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On 5 and 7 descents for elliptic curvesThomas Anthony Fisher, Clare CollegeA dissertation submitted for the degree of Doctor of Philosophyat the University of Cambridge, August 2000 Abstract 
We perform descent calculations for the families of elliptic curvesover Q with a rational point of order n = 5 or 7. Thesecalculations give an estimate for the Mordell-Weil rank which we relateto the parity conjecture. We exhibit explicit elements of theTate-Shafarevich group of order 5 and 7, and show that the 5-torsionof the Tate-Shafarevich group of an elliptic curve over Qmay become arbitrarily large. In a special case, namely when the 5-torsion of our elliptic curve splitsas µ5 × Z / 5 Z, 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(µn) and finally make some calculationsfor the curve X1(11) over its field of 5-division points.On 5 and 7 descents for elliptic curves (136 pages) 
PhD conferred March 2001