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Details of Grant 

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 Co-Investigators:
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:
Algebra & Geometry
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 number-theoretic consequences of such a result, for example one will be able to establish new cases of the Bloch-Kato 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 number-theoretic consequences.
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Organisation Website: http://www.imperial.ac.uk