GalRepsDiophantine · Galois Representations and Diophantine Problems
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2018-03-01 → 2020-02-29
- EU contribution
- €183,455
- Participants
- 1
- Scheme
- MSCA-IF
Lines connect the coordinator with its partners.
Results in brief
Galois Representations and Diophantine Problems
For over 350 years Fermat's Last Theorem was the most famous open problem in mathematics, and was finally resolved by Andrew Wiles in 1994. Whilst Wiles' proof had dramatically succeeded in resolving the Fermat equation over the rationals, for many other Diophantine problems (including the Fermat equation over number fields), the proof strategy is insufficient. Indeed, the approach in Wiles' proof, building on ideas of Frey, Serre and Ribet, associates a putative solution of certain Diophantine equations to a Frey elliptic curve, and then predicts that the residual Galois representation of that elliptic curve comes from a finite computable set of modular Galois representations. This project is concerned with the following two objectives. 1. Distinguishing Galois representations. The idea is to look for finer invariants of underlying objects that will enable us to conclude a contradiction and deduce that the original equation has no solutions. 2. The Darmon programme. Henri Darmon proposed a far reaching generalization of Frey elliptic curves. He associates to certain ternary Diophantine problems hypergeometric abelian varieties. This objective is concerned with making the Darmon programme sufficiently practical to enable the resolution of particular Diophantine equations.
Data: CORDIS, © European Union
Project objective
Wiles' remarkable proof of Fermat's Last Theorem paved the way for the modular approach to Diophantine equations. This associates a Frey elliptic curve to a putative solution of a Diophantine equation and studies it using Galois representations and modularity. This proposal is organized around two research programmes, both of which develop new tools for the modular approach. The first is concerned with distinguishing Galois representations; this is currently the most frequent obstruction to the success of the approach. The second aims to prove modularity and irreducibility theorems for abelian varieties of GL2 type. Such theorems are of tremendous independent interest, but will also allow the replacement of Frey elliptic curves with Frey abelian varieties giving the modular approach immense flexibility.The University of Warwick has a strong and active number theory group, making it a natural host for the project. The Supervisor, Professor Siksek, is a leading expert on curves, rational points, Diophantine equations and modularity, with considerable experience in supervising research including eight postdocs and ten completed PhD students.The Researcher, Dr Freitas, did his undergraduate studies in Lisbon, and his PhD at the University of Barcelona. He has worked for almost three years in Germany (Bonn and Bayreuth), and is now a postdoctoral fellow at the University of British Columbia (Vancouver). He has a successful track record of research in modularity and Diophantine equations, with 12 papers already published or accepted in excellent journals. He was awarded the prestigious 2014 Jose Luis Rubio de Francia prize by the Spanish Mathematical Society. The envisioned research will make the Researcher influential in Diophantine equations and adjacent subjects. The project will reintegrate him into the European research environment, and allow him to realize his ambition of becoming an independent researcher at a leading European institution.
Original text from CORDIS.
Participants
- UNIVERSITY OF WARWICK · COVENTRYCoordinatorUnited Kingdom
Links
Data: CORDIS, © European Union