Abstract
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.
Résumé
Nous décrivons complètement K\(_{1}(\mathbb {Z}_p[\![\mathcal {G}_{\infty }]\!])\)et ses localisations en utilisant une famille infinie de congruencesp-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.
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Acknowledgements
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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Qin Chao: To form a part of this author’s PhD thesis.
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Delbourgo, D., Chao, Q. \(\hbox {K}_{1}\)-congruences for three-dimensional Lie groups. Ann. Math. Québec 43, 161–211 (2019). https://doi.org/10.1007/s40316-018-0100-y
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DOI: https://doi.org/10.1007/s40316-018-0100-y