dc.contributor.advisor | Omar, Rafiq | |
dc.contributor.author | Smith, Duncan | |
dc.date.accessioned | 2015-08-27T14:49:05Z | |
dc.date.available | 2015-08-27T14:49:05Z | |
dc.date.issued | 2014 | |
dc.identifier.uri | http://hdl.handle.net/11394/4452 | |
dc.description | >Magister Scientiae - MSc | en_US |
dc.description.abstract | The 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. | en_US |
dc.language.iso | en | en_US |
dc.publisher | University of the Western Cape | en_US |
dc.subject | Elliptic curve | en_US |
dc.subject | Birch and Swinnerton-Dyer conjecture | en_US |
dc.subject | L-function | en_US |
dc.title | The Birch and Swinnerton-Dyer Conjecture for elliptic curves. | en_US |
dc.rights.holder | University of the Western Cape | en_US |