$$\hbox {K}_{1}$$ K 1 -congruences for three-dimensional Lie groups
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K1 -congruences for three-dimensional Lie groups Daniel Delbourgo1
· Qin Chao1
Received: 22 August 2017 / Accepted: 21 March 2018 © Fondation Carl-Herz and Springer International Publishing AG, part of Springer Nature 2018
Abstract We completely describe K1 (Z p [[G∞ ]]) and its localisations by using an infinite family of p-adic congruences, where G∞ is any solvable p-adic Lie group of dimension 3. This builds on earlier work of Kato when dim(G∞ ) = 2, and of the first named author and d × Lloyd Peters when G∞ ∼ = Z× p Z p with a scalar action of Z p . The method exploits the classification of 3-dimensional p-adic Lie groups due to González-Sánchez and Klopsch, as well as the fundamental ideas of Kakde, Burns, etc. in non-commutative Iwasawa theory. Résumé Nous décrivons complètement K1 (Z p [[G∞ ]]) et ses localisations en utilisant une famille infinie de congruences p-adiques, où G∞ est un groupe de Lie résoluble de dimension trois. Ce travail s’appuie sur les résultats de Kato lorsque dim(G∞ ) = 2, et du premier auteur d × et Lloyd Peters lorsque G∞ ∼ = Z× p Z p avec une action scalaire de Z p . La méthode exploite la classification des groupes de Lie de dimension trois due à González-Sánchez et Klopsch, ainsi que les idées fondamentales de Kakde, Burns etc. en théorie d’Iwasawa non-commutative. Keywords Iwasawa theory · K -theory · p-adic L-functions · Galois representations Mathematics Subject Classification 11R23, 11G40, 19B28
Contents 1 Introduction . . . . . . . . . . . . . 1.1 Preliminaries . . . . . . . . . . 1.2 The main results . . . . . . . . . 1.3 Some arithmetic examples . . . 2 The general set-up in dimension three
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Qin Chao: To form a part of this author’s PhD thesis.
B 1
Daniel Delbourgo [email protected] Department of Mathematics, University of Waikato, Hamilton 3240, New Zealand
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D. Delbourgo, Q. Chao 2.1 Determining the stabilizer of a character on H∞ . 2.2 A “coarse but clean” system of subgroups . . . . 2.3 Maps between the abelianizations of Um,n . . . . 3 The additive calculations . . . . . . . . . . . . . . . . (m,n) . 3.1 The image of under the characters on H∞ 3.2 A transfer-compatible basis for the set Rm,n . . . 4 The multiplicative calculations . . . . . . . . . . . . 4.1 Convergence of the logarithm on Im(σm ) . . . . 4.2 Interaction of the theta-maps with both ϕ and log 4.3 The image of the Taylor-Oliver logarithm . . . . 4.4 A proof of Theorems 1 and 2 . . . . . . . . . . . 5 Computing the terms in Theorems 1 and 2 . . . . . . 5.1 A worked example for Case (II) . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .
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