HMF · Hilbert Modular Forms and Diophantine Applications
FP7 — People (Marie Curie Actions)
- Duration
- 2009-07-20 → 2011-07-19
- EU contribution
- €161,793
- Participants
- 1
- Scheme
- MC-IIF
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Results in brief
Hilbert modular forms and Diophantine applications
For over 350 years Fermat's last theorem was the most exciting unsolved problem in mathematics. The problem was finally resolved by Andrew Wiles in 1994, building on the ideas of Frey, Ribet and Serre. The HMF project was concerned with laying the computational foundations for possible extensions of the work of Wiles and others to the setting of Hilbert modular forms and automorphic forms. In particular, the project resulted in robust algorithms for computing Hilbert modular forms which were implemented and included in the computer algebra package MAGMA. The researcher also played a crucial role in resolving the long-standing Gross conjecture, an important open problem in the subject that was posed by Prof. Benedict Gross at Harvard. Furthermore, one very active area of research in mathematics is the subject of automorphic forms and the Langlands programme. This is a vast web of conjectures that connect number theory, algebraic geometry, analysis and representation theory. The researcher's work was instrumental in providing key evidence, along with a precise formulation for conjectures, in the area of mod p Langlands and Serre's conjectures over totally real fields. The researcher's website was http://www.warwick.ac.uk/staff/L.Dembele/.
Data: CORDIS, © European Union
Project objective
The ideas of Frey, Serre, Ribet and Wiles connect Diophantine equations to Galois representations arising from automorphic forms. The most spectacular success in this direction is Wiles' amazing proof of Fermat's Last Theorem. The proof relates hypothetical solutions of the Fermat equation with elliptic modular forms which are the most basic (and best understood) of automorphic forms. It has become clear however, thanks to the work of Darmon and of Jarvis and Meekin, that the resolution of many other Diophantine problems lies through an explicit understanding of the more difficult Hilbert modular and automorphic forms. This project has the following aims: 1. Develop and improve algorithms for Hilbert modular forms and for automorphic forms on unitary groups. 2. To solve several cases of the generalized Fermat equation after making explicit the strategies of Darmon and of Jarvis and Meekin and computing/studying the relevant Hilbert modular forms. 3. To make precise and explicit certain instances of the Langlands programme.
Original text from CORDIS.
Participants
- UNIVERSITY OF WARWICK · COVENTRYCoordinatorUnited Kingdom
Links
Data: CORDIS, © European Union