\(\ell\)-adic embeddings and \(\ell\)-adic Weil numbers
type: Item
instance of: scholarly article
title: \(\ell\)-adic embeddings and \(\ell\)-adic Weil numbers
zbMATH Open document ID: 1173.11055
author: Jean-François Jaulent
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2009-04-14T00:00:00Z
full work available at URL: https://eudml.org/doc/10839
review text: The classical Weil numbers are elements of the algebraic closure \(\overline{\mathbb{Q}}\) of \(\mathbb{Q}\) whose images under all embeddings into \(\mathbb{C}\) have absolute value \(1\). These are also referred to as Weil numbers `at infinity' and their connexions with totally real and CM (complex multiplication) fields are reviewed in the first part of this article. The author then defines \(\ell\)-adic analogues of the classical Weil numbers. In addition, he defines the \(\ell\)-adic counterparts to totally real and CM fields, and discusses their basic properties. As an application, some consequences in Iwasawa theory are given. In particular, the conjectures of Gross and Leopoldt come into play and, under certain (strong) hypotheses, a lower bound for for Iwasawa's \(\lambda\)-invariant attached to the \(\ell\)-parts of ideal class groups in a cyclotomic \(\mathbb{Z}_{\ell}\)-extension is given.
reviewed by: Henri Johnston
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 5543165
zbMATH Keywords: Weil numbers
zbMATH Keywords: totally real fields
zbMATH Keywords: CM fields
zbMATH Keywords: Iwasawa theory
MaRDI profile type: Publication
OpenAlex ID: W1648893613
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DOI: 10.5802/JTNB.630
description: scientific article; zbMATH DE number 5543165