Type of Document	Dissertation
Author	Biswas, Saikat
Author's Email Address	sb07c@fsu.edu
URN	etd-07202011-222418
Title	Constructing Non-trivial Elements of the Shafarevich-Tate Group of an Abelian Variety
Degree	Doctor of Philosophy
Department	Mathematics, Department of
Advisory Committee	Advisor Name Title Amod Agashe Committee Chair Eriko Hironaka Committee Member Ettore Aldrovandi Committee Member Mark Van Hoeij Committee Member Sudhir Aggarwal University Representative
Keywords	arithmetic geometry number theory component group Shafarevich-Tate Elliptic curve
Date of Defense	2011-06-28
Availability	unrestricted
Abstract 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.
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