2026-09-25T13:02:33Zhttps://researchcommons.waikato.ac.nz/server/oai/request

oai:researchcommons.waikato.ac.nz:10289/122502018-12-18T23:25:08Zcom_10289_2222col_10289_2223

Iwasawa theory over solvable three-dimensional p-adic Lie extensions Qin, Chao Delbourgo, Daniel Iwasawa theory is a powerful tool which describes the mysterious relationship between arithmetic objects (motives) and the special values of L-functions. A precise form of this relationship is neatly encoded in the so-called "Iwasawa Main Conjecture". Classically the Main Conjecture (as formulated by Iwasawa himself) involved the behaviour of ideal class groups over cyclotomic Zp-extensions, and related this to the Kubota-Leopoldt p-adic zeta-function. During the last two decades, the main conjecture has been greatly generalized to admissible p-adic Lie extensions, and provides a conjectural relationship between L-values of motives and their associated Selmer groups. A key component of the “Non-commutative Iwasawa Main Conjecture” in [CFK+05] predicts the existence of an analytic p-adic L-function L an M inside K1 ( Zp[[G∞]]S∗ ) . To establish the existence of such an object, we need to be able to do two things: (1) describe K1 ( Zp[[G∞]]S∗ ) in terms of the Artin representations factoring through G∞ using p-adic congruences, and then (2) show for each motive that the abelian fragments satisfy these congruences. This thesis provides a complete answer to the first task (1), in the specific situation where the pro-p-group G∞ has dimension ≤ 3 and is torsion-free. We completely describe K1(Zp[[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 Daniel Delbourgo and Lloyd Peters when G∞ ∼= Z × p ⋉Zd 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. We also undertake a short study of elliptic curves over GL2(Fp)-extensions, and compile some numerical evidence in support of the first layer congruences predicted by Kakde [Kak17] for non-CM curves. The University of Waikato 2018 Thesis en https://hdl.handle.net/10289/12250 https://researchcommons.waikato.ac.nz/bitstreams/9f9d122e-8f09-4d84-b967-b60ea86bdd7b/download e6cd0f333800ee8dd97475accde8427d https://researchcommons.waikato.ac.nz/bitstreams/8ec6bf2f-b153-482d-b386-fe00218576ad/download e14202ab27e47ddb00d33097327ba050 https://researchcommons.waikato.ac.nz/bitstreams/a2c82f08-9882-4eca-96df-4ea3b15657be/download 9a2b4a42b4fdc0164f92746f0bc5a10d All items in Research Commons are provided for private study and research purposes and are protected by copyright with all rights reserved unless otherwise indicated. Iwasawa theory K-theory p-adic L-functions Galois representations