| 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: |
|
| 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.
|
|
Key Findings |
|
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
|
|
Potential use in non-academic contexts |
|
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
|
|
Impacts |
|
| Description |
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk |
| Summary |
|
| Date Materialised |
|
|
|
|
Sectors submitted by the Researcher |
|
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
|
| Project URL: |
|
| Further Information: |
|
| Organisation Website: |
http://www.imperial.ac.uk |