| EPSRC Reference: |
GR/M95707/01 |
| Title: |
DESCENT ON ELLIPTIC CURVES |
| Principal Investigator: |
Cremona, Professor J |
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| Department: |
Sch of Mathematical Sciences |
| Organisation: |
University of Nottingham |
| Scheme: |
Standard Research (Pre-FEC) |
| Starts: |
14 September 1999 |
Ends: |
13 October 1999 |
Value (£): |
1,900
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| No relevance to Underpinning Sectors |
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Summary on Grant Application Form |
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One key problem is the theory of Diophantine equations is to find by an effective method all rational points on an elliptic curve or more generally on the Jacobian of a curve of higher genus. The methods used for this are known as descent methods, and date back many years. In the 1960s Birch and Swinnerton-Dyer carried out 2-descents systematically before formulating their celebrated (and still unproved) conjectures. Cremona has developed 2-descent methods for elliptic curves defined over number fields, while Stoll has done significantly work on 2-descent for Jacobians of curves of higher genus and on p-descent for p>2. Various questions concerning explicit descents on elliptic curves will be studied, employing techniques which will eventually apply over general number fields, though initially we will work over Q. These will include the following: Lifting 2-descents to 4-descents: by effectively computing the Cassels-Tate pairing on the 2-Selmer group, then employing Siksek's methods for explicit models for the associated homogeneous spaces. p-descents: Stoll has already developed an algorithm for p-descent which allows explicit computation of the Selmer group. We will investigate the use of explicit models for the associated homogeneous spaces when p=3, using classical reduction theory, in order to be able to search effectively for rational points on these and hence on the original elliptic curve.
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Key Findings |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Potential use in non-academic contexts |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Impacts |
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| Description |
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk |
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Sectors submitted by the Researcher |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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| Project URL: |
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| Further Information: |
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| Organisation Website: |
http://www.nottingham.ac.uk |