EPSRC Reference: 
GR/M95707/01 
Title: 
DESCENT ON ELLIPTIC CURVES 
Principal Investigator: 
Cremona, Professor J 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Sch of Mathematical Sciences 
Organisation: 
University of Nottingham 
Scheme: 
Standard Research (PreFEC) 
Starts: 
14 September 1999 
Ends: 
13 October 1999 
Value (£): 
1,900

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


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Panel History: 

Summary on Grant Application Form 
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 SwinnertonDyer carried out 2descents systematically before formulating their celebrated (and still unproved) conjectures. Cremona has developed 2descent methods for elliptic curves defined over number fields, while Stoll has done significantly work on 2descent for Jacobians of curves of higher genus and on pdescent 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 2descents to 4descents: by effectively computing the CasselsTate pairing on the 2Selmer group, then employing Siksek's methods for explicit models for the associated homogeneous spaces. pdescents: Stoll has already developed an algorithm for pdescent 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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Organisation Website: 
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