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Title: On 5 and 7 descents for elliptic curves
Author: Fisher, T. A.
Awarding Body: University of Cambridge
Current Institution: University of Cambridge
Date of Award: 2001
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We perform descent calculations for the families of elliptic curves over Q with a rotational 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 a elliptic curve over Q may become arbitrarily large. In a special case, namely when the 5-torsion of our elliptic curve splits as μ5Z/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(μn) and finally make some calculations for the curve X1(11) over its field of 5-division points.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available