
Thomas Anthony Fisher, Clare College
A dissertation submitted for the degree of Doctor of Philosophy
at the University of Cambridge, August 2000
We perform descent calculations for the families of elliptic curves over Q with a rational point of order n = 5 or 7. These calculations give an estimate for the MordellWeil rank which we relate to the parity conjecture. We exhibit explicit elements of the TateShafarevich group of order 5 and 7, and show that the 5torsion of the TateShafarevich group of an elliptic curve over Q may become arbitrarily large.
In a special case, namely when the 5torsion of our elliptic curve splits as µ_{5} × Z / 5 Z, we improve our estimate for the MordellWeil rank by using the CasselsTate pairing to perform a full 5descent. We generalise our results to curves over Q(µ_{n}) and finally make some calculations for the curve X_{1}(11) over its field of 5division points.