The Birch and Swinnerton-Dyer Conjecture for elliptic curves
type: CreativeWork
description: 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: 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.
author: Smith, Duncan
keywords: Elliptic curve, Birch and Swinnerton-Dyer conjecture, L-function, modern Mathematics, Algebraic Number Theory
license: https://uwcscholar.uwc.ac.za/bitstreams/5926f36a-ddf0-465b-9353-85dab69c5887/download
date published: 2014
date modified: 2026-07-24T06:00:42Z
provider: University of the Western Cape
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of the Western Cape