Persistent Link:
http://hdl.handle.net/10150/146586
Title:
On Elliptic Curves of Conductor N=PQ
Author:
Howe, Sean
Issue Date:
May-2010
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
We study elliptic curves with conductor N = pq for p and q prime. By studying the 2-torsion field we obtain that for N a product of primes satisfying some congruency conditions and class number conditions on related quadratic fields, any elliptic curve of conductor N has a rational point of order 2. By studying a minimal Weierstrass equation and its discriminant we obtain a solution to some Diophantine equation from any curve with conductor N = pq and a rational point of order 2. Under certain congruency conditions, this equation has no solutions, and so we conclude that in this situation there is no elliptic curve of conductor N with a rational point of order 2. Combining these two results, we prove that for a family of N = pq satisfying more specific congruency conditions and class number conditions on related quadratic fields, there are no elliptic curves of conductor N. We use a computer to find all N < 10^7 satisfying these conditions, of which there are 67. This work is similar to and largely inspired by past work on conductors p by Ogg [14, 15], Hadano [9], Neumann [13], Setzer [17], and Brumer and Kramer [4].
Type:
text; Electronic Thesis
Degree Name:
B.S.
Degree Level:
bachelors
Degree Program:
Honors College; Mathematics
Degree Grantor:
University of Arizona

Full metadata record

DC FieldValue Language
dc.language.isoenen_US
dc.titleOn Elliptic Curves of Conductor N=PQen_US
dc.creatorHowe, Seanen_US
dc.contributor.authorHowe, Seanen_US
dc.date.issued2010-05-
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractWe study elliptic curves with conductor N = pq for p and q prime. By studying the 2-torsion field we obtain that for N a product of primes satisfying some congruency conditions and class number conditions on related quadratic fields, any elliptic curve of conductor N has a rational point of order 2. By studying a minimal Weierstrass equation and its discriminant we obtain a solution to some Diophantine equation from any curve with conductor N = pq and a rational point of order 2. Under certain congruency conditions, this equation has no solutions, and so we conclude that in this situation there is no elliptic curve of conductor N with a rational point of order 2. Combining these two results, we prove that for a family of N = pq satisfying more specific congruency conditions and class number conditions on related quadratic fields, there are no elliptic curves of conductor N. We use a computer to find all N < 10^7 satisfying these conditions, of which there are 67. This work is similar to and largely inspired by past work on conductors p by Ogg [14, 15], Hadano [9], Neumann [13], Setzer [17], and Brumer and Kramer [4].en_US
dc.typetexten_US
dc.typeElectronic Thesisen_US
thesis.degree.nameB.S.en_US
thesis.degree.levelbachelorsen_US
thesis.degree.disciplineHonors Collegeen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US
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