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dc.contributor.authorDonnelly, Stephen Robert
dc.description.abstractThe Tate-Shafarevich group of an elliptic curve over a number field K measures the obstruction to determing the K-rational points by the standard method, which is known as ‘descent’. During the last two decades, a focal point of research in arithmetic geometry has been to better understand the structure and behaviour of Tate-Shafarevich groups. Selmer groups provide a stepping stone towards studying Tate-Shafarevich groups; the Selmer groups for the various isogenies defined on the curve encapsulate the information obtainable from local calculations about the set of K-rational points and about the Tate-Shafarevich group. We give a theorem that describes the Selmer groups for a certain kind of isogeny. We then give a method for constructing elliptic curves defined over quadratic fields that have many elements of order 7 in their Tate-Shafarevich groups; the construction shows that there can be arbitrarily many assuming a technical arithmetic hypothesis. As an ingredient in the construction, we find a model of the moduli space X0(14) with certain convenient properties.
dc.subjectAlgebraic geometry
dc.subjectArithmetic geometry
dc.subjectElliptic curves,Tate-Shafarevich group
dc.subjectSelmer groups
dc.subjectMordell-Weil group
dc.titleFinding elements of given order in Tate-Shafarevich groups of elliptic curves
dc.description.advisorRobert Rumely
dc.description.committeeRobert Rumely
dc.description.committeeMatt Baker
dc.description.committeeSybilla Beckman-Kazez
dc.description.committeeRobert Varley
dc.description.committeeRoy Smith

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