{"id":{"repo_id":"waikato-masters","oai_identifier":"oai:researchcommons.waikato.ac.nz:10289/12250"},"canonical_url":"https://search.dev.ndltd.org/etd/waikato-masters/oai:researchcommons.waikato.ac.nz:10289/12250","repository":{"repo_id":"waikato-masters","name":"University Waikato","base_url":"https://researchcommons.waikato.ac.nz/server/oai/request"},"display":{"title":"Iwasawa theory over solvable three-dimensional p-adic Lie extensions","abstract":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Qin, Chao"],"institution":"The University of Waikato","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Delbourgo, Daniel"],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018","date_published":"2018","updated_at":"2026-07-24T05:57:46Z","subjects":["Iwasawa theory","K-theory","p-adic L-functions","Galois representations"],"languages":[],"rights":["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."],"rights_urls":["https://researchcommons.waikato.ac.nz/bitstreams/8ec6bf2f-b153-482d-b386-fe00218576ad/download"],"identifier_entries":[]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Delbourgo, Daniel"]},{"key":"dc:creator","label":"Author","values":["Qin, Chao"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2018"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["The University of Waikato"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["https://hdl.handle.net/10289/12250"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Iwasawa theory","K-theory","p-adic L-functions","Galois representations"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["https://researchcommons.waikato.ac.nz/bitstreams/8ec6bf2f-b153-482d-b386-fe00218576ad/download","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."]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://researchcommons.waikato.ac.nz/bitstreams/9f9d122e-8f09-4d84-b967-b60ea86bdd7b/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Iwasawa theory is a powerful tool which describes the mysterious relationship between arithmetic objects (motives) and the special values of L-functions. 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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. 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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. 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