Systèmes d'Euler \(p\)-adiques et théorie d'Iwasawa. (\(p\)-adic Euler systems and Iwasawa theory.)
type: Item
instance of: scholarly article
title: Systèmes d'Euler \(p\)-adiques et théorie d'Iwasawa. (\(p\)-adic Euler systems and Iwasawa theory.)
zbMATH Open document ID: 0930.11078
author: Bernadette Perrin-Riou
publication date: 1998-11-29T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75319
review text: In number theory, the importance of the notion of Euler systems introduced by \textit{V. A. Kolyvagin} in 1988 [Math. USSR, Izv. 33, 473-499 (1989); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 52, 1154-1180 (1988; Zbl 0681.14016)] can hardly be underestimated. In this paper, the author concentrates on the study of \(p\)-adic Euler systems, that is Euler systems attached to \(p\)-adic representations \(V\) of the absolute Galois group \(G_\mathbb{Q}\). The traditional congruences in cyclotomic Euler systems (namely \(1-\xi_{m\ell}\equiv 1-\xi_m\) modulo the places dividing the prime number \(\ell)\) are replaced by trace relations in cyclotomic extensions \(\mathbb{Q}(\xi_{m\ell})/\mathbb{Q}(\xi_m)\). These can be translated into relations between \(p\)-adic ``functions'' Gal\((\mathbb{Q}(\xi_{mp^\infty})/\mathbb{Q})\to{\mathbf D}_p(V)\), relations which in turn reflect the factorization of classical \(L\)-functions by means of Euler products. Modulo certain technical conditions on the representation \(V\), the author proves the ``Leopoldt conjecture for \(V\)'' (which asserts that a certain Iwasawa module attached to \(V\) is \(\Lambda\)-torsion), as well as some results on the divisibility of certain characteristic series in Iwasawa theory. Actually, the paper contains a host of informations on \(p\)-adic Euler systems, such as local properties of Euler systems, Kolyvagin's derivation, applications of Chebotarev's theorem \(\dots\), but the technicality inherent to the subject makes it difficult to give a fair summary.
reviewed by: Thong Nguyen Quang Do
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 1227211
zbMATH Keywords: divisibility of characteristic series
zbMATH Keywords: \(p\)-adic Euler systems
zbMATH Keywords: Iwasawa module
zbMATH Keywords: Leopoldt conjecture
zbMATH Keywords: trace relations in cyclotomic extensions
zbMATH Keywords: Kolyvagin's derivation
zbMATH Keywords: applications of Chebotarev's theorem
MaRDI profile type: Publication
OpenAlex ID: W2026576169
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DOI: 10.5802/AIF.1655
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1227211