# Finding elements of given order in Tate-Shafarevich groups of elliptic curves

Abstract (Summary)

The 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 â€˜descentâ€™. 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.
Index words: Algebraic geometry, Arithmetic geometry, Elliptic curves,
Tate-Shafarevich group, Descent, Selmer groups,
Mordell-Weil group
Bibliographical Information:

Advisor:

School:The University of Georgia

School Location:USA - Georgia

Source Type:Master's Thesis

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