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The main conjecture of Iwasawa theory for totally real fields

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Let p be an odd prime. Let \(\mathcal{G}\) be a compact p-adic Lie group with a quotient isomorphic to ℤ p . We give an explicit description of K 1 of the Iwasawa algebra of \(\mathcal{G}\) in terms of Iwasawa algebras of Abelian subquotients of \(\mathcal{G}\). We also prove a result about K 1 of a certain canonical localisation of the Iwasawa algebra of \(\mathcal{G}\), which occurs in the formulation of the main conjectures of noncommutative Iwasawa theory. These results predict new congruences between special values of Artin L-functions, which we then prove using the q-expansion principle of Deligne-Ribet. As a consequence we prove the noncommutative main conjecture for totally real fields, assuming a suitable version of Iwasawa’s conjecture about vanishing of the cyclotomic μ-invariant. In particular, we get an unconditional result for totally real pro-p p-adic Lie extension of Abelian extensions of ℚ.

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Acknowledgements

I have accumulated quite a debt of gratitude in writing this paper. Most of all to my teacher Professor John Coates for introducing me to this problem, many invaluable suggestions and discussions and for constant inspiration. To Professor Kazuya Kato for generously sharing his ideas on the main conjecture with me while I was a graduate student in Cambridge. To Professor David Burns for motivating discussions and much needed encouragement towards the end of this paper. I would like to thank Professor Peter Schneider and Professor Otmar Venjakob for carefully reading an earlier version of the manuscript and pointing out several errors. Much of this work was done while I was visiting Newton Institute for the programme on “Non-Abelian Fundamental Groups in Arithmetic Geometry” and I thank the organisers, especially Professor Minhyong Kim, for inviting me and providing a very stimulating environment. I would like to thank the anonymous referee for careful reading of the manuscript and making many helpful remarks.

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  1. Department of Mathematics, King’s College London, London, WC2R 2LS, UK

    Mahesh Kakde

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Kakde, M. The main conjecture of Iwasawa theory for totally real fields. Invent. math. 193, 539–626 (2013). https://doi.org/10.1007/s00222-012-0436-x

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