On 5 and 7 Descents for Elliptic Curves

Thomas Anthony Fisher

Summary

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 Mordell-Weil rank which we relate to the parity conjecture. We exhibit explicit elements of the Tate-Shafarevich group of order 5 and 7, and show that the 5-torsion of the Tate-Shafarevich group of an elliptic curve over Q may become arbitrarily large.

In a special case, namely when the 5-torsion of our elliptic curve splits as μ_{5} ⊕ Z/5Z, we improve our estimate for the Mordell-Weil rank by using the Cassels-Tate pairing to perform a full 5-descent. 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 5-division points.