= \text{Log}_{Iw}\). This logarithmic valuation \(\widetilde {v}_{\mathfrak p}\) is actually proportional to the traditional valuation \(v_{\mathfrak p}\) for \({\mathfrak p}\) not dividing \(\ell\). It is a \(\mathbb{Z}_\ell\)-epimorphism of the \(\ell\)-adic compactification \({\mathcal K}^\times_{\mathfrak p}:= \varprojlim K^\times_{\mathfrak p}/ K_{\mathfrak p}^{\times \ell^n}\) of \(K^\times_{\mathfrak p}\), with kernel given by the submodule \({\mathcal K}^*_{\mathfrak p}\) of the cyclotomic norms in \({\mathcal K}^\times_{\mathfrak p}\). One also defines a logarithmic (absolute) ramification index \(\widetilde {e}_{\mathfrak p}\), the logarithmetic inertial degree \(\widetilde {f}_{\mathfrak p}\) of \(K_{\mathfrak p}\) and the \(\ell\)-adic degree \(\deg {\mathfrak p}\) of the ideal \({\mathfrak p}\). Then, by definition, \(\widetilde {v}_{\mathfrak p}=- \text{Log}|\;|_p/ \deg {\mathfrak p}\), where \(|\;|_{\mathfrak p}\) is the usual absolute value on \(K^\times_{\mathfrak p}\). One defines the \(\ell\)-group of logarithmic divisors \({\mathcal D} \ell_K\) of \(K\) as the free \(\mathbb{Z}_\ell\)-module on the finite places \({\mathfrak p}\) of \(K\): \({\mathcal D}\ell_K:= \bigoplus_{{\mathfrak p}\text{ finite}} \mathbb{Z}_\ell {\mathfrak p}\). By linear extension of the \(\ell\)-adic degree map one gets a degree map on \({\mathcal D} \ell_K\). One writes \(\widetilde {{\mathcal D} \ell}_K\) for the degree zero submodule of \({\mathcal D} \ell_K\). Write \({\mathcal R}_K= \mathbb{Z}_\ell \otimes_\mathbb{Z} K^\times\); then the \({\mathcal D} \ell_K\)-valued map \(\widetilde{\text{div}}_K: …
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 12J20
zbMATH DE Number: 754095
zbMATH Keywords: logarithmic classes
zbMATH Keywords: logarithmic genera
reviewed by: W. W. J. Hulsbergen
MaRDI profile type: Publication
OpenAlex ID: W2008501740
cites work: Regulators and Iwasawa modules
cites work: Q3735801
cites work: The Stark conjectures on Artin \(L\)-functions at \(s=0\). Lecture notes of a course in Orsay edited by Dominique Bernardi and Norbert Schappacher.
DOI: 10.5802/JTNB.117
description: scientific article; zbMATH DE number 754095