\(\hbox {K}_{1}\)-congruences for three-dimensional Lie groups

  • Daniel Delbourgo
  • Qin Chao


We completely describe \(\hbox {K}_{1}({\mathbb {Z}}_p[\![{\mathcal {G}}_{\infty }]\!])\) and its localisations by using an infinite family of p-adic congruences, where \({\mathcal {G}}_{\infty }\) is any solvable p-adic Lie group of dimension 3. This builds on earlier work of Kato when \(\hbox {dim}({\mathcal {G}}_{\infty })=2\), and of the first named author and Lloyd Peters when \({\mathcal {G}}_{\infty } \cong {\mathbb {Z}}_p^{\times }\ltimes {\mathbb {Z}}_p^d\) with a scalar action of \({\mathbb {Z}}_p^{\times }\). 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.


Iwasawa theory K-theory p-adic L-functions Galois representations 


Nous décrivons complètement K\(_{1}(\mathbb {Z}_p[\![\mathcal {G}_{\infty }]\!])\) et ses localisations en utilisant une famille infinie de congruences p-adiques, où \(\mathcal {G}_{\infty }\) est un groupe de Lie résoluble de dimension trois. Ce travail s’appuie sur les résultats de Kato lorsque \({\hbox {dim}}(\mathcal {G}_{\infty })=2\), et du premier auteur et Lloyd Peters lorsque \(\mathcal {G}_{\infty }\cong \mathbb {Z}_p^{\times }\ltimes \mathbb {Z}_p^d\) avec une action scalaire de \(\mathbb {Z}_p^{\times }\). 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.

Mathematics Subject Classification

11R23 11G40 19B28 



The authors are grateful to both Antonio Lei and Lloyd Peters for numerous discussions about non-commutative congruences. They were also hugely inspired by the work of Mahesh Kakde, to which many arguments in this paper owe a great debt. Lastly they thank Ian Hawthorn for his friendly guidance during some difficult times.


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© Fondation Carl-Herz and Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WaikatoHamiltonNew Zealand

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