AuthorDeLorme, Cheryl Lynn, 1969-
AdvisorMcCallum, William G.
MetadataShow full item record
PublisherThe University of Arizona.
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.
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).
Degree ProgramGraduate College