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Hilbert Modular Forms and Diophantine Applications

Final Report Summary - HMF (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/.