| EPSRC Reference: |
GR/R11872/01 |
| Title: |
Modularity of Elliptic Curves Over Totally Real Fields |
| Principal Investigator: |
Jarvis, Dr AF |
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| Department: |
Mathematics and Statistics |
| Organisation: |
University of Sheffield |
| Scheme: |
Fast Stream |
| Starts: |
01 October 2001 |
Ends: |
31 May 2003 |
Value (£): |
62,067
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| No relevance to Underpinning Sectors |
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Summary on Grant Application Form |
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The key point of the proof of Fermat's Last Theorem (1994) was the proof of the statement that every semistable elliptic curve over the rationals should be modular, finally proven in 1999 by Breuil, Conrad, Diamond and Taylor. This result is a very substantial piece of work (two papers, totalling about 130 pages). The experience of the last 15 years is that all results for elliptic curves and molecular forms which are true over the rationals should also have analogues for totally real number fields. One would therefore expect that every elliptic curve over a totally real field number should be molecular. The proof should be a fairly direct generalisation other proof over the rationals, although it is likely that a few extra subtleties will arise. The principal investigator (and others) have been working on proving the necessary results involving modular forms over the past few years; a little work remains to be done on this side, but it is the applications to elliptic curves which remain to be studied in depth. As outlined above, the steps are cumulative; the aim is to find further fields over which all elliptic curves are modular. One would start by extending the results of Wiles and Taylor-Wiles (mostly already done by Fujiwara) to find fields over which all semistable elliptic curves are modular. Next, one would hope to find fields over which one can show that all elliptic curves are modular. If time permits, one would hope to try to extend the class of fields for which this result may be demonstrated.
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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.shef.ac.uk |