On the integrality of characteristic elements in nonabelian Iwasawa theory
type: Item
instance of: scholarly article
title: On the integrality of characteristic elements in nonabelian Iwasawa theory
zbMATH Open document ID: 1149.11051
author: Ling Quan Kong
publication date: 2008-06-11T00:00:00Z
review text: For a compact \(p\)-adic Lie group \( G \) having a closed normal subgroup \(H\) such that \(G / H\) is isomorphic to \(\mathbb Z_p\), and a canonical Ore set \( S^{*} \) in the Iwasawa algebra \(\Lambda(G)\), in this paper the author gives several conjectures on the integrality of elements in \(K_1(\Lambda(G)_{S^{*}})\), and prove certain part of them. These conjectures are almost equivalent to the previous one formulated by \textit{J.~Coates} et al. in [Publ. Math., Inst. Hautes Étud. Sci. 101, 163--208 (2005; Zbl 1108.11081)]. Let \(K\) be a finite extension of \(\mathbb Q_p\), \(\mathbb O\) the ring of integers of \(K\). For a \(p\)-adic representation \(\rho: G \to \text{GL}_n(\mathbb O)\) of \(G\), define the induced homomorphisms \(\Phi _{\rho}: \Lambda (G) \to M_n(\Lambda _{\mathbb O}(\Gamma))\) and \(\Phi' _{\rho}: K_1(\Lambda(G)_{S^{*}}) \to \mathbb Q_{\mathbb O}(\Gamma)^{\times}\) \((\Gamma = G/H \simeq \mathbb Z_p)\). We also denote \(A_1 = \Lambda (G)_S\), \(A_2 = \Lambda (G) [\frac{1}{p}]\), \(A_3 = \Lambda (G)_{S^{*}}\), \(R_1 = \Lambda (\Gamma)_{(p)}\), \(R_2 = \Lambda (\Gamma) [\frac{1}{p}]\), \(R_3 = \mathbb Q(\Gamma)\), \(R_{i, \mathbb O} = R_{i} \otimes _{\mathbb Z_p} \mathbb O\) \((i= 0, 2, 3)\), \(R_{1, \mathbb O} = \) the localization of \(\Lambda _{\mathbb O} (\Gamma)\) at the prime \((\pi_K)\). Conjecture 1.1 states that if \(G\) has no elements of order \(p\), then in each of the following new cases, the assertions (a), (b) and (c) are equivalent for any element \(\xi \in K_1 (A_3)\): Case V: (a) \(\xi \in \alpha (A_1 \cap A_3^{\times})\), (b) \(\Phi' _{\rho}(\xi) \in R_{1, \mathbb O}\) for all \(\rho\), (c) \(\Phi' _{\rho}(\xi) \in R_{1, \mathbb O}\) for all Artin representations \(\rho\). Case VI: (a) \(\xi\in \alpha (A_{1}^{\times})\), (b) \( \Phi _{\rho }^{'} (\xi) \in R_{1, \mathbb O}^{\times}\) for all \(\rho\), (c) \(\Phi '_{\rho} (\xi ) \in R_{1, \mathbb O}^{\times}\) for all Artin representations \(\rho\) (where \(\alpha : A_{3}^{\…
reviewed by: Derong Qiu
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 5286517
zbMATH Keywords: \(p\)-adic Lie group
zbMATH Keywords: \(K\) group
zbMATH Keywords: characteristic element
zbMATH Keywords: Iwasawa theory
MaRDI profile type: Publication
full work available at URL: s10114-007-1006-9
OpenAlex ID: W2112111409
cites work: On the conjecture of Birch and Swinnerton-Dyer
cites work: Q3972988
cites work: On \(\mathbb Z_{\ell}\)-extensions of algebraic number fields
cites work: The \(\text{GL}_2\) main conjecture for elliptic curves without complex multiplication
DOI: 10.1007/S10114-007-1006-9
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 5286517