EPSRC Reference: 
GR/R22377/01 
Title: 
Deformation Rings and Hecke Rings Associated To Families of Galois Representations 
Principal Investigator: 
Buzzard, Professor K 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematics 
Organisation: 
Imperial College London 
Scheme: 
Fast Stream 
Starts: 
01 October 2001 
Ends: 
31 March 2003 
Value (£): 
61,739

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
In 1996 Coleman and Mazur constructed the eigencurve, a geometric object whose points parameterise certain modular forms. One can associate a Galois representation to each of these forms, and hence the eigencurve can be thought of as a parameter space for a family of Galois representations. Recent work of Wiles and others shows that in many cases, the slope 0 part of this family can be thought of as the universal ordinary deformation of a modular mod p Galois representation. Guided by the case, we should expect that for a general point on the eigencurve of arbitrary sloe, there should be a universal deformation ring defined by local conditions, whose associated rigid space will be a neighbourhood of the point on the curve (an R=T theorem). The heart of the proposed research is to come up with a definition of such an R, generalising work of Mazur in the slope 0 case. There are strong numbertheoretic consequences of such a result, for example one will be able to establish new cases of the BlochKato conjecture for the symmetric square of a modular form inmany new cases. Even partial results, for example the construction of certain quotients of R, should still have these strong numbertheoretic consequences.

Key Findings 
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Summary 

Date Materialised 


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Organisation Website: 
http://www.imperial.ac.uk 