This thesis is primarily concerned with the rank of elliptic curves over the rationals. It establishes a new long exact sequence of groups related to an elliptic curve <i>E </i>over <i>Q. </i> The sequence appears to have strong connections with the conjectures of Birch and Swinnerton-Dyer. Using Haar and Tamagawa measures in the case of the field <i>Q, </i>the algebraic rank and the analytic rank respectively are shown to be closely related to the first two groups. This approach also offers explanations, but not proofs, of the constants of the conjectures. The first part of the thesis ends with a discussion of the possible generalisation of the approach to a number field <i>K. </i> The second part of the thesis considers conductor related bounds on rank and also develops a theory of the distribution of rank for the set of all elliptic curves over <i>Q, </i>ordered by the size of coefficients. Published experimental results are compared with this theory. I conclude with algorithmic methods of identifying high rank elliptic curves.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:662586 |
| Date | January 2000 |
| Creators | Suess, Nigel Marcus |
| Publisher | University of Edinburgh |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/1842/14511 |
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