>Magister Scientiae - MSc / 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.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:uwc/oai:etd.uwc.ac.za:11394/4452 |
| Date | January 2014 |
| Creators | Smith, Duncan |
| Contributors | Omar, Rafiq |
| Publisher | University of the Western Cape |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Rights | University of the Western Cape |
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