Euler systems for higher \(K\)-theory of number fields
type: Item
instance of: scholarly article
title: Euler systems for higher \(K\)-theory of number fields
zbMATH Open document ID: 0851.19003
author: Grzegorz Banaszak
author: Wojciech Gajda
published in: Journal of Number Theory
publication date: 1996-11-18T00:00:00Z
review text: The general notion of Euler systems attached to number fields was introduced by Kolyvagin, and special Euler systems of cyclotomic elements or Gauss sums, etc. \(\dots\) were used by Kolyvagin, Rubin and others to prove some important finiteness theorems in algebraic number theory. In Compos. Math. 81, No. 2, 223-236 (1992; Zbl 0747.11055), \textit{M. Kurihara} showed that the Deligne-Soulé cyclotomic elements \(c (\eta)\), where \(\eta\) is a root of unity of order not divisible by the prime number \(p\), constitute Euler systems in the étale cohomology groups \(H^1_{\text{ét}} (\mathbb{Z} [1/p, \eta], \mathbb{Z}_p (n + 1))\), and he used them to prove the finiteness of the groups \(H^2_{\text{ét}} (\mathbb{Z} [1/p],\;\mathbb{Z}_p (n + 1))\), for even \(n \geq 2\) (independently of Borel's results on \(K\)-groups; but note that Kurihara's proof, as published, is not quite correct). In a similar way, \textit{K. Kato} has recently constructed Euler systems for \(K_2\) of modular curves [Kodai Math. J. 16, No. 1, 1-31 (1993; Zbl 0798.11050)]. In the present paper, the authors construct Euler systems in higher odd \(K\)-groups of number fields. Fix an odd prime number \(p\) and a natural integer \(m\). Let \(S\) be the set of square free natural integers \(L\) whose prime factors split completely in a number field \(F\) and are congruent to 1 modulo \(p^m\). Let \(\{\alpha (L) \in F (\mu_{p^mL})\); \(L \in S\}\) be an Euler system in the sense of \textit{K. Rubin} [J. Reine Angew. Math. 425, 141-154 (1992; Zbl 0752.11045)]. Multiplying by special Bott elements and taking transfer from \(F (\mu_{p^m L})\) to \(F (\mu_L)\), the authors construct an Euler system of elements \(\wedge (L)\) living in \(K_{2n + 1} (F (\mu_L), \mathbb{Z}/p^m)\). Choosing interesting \(\alpha (L)\), one gets interesting Euler systems in \(K\)-theory and applications to number theory: 1) Taking \(\alpha (L)\) to be special cyclotomic elements and applying the Dwyer-Friedlander map, one recovers…
Mathematics Subject Classification ID: 19F27
Mathematics Subject Classification ID: 11R70
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11S40
zbMATH DE Number: 894605
zbMATH Keywords: \(p\)-adic \(L\)-functions
zbMATH Keywords: main theorem of Iwasawa theory
zbMATH Keywords: \(S\)-units
zbMATH Keywords: Euler systems
zbMATH Keywords: étale cohomology
zbMATH Keywords: Bott elements
zbMATH Keywords: cyclotomic elements
zbMATH Keywords: Gauss sums
zbMATH Keywords: Iwasawa theory
zbMATH Keywords: Lichtenbaum conjecture
zbMATH Keywords: wild kernel
zbMATH Keywords: class number formulas
reviewed by: Thong Nguyen Quang Do
MaRDI profile type: Publication
full work available at URL: jnth.1996.0075
OpenAlex ID: W2053341394
DOI: 10.1006/JNTH.1996.0075
description: scientific article; zbMATH DE number 894605