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Constructing Non-Trivial Elements of the Shafarevich-Tate Group of an Abelian Variety

Title: Constructing Non-Trivial Elements of the Shafarevich-Tate Group of an Abelian Variety.
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Name(s): Biswas, Saikat, author
Agashe, Amod, professor directing dissertation
Aggarwal, Sudhir, university representative
Hironaka, Eriko, committee member
Van Hoeij, Mark, committee member
Aldrovandi, Ettore, committee member
Department of Mathematics, degree granting department
Florida State University, degree granting institution
Type of Resource: text
Genre: Text
Issuance: monographic
Date Issued: 2011
Publisher: Florida State University
Place of Publication: Tallahassee, Florida
Physical Form: computer
online resource
Extent: 1 online resource
Language(s): English
Abstract/Description: The Shafarevich-Tate group of an elliptic curve is an important invariant of the curve whose conjectural finiteness can sometimes be used to determine the rank of the curve. The second part of the Birch and Swinnerton-Dyer (BSD) conjecture gives a conjectural formula for the order of the Shafarevich-Tate group of a elliptic curve in terms of other computable invariants of the curve. Cremona and Mazur initiated a theory that can often be used to verify the BSD conjecture by constructing non-trivial elements of the Shafarevich-Tate group of an elliptic curve by means of the Mordell-Weil group of an ambient curve. In this thesis, we extract a general theorem out of Cremona and Mazur's work and give precise conditions under which such a construction can be made. We then give an extension of our result which provides new theoretical evidence for the BSD conjecture. Finally, we prove a theorem that gives an alternative method to potentially construct non-trivial elements of the Shafarevich-Tate group of an elliptic curve by using the component groups of a second curve.
Identifier: FSU_migr_etd-3717 (IID)
Submitted Note: A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Degree Awarded: Summer Semester, 2011.
Date of Defense: June 28, 2011.
Keywords: arithmetic geometry, number theory, component group, Shafarevich-Tate, Elliptic curve
Bibliography Note: Includes bibliographical references.
Advisory Committee: Amod Agashe, Professor Directing Dissertation; Sudhir Aggarwal, University Representative; Eriko Hironaka, Committee Member; Mark Van Hoeij, Committee Member; Ettore Aldrovandi, Committee Member.
Subject(s): Mathematics
Persistent Link to This Record: http://purl.flvc.org/fsu/fd/FSU_migr_etd-3717
Owner Institution: FSU

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Biswas, S. (2011). Constructing Non-Trivial Elements of the Shafarevich-Tate Group of an Abelian Variety. Retrieved from http://purl.flvc.org/fsu/fd/FSU_migr_etd-3717