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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 |
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| 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 |
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| 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 |
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
