{"id":{"repo_id":"western-cape","oai_identifier":"oai:uwcscholar.uwc.ac.za:10566/22683"},"canonical_url":"https://search.dev.ndltd.org/etd/western-cape/oai:uwcscholar.uwc.ac.za:10566/22683","repository":{"repo_id":"western-cape","name":"University of the Western Cape","base_url":"https://uwcscholar.uwc.ac.za:8443/server/oai/request"},"display":{"title":"The Birch and Swinnerton-Dyer Conjecture for elliptic curves","abstract":"he aim of this dissertation is to provide an exposition of the Birch and Swinnerton-Dyer Conjecture, considered by many to be one of the most important unsolved problems in modern Mathematics. A review of topics in Algebraic Number Theory and Algebraic Geometry is provided in order to provide a characterisation for elliptic curves over rational numbers. We investigate the group structure of rational points on elliptic curves, and show that this group is finitely generated by the Mordell-Weil Theorem. The Shafarevich-Tate group is introduced by way of an example. Thereafter, with the use of Galois Cohomology, we provide a general definition of this mysterious group. We also discuss invariants like the regulator and real period, which appear in the Birch and Swinnerton-Dyer Conjecture. After defining the L-function, we state the Birch and Swinnerton-Dyer Conjecture and discuss results which have been proved and some consequences. We discuss numerical verification of the Conjecture, and show some computations, including an example of our own.","abstract_html":"he aim of this dissertation is to provide an exposition of the Birch and Swinnerton-Dyer Conjecture, considered by many to be one of the most important unsolved problems in modern Mathematics. A review of topics in Algebraic Number Theory and Algebraic Geometry is provided in order to provide a characterisation for elliptic curves over rational numbers. We investigate the group structure of rational points on elliptic curves, and show that this group is finitely generated by the Mordell-Weil Theorem. The Shafarevich-Tate group is introduced by way of an example. Thereafter, with the use of Galois Cohomology, we provide a general definition of this mysterious group. We also discuss invariants like the regulator and real period, which appear in the Birch and Swinnerton-Dyer Conjecture. After defining the L-function, we state the Birch and Swinnerton-Dyer Conjecture and discuss results which have been proved and some consequences. We discuss numerical verification of the Conjecture, and show some computations, including an example of our own.","abstract_has_math":false,"creators":["Smith, Duncan"],"institution":"University of the Western Cape","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014","date_published":"2014","updated_at":"2026-07-24T06:00:42Z","subjects":["Elliptic curve","Birch and Swinnerton-Dyer conjecture","L-function","modern Mathematics","Algebraic Number Theory"],"languages":[],"rights":[],"rights_urls":["https://uwcscholar.uwc.ac.za/bitstreams/5926f36a-ddf0-465b-9353-85dab69c5887/download"],"identifier_entries":[]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Smith, Duncan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2014"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of the Western Cape"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["https://hdl.handle.net/10566/22683"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Elliptic curve","Birch and Swinnerton-Dyer conjecture","L-function","modern Mathematics","Algebraic Number Theory"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["https://uwcscholar.uwc.ac.za/bitstreams/5926f36a-ddf0-465b-9353-85dab69c5887/download"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://uwcscholar.uwc.ac.za/bitstreams/c12909d1-9cb4-42c1-a5d2-cfa0003829be/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["he aim of this dissertation is to provide an exposition of the Birch and Swinnerton-Dyer Conjecture, considered by many to be one of the most important unsolved problems in modern Mathematics. A review of topics in Algebraic Number Theory and Algebraic Geometry is provided in order to provide a characterisation for elliptic curves over rational numbers. We investigate the group structure of rational points on elliptic curves, and show that this group is finitely generated by the Mordell-Weil Theorem. The Shafarevich-Tate group is introduced by way of an example. Thereafter, with the use of Galois Cohomology, we provide a general definition of this mysterious group. We also discuss invariants like the regulator and real period, which appear in the Birch and Swinnerton-Dyer Conjecture. After defining the L-function, we state the Birch and Swinnerton-Dyer Conjecture and discuss results which have been proved and some consequences. We discuss numerical verification of the Conjecture, and show some computations, including an example of our own."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["8fc7000bdadd9e1e3e7c1d4ed62592d6","bb9bdc0b3349e4284e09149f943790b4","9a38b8e31330d1684300d9cab427dea6"]},{"key":"dc:title","label":"Title","values":["The Birch and Swinnerton-Dyer Conjecture for elliptic curves"]}]}],"canonical_facts":{"dc:creator":["Smith, Duncan"],"dc:date.issued":["2014"],"dc:description.abstract":["he aim of this dissertation is to provide an exposition of the Birch and Swinnerton-Dyer Conjecture, considered by many to be one of the most important unsolved problems in modern Mathematics. A review of topics in Algebraic Number Theory and Algebraic Geometry is provided in order to provide a characterisation for elliptic curves over rational numbers. We investigate the group structure of rational points on elliptic curves, and show that this group is finitely generated by the Mordell-Weil Theorem. The Shafarevich-Tate group is introduced by way of an example. Thereafter, with the use of Galois Cohomology, we provide a general definition of this mysterious group. We also discuss invariants like the regulator and real period, which appear in the Birch and Swinnerton-Dyer Conjecture. After defining the L-function, we state the Birch and Swinnerton-Dyer Conjecture and discuss results which have been proved and some consequences. We discuss numerical verification of the Conjecture, and show some computations, including an example of our own."],"dc:format.checksum.md5":["8fc7000bdadd9e1e3e7c1d4ed62592d6","bb9bdc0b3349e4284e09149f943790b4","9a38b8e31330d1684300d9cab427dea6"],"dc:identifier.uri":["https://uwcscholar.uwc.ac.za/bitstreams/c12909d1-9cb4-42c1-a5d2-cfa0003829be/download"],"dc:publisher.institution":["University of the Western Cape"],"dc:relation.isreferencedby":["https://hdl.handle.net/10566/22683"],"dc:rights":["https://uwcscholar.uwc.ac.za/bitstreams/5926f36a-ddf0-465b-9353-85dab69c5887/download"],"dc:subject":["Elliptic curve","Birch and Swinnerton-Dyer conjecture","L-function","modern Mathematics","Algebraic Number Theory"],"dc:title":["The Birch and Swinnerton-Dyer Conjecture for elliptic curves"],"dc:type":["Thesis"]},"updated_at":"2026-07-24T06:00:42Z"}