Persistent Link:
http://hdl.handle.net/10150/288706
Title:
On the Shafarevich-Tate group of an elliptic curve
Author:
DeLorme, Cheryl Lynn, 1969-
Issue Date:
1997
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:
This dissertation work concentrates on finding non-trivial elements in the Shafarevich-Tate group of an elliptic curve. The set of K-rational points on an elliptic curve, E, are known to form a finitely generated abelian group. My results are of interest when trying to find the rank of this group, which in general is a hard problem. The Selmer group of E,S(E/K), can be used to give a bound on this rank, and the obstruction to using this to find the exact rank is the Shafarevich-Tate group, scIII(E/K). There is a pairing on scIII(E/K), called the Cassels-Tate pairing, which is non-degenerate modulo the infinitely divisible subgroup of scIII(E/K). I compute the pairing in certain cases; in particular, I compute it on certain 5-torsion elements in scIII(E/K) for an infinite family of elliptic curves over Q described by Rubin and Silverberg and find examples where scIII₅(E/Q) is non-trivial. The curves in question have 5-torsion over Q isomorphic to Z/5Z(⊕) μ₅, and the elements of scIII(E/K) for which the pairing is trivial are those killed by φ: E → E', the isogeny with kernel μ₅. I also compute the pairing for a family of curves in the 2-torsion case, giving a new method for constructing curves with non-trivial 2-torsion in scIII(E/K).
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Mathematics.
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Mathematics
Degree Grantor:
University of Arizona
Advisor:
McCallum, William G.

Full metadata record

DC FieldValue Language
dc.language.isoen_USen_US
dc.titleOn the Shafarevich-Tate group of an elliptic curveen_US
dc.creatorDeLorme, Cheryl Lynn, 1969-en_US
dc.contributor.authorDeLorme, Cheryl Lynn, 1969-en_US
dc.date.issued1997en_US
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.abstractThis dissertation work concentrates on finding non-trivial elements in the Shafarevich-Tate group of an elliptic curve. The set of K-rational points on an elliptic curve, E, are known to form a finitely generated abelian group. My results are of interest when trying to find the rank of this group, which in general is a hard problem. The Selmer group of E,S(E/K), can be used to give a bound on this rank, and the obstruction to using this to find the exact rank is the Shafarevich-Tate group, scIII(E/K). There is a pairing on scIII(E/K), called the Cassels-Tate pairing, which is non-degenerate modulo the infinitely divisible subgroup of scIII(E/K). I compute the pairing in certain cases; in particular, I compute it on certain 5-torsion elements in scIII(E/K) for an infinite family of elliptic curves over Q described by Rubin and Silverberg and find examples where scIII₅(E/Q) is non-trivial. The curves in question have 5-torsion over Q isomorphic to Z/5Z(⊕) μ₅, and the elements of scIII(E/K) for which the pairing is trivial are those killed by φ: E → E', the isogeny with kernel μ₅. I also compute the pairing for a family of curves in the 2-torsion case, giving a new method for constructing curves with non-trivial 2-torsion in scIII(E/K).en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectMathematics.en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineMathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorMcCallum, William G.en_US
dc.identifier.proquest9806765en_US
dc.identifier.bibrecord.b37516036en_US
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