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

by 1974- Donnelly, Stephen

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 ‘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. Index words: Algebraic geometry, Arithmetic geometry, Elliptic curves, Tate-Shafarevich group, Descent, Selmer groups, Mordell-Weil group