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SOME COMPUTATIONAL PROBLEMS MOTIVATED BY THE BIRCH AND
SWINNERTON-DYER CONJECTURE
by
Iftikhar A. Burhanuddin
A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(COMPUTER SCIENCE)
August 2007
Copyright 2007 Iftikhar A. Burhanuddin
Object Description
| Title | Some computational problems motivated by the Birch and Swinnerton-Dyer conjecture |
| Author | Burhanuddin, Iftikhar A. |
| Author email | burhanud@usc.edu |
| Degree | Doctor of Philosophy |
| Document type | Dissertation |
| Degree program | Computer Science |
| School | Viterbi School of Engineering |
| Date defended/completed | 2007-05-03 |
| Date submitted | 2007 |
| Restricted until | Unrestricted |
| Date published | 2007-07-25 |
| Advisor (committee chair) |
Huang, Ming-Deh A. Stein, William A. |
| Advisor (committee member) |
Adleman, Leonard M. Kamienny, Sheldon |
| Abstract | This dissertation revolves around the BSD (Birch and Swinnerton-Dyer) conjecture for elliptic curves defined over the rational numbers, a famous problem that has been open for over forty years and one of the seven Millennium Prize problems. The BSD conjecture is considered to be the first nontrivial number theoretic problem put forth as a result of explicit machine computation --- in the late '50s at Cambridge University. The BSD conjecture relates the rank of the Mordell-Weil group, the group of rational points of an elliptic curve, a quantity which seems to be difficult to pin down, to the order of vanishing of the L-function of the elliptic curve at its central point.; We make algorithmic and theoretical advances with regards to some of the terms appearing in the BSD formula, namely the sizes of the torsion subgroup and the Shafarevich-Tate group.; Firstly, we introduce an algorithm to compute elliptic curve torsion subgroup. The randomized version of this procedure runs in expected time which is essentially linear in the number of bits required to write down the equation of the elliptic curve.; Next, we discuss a conjecture of Hindry, who proposed a Brauer-Siegel type formula for elliptic curves. Driven by a suggestion of Hindry, we prove assuming standard conjectures that there are infinitely many elliptic curves with Shafarevich-Tate group of size about as large as the square root of the minimal discriminant of the curve. This improves on a result of de Weger.; Thirdly, we consider certain quartic twists of an elliptic curve. We establish a reduction between the problem of factoring integers of a certain form and the problem of computing rational points on these twists. We illustrate that the size of Shafarevich-Tate group of these curves will make it computationally expensive to factor integers by computing rational points via the Heegner point method.; Finally, we sketch existing algorithms that compute the quantities appearing in the BSD formula and introduce stategies to parallelize them. |
| Keyword | elliptic curves; algorithms; computation; BSD conjecture |
| Language | English |
| Part of collection | University of Southern California dissertations and theses |
| Publisher (of the original version) | University of Southern California |
| Place of publication (of the original version) | Los Angeles, California |
| Publisher (of the digital version) | University of Southern California. Libraries |
| Type | texts |
| Legacy record ID | usctheses-m675 |
| Rights | Burhanuddin, Iftikhar A. |
| Repository name | Libraries, University of Southern California |
| Repository address | Los Angeles, California |
| Repository email | http://www.usc.edu/isd/libraries/services/ask_a_librarian/email/ |
| Filename | etd-Burhanuddin-20070725 |
| Archival file | uscthesesreloadpub_Volume26/etd-Burhanuddin-20070725.pdf |
Description
| Title | Page 1 |
| Full text | SOME COMPUTATIONAL PROBLEMS MOTIVATED BY THE BIRCH AND SWINNERTON-DYER CONJECTURE by Iftikhar A. Burhanuddin A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (COMPUTER SCIENCE) August 2007 Copyright 2007 Iftikhar A. Burhanuddin |
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